Arens-Michael Envelopes, Homological Epimorphisms, And

Home , Free product, Topological algebra

Trudy Moskov. Matem. Obw. Trans. Moscow Math. Soc. Tom 69 (2008) 2008, Pages 27–104 S 0077-1554(08)00169-6 Article electronically published on November 19, 2008

ARENS–MICHAEL ENVELOPES, HOMOLOGICAL EPIMORPHISMS, AND RELATIVELY QUASI-FREE ALGEBRAS

ALEXEI YUL’EVICH PIRKOVSKII

Abstract. We describe and investigate Arens–Michael envelopes of associative algebras and their homological properties. We also introduce and study analytic analogs of some classical ring-theoretic constructs: Ore extensions, Laurent extensions, and tensor algebras. For some ﬁnitely generated algebras, we explicitly describe their Arens–Michael envelopes as certain algebras of noncommutative power series, and we also show that the embeddings of such algebras in their Arens–Michael envelopes are homological epimorphisms (i.e., localizations in the sense of J. Taylor). For that purpose we introduce and study the concepts of relative homological epimorphism and relatively quasi-free algebra. The above results hold for multiparameter quantum aﬃne spaces and quantum tori, quantum Weyl algebras, algebras of quantum (2 × 2)-matrices, and universal enveloping algebras of some Lie algebras of small dimensions.

1. Introduction Noncommutative geometry emerged in the second half of the last century when it was realized that several classical results can be interpreted as some kind of “duality” between geometry and algebra. Informally speaking, those results show that quite often all the information about a geometric “space” (topological space, smooth manifold, affine algebraic variety, etc.) is in fact contained in a properly chosen algebra of functions (continuous, smooth, polynomial. . . ) defined on that space. More precisely, this means that assigning to a space X the corresponding algebra of functions, one obtains an anti- equivalence between a category of “spaces” and a category of commutative algebras. In some cases, the resulting category of commutative algebras admits an abstract description, which apriorihas nothing to do with functions on any space. A typical example is the first Gelfand–Naimark theorem, asserting that the functor sending a locally compact topological space X to the algebra C0(X) of continuous functions vanishing at infinity is an anti-equivalence between the category of locally compact spaces and the category of commutative C∗-algebras. Another classical result of this nature is the categorical form of Hilbert’s Nullstellensatz asserting that, over an algebraically closed field k,the functor sending an affine algebraic variety X to the algebra of regular (polynomial) functions on X is an anti-equivalence between the category of affine varieties over k and the category of finitely generated commutative k-algebras without nilpotents. One of the basic principles of noncommutative geometry states that an arbitrary (noncommutative) algebra should be viewed as an “algebra of functions on a noncommutative space”. The reason for this is the fact that many important geometric notions and theorems, when translated into the “dual” algebraic language, are also true for noncommutative algebras. These kinds of results gave rise to such branches of mathematics

2000 Mathematics Subject Classiﬁcation. Primary 46M18. The author was supported by the RFFI grant No. 05-01-00982 and No. 05-01-00001 and the President of Russia’s grant MK-2049.2004.1.

c 2008 American Mathematical Society 27

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as algebraic and operator K-theories, the theory of quantum groups, noncommutative dynamics, etc. In fact, the term “noncommutative geometry” (as well as the term “geometry”) is a name not for one but for several branches of mathematics which share many similar ideas but differ in their objects of study. For example, the natural objects of study in noncommutative measure theory are von Neumann algebras (since commutative von Neumann algebras are exactly the algebras of essentially bounded measurable functions on measure spaces), whereas in noncommutative topology one studies C∗-algebras (in concordance with the mentioned Gelfand–Naimark theorem). Similarly, noncommutative affine algebraic geometry mainly deals with finitely generated algebras (without topology) and noncommutative projective algebraic geometry studies graded algebras. Finally, in noncommutative differential geometry one uses dense subalgebras of C∗-algebras with some special “differential” properties (see [4, 33]). Notice that the above list of geometric theories misses an important discipline which should be placed “between” differential and algebraic geometries, namely, complex- analytic geometry. To the best of the author’s knowledge, there is no theory at the moment that could claim the name of “noncommutative complex-analytic geometry”. Perhaps, one of the main reasons for this is that it is not clear what kind of algebras should be viewed as noncommutative generalizations of the algebras of holomorphic functions on complex varieties. Of the few papers related to these issues we mention J. Tay- lor’s [67, 68], which consider various completions of a free algebra, and D. Luminet’s [37, 38], dealing with analytic algebras with polynomial identities. Some closely related problems are considered in a series of papers by L. L. Vaksman, S. D. Sinel’shchikov, and D. L. Shklyarov (see, in particular, [71, 63, 70] and [35]), dealing with noncommutative generalizations of function theory in classical domains and using methods from the theory of quantum groups. In particular, in [70], the author introduces a Banach algebra which is a “quantum” analog of the algebra A(B) of continuous functions on the closed unit ball B which are holomorphic in its interior, and computes its Shilov boundary. A somewhat different approach to “noncommutative complex analysis”, closely related to operator theory in the spirit of Szekefalvi-Nagy and Fojaˇs, can be traced back to W. Arveson’s classical papers [2, 3]. It is being systematically developed G. Popescu, K. Davidson, and others (see, for example, [11, 12, 50, 13, 51]). The algebras considered in the present paper may, for several reasons, be viewed as noncommutative generalizations of the algebras of holomorphic functions on complex affine algebraic varieties. They are formally defined as the Arens–Michael envelopes of finitely generated associative algebras. By definition, the Arens–Michael envelope of a C-algebra A is its completion A with respect to the system of all submultiplicative prenorms. The Arens–Michael envelopes were introduced by J. Taylor [66, 67] in relation to a problem of multi-operator holomorphic functional calculus. The reason why those algebras may be relevant to noncommutative complex-analytic geometry is that the Arens–Michael envelope of the algebra Oalg(V ) of regular (polynomial) functions on an affine algebraic variety V coincides with the algebra O(V ) of holomorphic functions on V endowed with the standard compact-open topology (see Example 3.6 below). Therefore, if one adopts the point of view of noncommutative algebraic geometry and considers a finitely generated algebra (for example, some algebra of noncommutative polynomials1) as an algebra of “regular functions on a noncommutative affine variety”, then its Arens–Michael envelope should be considered as the algebra of “holomorphic

1Following [20], we use the term “algebra of noncommutative polynomials” not as a synonym of the term “free algebra”, but in a diﬀerent and somewhat informal sense, meaning an algebra obtained from the usual algebra of polynomials in several indeterminates by “deforming” its multiplication.

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functions” on the same “variety”. This approach is convenient, in particular, because it is functorial: the correspondence A → A is a functor from the category of algebras to the category of Arens–Michael algebras, which is left adjoint to the forgetful functor acting in the opposite direction. Moreover, this functor extends to the category of A-modules and may therefore be considered as a noncommutative affine analog of the analytization functor [62], widely used in algebraic geometry. The present paper has two goals. The first is to explicitly compute Arens–Michael envelopes for some standard algebras of noncommutative polynomials, frequently used in noncommutative algebraic geometry. Those include, in particular, Yu. I. Manin’s quantum plane, the quantum torus, their higher-dimensional generalizations, quantum Weyl algebras, algebras of quantum matrices, universal enveloping algebras, etc. We will show that for many of those algebras of noncommutative polynomials their Arens– Michael envelope can be described as an algebra of noncommutative power series whose coefficients satisfy certain vanishing conditions at infinity. An interesting point here is that quite often for algebras of noncommutative polynomials which may be isomorphic as vector spaces (but not as algebras), their Arens–Michael envelopes are not isomorphic as topological vector spaces. The second goal of this paper, speaking informally, stems from the desire to answer the natural question of what are the common properties for an algebra of “noncommutative polynomial functions” and the corresponding algebra of “noncommutative holomorphic functions”? In other words, what properties of a finitely generated algebra A are inherited, in one sense or another, by its Arens–Michael envelope? Of course, there should be more than one approach to this question, depending on which properties of A are singled out and on what should be understood by the word “inherited”. The approach used in this paper was suggested by J. Taylor [67] as an essential element of his general view of “noncommutative holomorphic functional calculus”. It is based on the notion of homological epimorphism (or, in Taylor’s own terminology, localization) and allows us to compare homological properties of A and of A, including in particular their Hochschild (co)homology groups. By definition, a homomorphism of topological algebras A → B is called a homological epimorphism if the induced functor Db(B) → Db(A) between the bounded derived categories of topological modules is fully faithful. Informally this means that all the homological relations between topological B-modules are preserved when they are considered as A-modules under the change of rings A → B. The notion of homological epimorphism (localization) was introduced by Taylor in [67] and played a key role in the construction of the holomorphic calculus for a set of commuting operators in a Banach space. It looks like, in a purely algebraic setting, homological epimorphisms first appeared almost 20 years later in a paper by W. Geigle and H. Lenzing [17], independently of Taylor’s work and in connection with some questions of representation theory of finite-dimensional algebras. In particular, the term “homological epimorphism” was used for the first time in [17].2 Ten years later homological epimorphisms were independently rediscovered in a preprint [46] by A. Neeman and A. Ranicki, again in a purely algebraic context, and were used to solve some problems of algebraic K-theory. Notice that [46] uses a different terminology: when A → B is a homological epimorphism in the sense of [17], the authors of [46] say that “B is stably flat over A” (see also [48, 49]). Finally, at the end of 2004, R. Meyer [44] introduced, also independently, homological epimorphisms (therein called “isocohomological morphisms”) in the context of bornological algebras. The motivation came from the problem of computing Hochschild cohomology groups of various bornological algebras of functions of tempered growth on groups (with convolution multiplication) and more general crossed products.

2To the best of the author’s knowledge, this term was suggested by W. Crawley-Boevey.

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Thus homological epimorphisms were introduced independently in at least four papers in relation with various problems. That, we believe, indicates that this notion is natural and is worth further study. A basic example of homological epimorphism, which motivated the second half of this paper, is the canonical embedding of the polynomial algebra C[x1,...,xn] (viewed as a topological algebra with respect to the strongest locally convex topology) in the Fréchet algebra of entire functions O(Cn). The fact that this embedding is indeed a homological epimorphism (and remains such if Cn is replaced by an arbitrary polydomain U ⊂ Cn), was proved by Taylor in [67] and was substantially used in his construction of the holomorphic calculus of commuting operators. Since the algebra of entire functions is precisely the Arens–Michael envelope of the polynomial algebra, the above result brought Taylor to the following natural question: for which algebras A is the canonical homomorphism A → A a homological epimorphism? In fact, Taylor considered that question as a first step in his general program of constructing analogs of the holomorphic calculus of noncommuting operators. He found that the answer to that question is in the positive when A is a free algebra, and is in the negative when A is the universal enveloping algebra of a semisimple finite-dimensional Lie algebra. The case of universal enveloping algebras was also considered by A. A. Dosiev [16, 15] and the author [48, 49]. In particular, it was shown in [16] that the embedding of the universal enveloping algebra U(g) in its Arens–Michael envelope is a homological epimorphism, when g is metabelian (i.e., [g, [g, g]] = 0). Later, A. A. Dosiev generalized this result to nilpotent Lie algebras satisfying a certain condition of “normal growth” [15], and the author generalized it to positively graded Lie algebras [48]. On the other hand, it was shown in [49] that the embedding of U(g) in its Arens–Michael envelope can be a homological epimorphism only when g is solvable. In this paper we offer a general method allowing, for many finitely generated algebras A, to prove that the canonical homomorphism A → A is a homological epimorphism. It is based on the observation that if A is relatively quasi-free overasubalgebraR and satisfies certain additional finiteness and projectivity conditions (which can easily be checked in concrete situations), then A → A is a homological epimorphism whenever R → R is. This allows us to use induction in problems of that sort. More precisely, if the given algebra A has a chain of subalgebras

C = R0 ⊂ R1 ⊂···⊂Rn = A,

where each algebra Ri is relatively quasi-free over Ri−1 and satisfies the above additional finiteness and projectivity conditions, then it follows that A → A is a homological epimorphism. This method can be applied to many of the above algebras, in particular, the quantum plane, the quantum torus, some of their multi-dimensional generalizations, quantum Weyl algebras, the algebra of quantum (2 × 2)-matrices, the free algebras, and also, with some modifications, universal enveloping algebras of some solvable Lie algebras of small dimensions. Notice that the notion of quasi-free algebra, which plays a key role in the described method, was introduced and studied, in the “absolute” case (i.e., when R = C), by J. Cuntz and D. Quillen. A “relative” version of this notion, which technically differs little from the absolute case, is introduced and studied here in Section 7. Now we describe the contents of the paper. In Section 2 we collect basic facts about topological algebras and modules, and also describe a relative version of homology theory of topological algebras (for the “absolute” version, we refer the reader to [25]; for a purely algebraic version of the “relative” theory, see [28]). We also prove, for later use, some

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general results about acyclic objects. The Arens–Michael envelopes of topological algebras and modules are defined in Section 3, where we also establish their basic properties and give the first examples. In Section 4 we introduce some constructions of locally convex algebras which are “analytic analogs” of the classical constructions from ring theory: Ore extensions, skew Laurent extensions, and tensor algebras of bimodules. In Section 5 we establish connections between the algebraic and analytic constructions from the preceding section. We prove that, under some additional assumptions, the Arens–Michael envelope of an Ore extension (resp., skew Laurent extension) is an analytic Ore extension (resp., analytic skew Laurent extension), and that the Arens–Michael envelope of the tensor algebra of a bimodule M is an analytic tensor algebra of its Arens–Michael envelope M. As a consequence, we give explicit descriptions of the Arens–Michael envelopes for a number of algebras of noncommutative polynomials: the universal enveloping algebras of the two-dimensional solvable Lie algebra and of the three-dimensional Heisenberg Lie algebra, multiparameter quantum affine spaces and quantum tori, quantum Weyl algebras, and the algebra of quantum (2 × 2)-matrices. In Section 6 we introduce several versions of the notion of homological epimorphism: three “relative” ones (left, right, and two-sided) and an absolute one (which coincides with Taylor’s absolute localization from [67]). We give various characterizations of these notions in the spirit of those from [17]. In Section 7 we introduce relatively quasi-free algebras by giving several equivalent definitions (of which the most convenient one is probably the one in the language of noncommutative differential forms, cf. [7]) and establish some of their general properties. In particular, we study the action of the Arens–Michael functor on noncommutative differential forms and on relatively quasi-free algebras, and establish connections with the notion of two-sided relative homological epimorphism. We also compute the bimodules of relative differential forms for the algebras considered in Sections 4–5, and prove that those algebras are relatively quasi-free. Section 8 is of a technical nature and is devoted to various finiteness conditions for topological algebras and modules, which will be used later in the paper. The main results of the paper are contained in Section 9. There we state and prove a general theorem on sufficient conditions for the canonical epimorphism from A to its Arens–Michael envelope A to be a homological homomorphism. We also discuss its various consequences and provide concrete applications (see above). Finally, Section 10 is a detailed appendix containing basic facts about derived categories which are necessary for understating the present paper. We decided to include this appendix because all standard texts on derived categories deal with those of abelian categories, which is not enough for our purposes. For details, see [32].

2. Preliminaries All vector spaces and algebras in this paper are over the field of complex numbers. They are automatically assumed to be associative and unital (i.e., with identity), and all algebra homomorphisms will preserve identity elements. We do not require that 1 = 0; in other words, the zero algebra is by definition unital. Modules over algebras are also assumed to be unital (i.e., the identity of the algebra acts on modules by identity transformations). For concepts pertaining to the theory of topological vector spaces see, for example, [60]. Notice that in this paper, unlike [60], locally convex spaces (LCS) are not assumed to be Hausdorff. The completion of an LCS E is defined as the completion of the associated Hausdorff LCS E/{0} and is denoted either E or E ˜. The completed projective tensor product of locally convex spaces E and F is denoted E ⊗ F . By a topological algebra we understand a topological space A endowed with a structure of an associative algebra such that the multiplication operator A × A → A,(a, b) → ab

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is separately continuous. The category of topological algebras and continuous homomorphisms will be denoted Topalg. When speaking of continuous homomorphisms between topological algebras we will often omit the word “continuous”. If A is locally convex and complete and the multiplication on A is jointly continuous, then it is called a ⊗-algebra. In this case the multiplication operator A × A → A gives rise to a continuous linear operator mA : A ⊗ A → A, which is uniquely determined by the condition a ⊗ b → ab. A locally convex algebra is called a Fréchet algebra if it is metrizable and complete, i.e., if its underlying LCS is a Fréchet space. Since any separately continuous bilinear map of Fréchet spaces is automatically jointly continuous (see [60, III.5.1]), any Fréchet algebra is a ⊗-algebra. A prenorm ·on an algebra A is said to be submultiplicative,ifab≤ab for all a, b ∈ A. This is equivalent to saying that the “unit ball” U = {a ∈ A : a≤1} with respect to this prenorm is an idempotent set, i.e., U 2 ⊂ U. Therefore submultiplicative prenorms are precisely Minkowski functionals of convex absorbing circled idempotent sets. In particular, it now easily follows that for any two-sided ideal I ⊂ A and any submultiplicative prenorm on A, its quotient prenorm on A/I is also submultiplicative. A locally convex algebra A is said to be locally multiplicatively-convex (or locally m-convex, or multinormed) if its topology can be defined by a system of submultiplicative prenorms (or, equivalently, if in A there is a basis of convex circled idempotent neighborhoods of zero). If A is also complete, then it is called an Arens–Michael algebra.AnyBanach algebra is an Arens–Michael algebra, and any Arens–Michael algebra is a ⊗-algebra. It follows from the above that a quotient algebra of a locally m-convex algebra (resp., the completion of a quotient algebra of an Arens–Michael algebra) is locally m-convex (resp., is an Arens–Michael algebra). Notice that any nonzero submultiplicative prenorm on A is equivalent to some submultiplicative prenorm · such that 1 = 1 (this is proved the same way as for norms; see, for example, [24, 1.2.5]). Therefore when speaking of nonzero submultiplicative prenorms we will always assume that they satisfy the above identity. To check whether an algebra is locally m-convex, it is often convenient to use the following lemma from [45]. Lemma 2.1. Let A be a locally convex topological algebra whose topology is defined by asystemofprenorms{ · ν : ν ∈ Λ}. Suppose that for any ν ∈ Λ there are µ ∈ Λ and n C>0 such that a1 ···anν ≤ C a1µ ···anµ for any a1,...,an ∈ A.ThenA is locally m-convex. Corollary 2.2. Let A be a locally convex topological algebra whose topology is defined by a system of prenorms { · ν : ν ∈ Λ}. Suppose that for any ν ∈ Λ there are µ ∈ Λ and C>0 such that abν ≤ Caµbν for any a, b ∈ A.ThenA is locally m-convex.

Let A be a topological algebra, and Mn(A) the algebra of (n × n)-matrices with n2 coefficients in A. Identifying Mn(A)withA , we may endow Mn(A) with the direct product topology, with respect to which, as can easily be checked, Mn(A) is a topological algebra. If A is locally convex, (resp., locally m-convex, is a Fréchet algebra, is an Arens– Michael algebra), then so is Mn(A). Indeed, if { · ν : ν ∈ Λ} is a system of prenorms defining the topology on A, then the topology on Mn(A) is determined by the system of prenorms n aν =max aijν a =(aij) ∈ Mn(A),ν∈ Λ . 1≤j≤n i=1 A trivial calculation (which we omit) shows that if the prenorms ·ν are submultiplicative, then so are ·ν.

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A left topological module over a topological algebra A is a topological vector space X endowed with a structure of a left A-module such that the operator A×X → X,(a, x) → a · x is separately continuous. If A is a ⊗-algebra, the underlying space of the module X is locally convex and complete, and the above operator is jointly continuous, then X is called a left A-⊗-module. In this case we have a well-defined continuous linear operator A ⊗ X → X, a ⊗ x → a · x. If X and Y are left topological A-modules, then a morphism from X to Y is a continuous linear map ϕ : X → Y, such that ϕ(a · x)=a · ϕ(x) for all a ∈ A and x ∈ X. The morphisms from X to Y form a vector space, denoted Ah(X, Y ). When Y = X that space will be denoted simply Ah(X). Similarly, one defines right topological modules, topological bimodules, and their morphisms. The space of morphisms from a right topological A-module X toarighttopolog- ical A-module Y (resp., from a topological A-bimodule X to a topological A-bimodule Y ) is denoted hA(X, Y )(resp.,AhA(X, Y )). Suppose X and Y are left topological A-modules. We endow the space Y X of all maps X from X to Y with the Tikhonov topology and consider Ah(X, Y ) as a subspace of Y . X The topology on Ah(X, Y ) induced from Y is called the topology of simple convergence. The subsets of the form (2.1) M(S, U)= ϕ ∈ Ah(X, Y ):ϕ(S) ⊂ U , where U ⊂ Y is a neighborhood of zero and S ⊂ X is a finite set, form a basis of neighborhoods of zero in that space. It is easy to check that the canonical isomorphism ∼ of vector spaces Ah(A, X) = X, ϕ → ϕ(1) is a topological isomorphism. Similarly, using Xn as notation for the direct sum of n copies of a module X (with the standard n ∼ n direct sum topology), we have a topological isomorphism Ah(A ,X) = X .When X = An this isomorphism can be viewed as an isomorphism of topological algebras n ∼ n Ah(A ) = Mn(A). In particular, if A is an Arens–Michael algebra, then so is Ah(A ). Later on we will give a slight generalization of the last observation (see Lemma 8.2). Suppose A is a ⊗-algebra, X arightA-⊗-module, and Y aleftA-⊗-module. A projective A-module tensor product X ⊗A Y is defined as the completion of the quotient space of X ⊗ Y by the closure of the linear hull of all elements of the form x·a⊗y−x⊗a·y (x ∈ X, y ∈ Y, a ∈ A). For any two continuous prenorms on X and, respectively, on Y , their projective tensor product is a prenorm on X ⊗ Y and, after factorization and extension by continuity, yields a prenorm on X ⊗A Y, which we shall also call a projective tensor product of the given prenorms. Notice that, as in a purely algebraic situation, the projective A-module tensor product admits an abstract categorical description (see [25, II.4.1]). Recall that the strongest locally convex topology on a vector space E is defined by the system of all prenorms on E. We will denote the space E endowed with this topology by Es. Clearly, if E is an LCS, then its topology is strongest locally convex if and only if any linear map from E to any LCS is continuous. Later in this paper we shall often use some basic properties of the strongest locally convex topology. For ease of reference, we collect those properties in a separate proposition. Proposition 2.3. (i) Any strongest LCS is complete. (ii) If E is a strongest LCS and F ⊂ E is a vector subspace, then the quotient topology on E/F is the strongest one. (iii) If E is a strongest LCS and F ⊂ E is a vector subspace, then F is closed and complemented in E and the topology on F induced from E is the strongest one. (iv) If E,F are strongest LCS of at most countable dimension, then any bilinear map from E × F to an arbitrary LCS is jointly continuous.

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(v) Any algebra A of at most countable dimension is a ⊗-algebra with respect to the strongest locally convex topology and any A-module of at most countable dimension is an A-⊗-module with respect to the strongest locally convex topology. (vi) Suppose A is a ⊗-algebra, X is a right A-⊗-module, Y is a left A-⊗-module. Suppose also that X and Y are of at most countable dimension and the topology on each one of them is strongest locally convex. Then the algebraic tensor product X ⊗A Y endowed with the strongest locally convex topology coincides with X ⊗A Y. Proof. (i) Just notice that E is isomorphic to the topological direct sum of one- dimensional spaces and use the fact that completeness is preserved under direct sums (see [60, II.6.2]). (ii) Just notice that the canonical map from E to (E/F)s is continuous and therefore factors through E/F. (iii) Just consider an arbitrary (and automatically continuous) projector from E to F and use (ii). (iv) See [5], Corollary A.2.8. (v), (vi) These follow from (iv) and (i).

Remark 2.1. Notice that (v) applies, in particular, to any finitely generated algebra A and any finitely generated A-module. Let A be an algebra, X an A-bimodule, and α an endomorphism of A. We define a new structure of an A-bimodule on X by setting a◦x = a·x and x◦a = x·α(a)fora ∈ A, x ∈ X. This bimodule is denoted Xα. In a similar way we define a bimodule αX.Itis easy to see that if A is a topological algebra (resp., ⊗-algebra), α is continuous, and X is a topological A-bimodule (resp., A-⊗-bimodule), then Xα and αX are also topological A-bimodules (resp., A-⊗-bimodules). Suppose R is an algebra. Recall that an R-algebra is a pair (A, ηA), where A is an algebra and ηA : R → A is a homomorphism. If, in addition, R and A are topological algebras and ηA is continuous, then (A, ηA) is called a topological R-algebra. Finally, if both A and R are ⊗-algebras, then we shall say that (A, ηA)isanR-⊗-algebra.When speaking of R-algebras we shall often abuse the language by saying that A is an R-algebra (without explicitly mentioning ηA); it should not lead to confusion. A similar convention applies to topological R-algebras and R-⊗-algebras. A homomorphism ϕ : A → B between two R-algebras is called an R-homomorphism if ϕηA = ηB.IfA is an R-algebra and B is an S-algebra, then an R-S-homomorphism from A to B is defined as a pair of homomorphisms f : A → B and g : R → S making the diagram

f / AO BO

(2.2) ηA ηB g / R S commute. If g is ﬁxed, then we shall sometimes say that f : A → B is an R-S-homomorphism. The category of topological R-algebras and continuous R-homomorphisms will be denoted R-Topalg. Every topological R-algebra (resp., R-⊗-algebra) A will be considered as a topological R-bimodule (resp., R-⊗-bimodule) with respect to the homomorphism ηA : R → A. We remark that R-homomorphisms are precisely algebra homomorphisms which are R- bimodule homomorphisms (or in this case, equivalently, homomorphisms of left or right R-modules).

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If A is a topological algebra, and X is a topological A-bimodule, then the direct product A × X becomes a topological algebra with respect to multiplication

(a, x)(b, y)=(ab, a · y + x · b).

Algebra thus obtained is called the semidirect product of A and X or the trivial extension of A by X. It is easy to see that if A is a ⊗-algebra, and X is an A-⊗-bimodule, then A × X is also a ⊗-algebra. If A is a topological R-algebra, then A × X also becomes a topological R-algebra with respect to the homomorphism

ηA×X : R → A × X, ηA×X (r)=(ηA(r), 0).

Notice that the projection p1 : A × X → A to the first factor is an R-homomorphism. Suppose A is an algebra and X is an A-bimodule. Recall that a linear map D : A → X is called a derivation if D(ab)=D(a) · b + a · D(b) for any a, b ∈ A. Henceforth all derivations of topological algebras with values in topological bimodules will be assumed to be continuous. For a topological algebra A and a topological A-bimodule X the vector space of all derivations from A to X is denoted Der (A, X). If A is a topological R-algebra, then a derivation D : A → X is called an R-derivation if DηA =0.Itisnot difficult to see that this means precisely that D is a morphism of R-bimodules (or in this case, equivalently, of left or right R-modules). The subspace of Der (A, X) consisting of R-derivations will be denoted Der R(A, X). There is a close relationship between R-derivations from A to X and R-homomorphisms from A to A × X. More precisely, as direct calculation shows, a continuous linear map ϕ: A → A × X of the form ϕ(a)=(a, D(a)) is an R-homomorphism if and only if D is an R-derivation. In other words, there is an isomorphism of vector spaces ∼ (2.3) Der R(A, X) = ϕ ∈ Hom R-Topalg (A, A × X): p1ϕ = 1A . Moreover, the isomorphism (2.3) is in fact an isomorphism between functors of X acting from the category of topological A-bimodules to the category of vector spaces. If A is a ⊗-algebra, then the category of left A-⊗-modules (resp., right A-⊗-modules, A-⊗-bimodules) and (continuous) morphisms between them will be denoted A-mod (resp., mod-A, A-mod-A). These categories are additive and have both kernels and cokernels (see [25, 0.4.1]). More precisely, for any morphism ϕ: X → Y itskernelisa pair (Ker ϕ, ker ϕ), where Ker ϕ = ϕ−1(0) (with the topology induced from X)and ker ϕ:Kerϕ → X is the canonical embedding. Similarly, the cokernel of ϕ is a pair (Coker ϕ, coker ϕ), where Coker ϕ is the completion of the quotient module Y/ϕ(X), and coker ϕ: Y → Coker ϕ is the quotient map. Suppose f : A → B is a homomorphism of ⊗-algebras. Recall that any left B-⊗- module X is a left A-⊗-module under the action a · x = f(a) · x (a ∈ A, x ∈ X). Clearly, any morphism of B-⊗-modules ϕ: X → Y is a morphism of A-⊗-modules. Thus we have a functor from B-mod to A-mod, which we will denote f • and call the restriction of scalars along f. A similar functor from mod-B to mod-A will also be denoted f •. Suppose now that (A, ηA)isanR-⊗-algebra. Set A = A-mod, R = R-mod and endow R with the split exact structure (see Appendix, Example 10.1). Let : A → R denote the restriction-of-scalars functor along ηA; clearly, it is additive and preserves kernels and cokernels. Hence A-mod can be made an exact category as in Example 10.3 (see Appendix). The exact category AR thus obtained will be denoted (A, R)-mod (cf. [28]). Thus the admissible sequences in (A, R)-mod are those that split in R-mod.Inparticular, when R = C, we recover the standard definition of an admissible (or C-split) sequence of A-⊗-modules used in the homological theory of topological algebras (see [25, 66]).

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For a ⊗-algebra A let Aop denote the opposite algebra and Ae the enveloping algebra A ⊗ Aop.IfA is an R-⊗-algebra and B is an S-⊗-algebra, then we set mod-(A, R)=(Aop,Rop)-mod, (A, R)-mod-(B,S)=(A ⊗ Bop,R⊗ Sop). The exact category (A, C)-mod will be denoted simply A-mod; this should not lead to confusion. We adopt a similar convention for the categories mod-A = mod-(A, C)and A-mod-A =(A, C)-mod-(A, C). Suppose as before that (A, ηA)isanR-⊗-algebra and let : A-mod → R-mod be the restriction-of-scalars functor along ηA. It has a left adjoint functor, namely, A ⊗R ( · ). Therefore, taking account of Example 10.8 (see Appendix), for any E ∈ R-mod the A-⊗-module A ⊗R E is projective in (A, R)-mod,(A, R)-mod has enough projectives, and a module X is projective in (A, R)-mod if and only if the multiplication morphism A ⊗R X → X is a retraction of left A-⊗-modules. Modules of the form A ⊗R E,whereE ∈ R-mod, will be called free in (A, R)-mod.For future reference we remark that, under the standard identiﬁcation of right A-⊗-modules with left Aop -⊗-modules and of A-⊗-bimodules with left Ae-⊗-modules, free objects in the exact categories of right A-⊗-modules and A-⊗-bimodules are of the form: E ⊗R A (E ∈ mod-R)—inmod-(A, R), A ⊗R E ⊗R A (E ∈ R-mod-R)—in(A, R)-mod-(A, R), M ⊗R A (M ∈ A-mod-R)—in(A, A)-mod-(A, R), A ⊗R M (M ∈ R-mod-A)—in(A, R)-mod-(A, A). Remark 2.2. Of course, all this has an obvious purely algebraic analog (see [28]). In particular, if A is an R-algebra, then the exact category of left A-modules (A, R)-algmod is the category of all left A-modules whose admissible sequences are precisely those that split as sequences of left R-modules. Notice that if A and the left A-module X are of at most countable dimension, then X is projective in (A, R)-algmod if and only if the strongest locally convex As-⊗-module Xs is projective in (As,Rs)-mod. This follows immediately from the description of projective objects in terms of free ones (see above) ⊗ ∼ ⊗ and from the isomorphism (A R X)s = As Rs Xs (see Prop. 2.3). Clearly, similar assertions are also true for right modules and for bimodules. Since the above categories have enough projective objects, it follows that any additive covariant (resp., contravariant) functor on those categories with values in an exact category has a left (resp., right) derived functor (see Appendix). In particular, let Vect be the abelian category of vector spaces and LCS the quasia- belian category of locally convex spaces (see Appendix, Example 10.2). Then, for a ﬁxed Y ∈ (A, R)-mod, the morphism functor

Ah( · ,Y): (A, R)-mod → Vect has a right derived functor − + RAh( · ,Y): D ((A, R)-mod) → D (Vect). n · The corresponding n-th classical derived functor is denoted Ext A,R( ,Y). Similarly, for aﬁxedX ∈ mod-(A, R), the tensor product functor X ⊗A ( · ): (A, R)-mod → LCS has a left derived functor ⊗ L · − → − X A( ): D ((A, R)-mod) D (LCS).

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A,R · C The corresponding n-th classical derived functor is denoted Tor n (X, ). When R = n · A · the constructed functors are denoted Ext A( ,Y)andTorn (X, ), respectively; see [25, III.4]. Remark 2.3. Since the forgetful functor from Vect to the category Ab of abelian groups preserves kernels and cokernels, it is easy to see that the underlying abelian group of n n Ext A,R(X, Y ) is naturally isomorphic to the group Ext (A,R)-mod(X, Y ) (see Appendix, Example 10.9). ∈ ∈ ⊗ L → ⊗ For X mod-(A, R)andY (A, R)-mod the canonical morphism X A Y X A Y A,R → (see Appendix) gives rise, when passing to cohomology, to a morphism Tor 0 (X, Y ) X ⊗A Y of LCS. It is an open map onto its image, which is dense in X ⊗A Y,andits A,R kernel is the closure of zero in Tor 0 (X, Y ). We omit the proof because it is identical to that in the case R = C; see [25, III.4.26]. Therefore, X ⊗A Y can be identiﬁed with A,R the completion of Tor 0 (X, Y ). If B is an S-⊗-algebra and X ∈ (B,S)-mod-(A, R), then X ⊗A ( · ) can be viewed as a functor from (A, R)-mod to (B,S)-mod. The corresponding left derived functor with − ⊗ L · values in D ((B,S)-mod) will be denoted by the same symbol X A( ) as before. It will not lead to confusion because we will always indicate the domain and codomain of the given functor. Remark 2.4. As in a purely algebraic situation (see, for example, [23, I.6] or [30, 1.10]), ⊗ L one can show that RAh and A extend to bifunctors − + + RAh: D ((A, R)-mod) × D ((A, R)-mod) → D (Vect), ⊗ L − × − → − A : D (mod-(A, R)) D ((A, R)-mod) D (LCS).

Remark 2.5. On the morphism spaces Ah(X, Y ) one can consider various topologies of uniform convergence on families of bounded subsets (for example, all bounded subsets, orprecompactsubsets,orﬁnitesubsets...);see[60,III.3]. ThisallowsustoviewRAh + n · · as a functor with values in D (LCS) and Ext A,R( , ) as a functor with values in LCS. We will not go into detail since for our purpose, we do not need such a level of generality. The relative homological dimension of an A-⊗-module X is by deﬁnition ∈ Z n+1 ∀ ∈ dhA,R X =min n + :ExtA,R (X, Y )=0 Y (A, R)-mod .

If such an n does not exist, then one sets dhA,R X =+∞.NoticethatX is projective in (A, R)-mod if and only if dhA,R X = 0. The relative homological dimension may also be deﬁned as the length of a shortest projective resolution (see Appendix) of X in (A, R)-mod. Finally, for any p>dhA,R X and any Y ∈ (A, R)-mod we have p Ext A,R(X, Y ) = 0. We omit the proofs of these assertions because they are identical to those in the case R = C (see [25, III.5]). The relative left global dimension of an R-⊗-algebra A is deﬁned as ∈ dgR A =sup dhA,R X : X (A, R)-mod .

When R = C the dimensions of modules and algebras just deﬁned are denoted dhA X and dg A, respectively; see [25, III.5]. We end this section with several results about acyclic objects (see Appendix) which will be needed later. Proposition 2.4. Let A, B and C be exact categories and F : A → B and G: B → C additive functors having left derived functors. Suppose that the conditions of the theorem

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on the composition of derived functors (see Appendix) hold for F and G.LetX ∈ Ob (A) be an F -acyclic object. Then X is GF -acyclic if and only if F (X) is G-acyclic. Proof. By the assumption, the canonical morphism LF (X) → F (X) is an isomorphism in D−(B). Therefore, (LG ◦ LF )(X) → LG(F (X)) is an isomorphism in D−(C). In the commutative diagram ∼ (LG ◦ LF )(X) /LG(F (X))

(2.4) L(G ◦ F )(X) /G(F (X)) the top horizontal and the left vertical arrows are isomorphisms. Therefore, the lower horizontal arrow is an isomorphism if and only if the right vertical arrow is an isomorphism. The rest is clear.

Corollary 2.5. Let

F1 / AO BO

G1 G2 R /S F2 be a commutative diagram of exact categories and additive functors having left derived functors. Suppose that the theorem on the composition of derived functors is true for (F2,G2) and (G1,F1).SupposeX ∈ Ob (R) is acyclic with respect to F2 and G1,and F2(X) is acyclic with respect to G2.ThenG1(X) is acyclic with respect to F1. Proposition 2.6. Any R-⊗-algebra A is acyclic with respect to the tautological functors (A, R)-mod-(A, R) → (A, A)-mod-(A, R), (A, R)-mod-(A, R) → (A, R)-mod-(A, A). Moreover, any projective resolution of A in (A, R)-mod-(A, R) is automatically a projective resolution in (A, A)-mod-(A, R) and (A, R)-mod-(A, A). Proof. Clearly, it suﬃces to consider the ﬁrst of the two functors. Fix a projective resolution

(2.5) 0 ← A ← P• in (A, R)-mod-(A, R). The augmented complex (2.5) in the exact category (A, R)-mod- (R, R) is admissible and all of its objects are projective Therefore, it splits in A-mod-R; i.e., it is admissible in (A, A)-mod-(A, R). This proves the acyclicity. The second assertion follows at once from the fact that the above functors send free modules to free modules and therefore projectives to projectives.

Proposition 2.7. Let A be an R-⊗-algebra, B an exact category, and F : A-mod-A → B an additive covariant functor. Suppose functors

F1 :(A, A)-mod-(A, R) → B,

F2 :(A, R)-mod-(A, R) → B,

F3 :(A, R)-mod-(A, A) → B ∼ ∼ − act the same way as F .ThenLF1(A) = LF2(A) = LF3(A) in D (B).

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Proof. Let I :(A, R)-mod-(A, R) → (A, A)-mod-(A, R) denote the tautological functor. − Clearly, F2 = F1 ◦ I.InD (B)wehaveachainofmorphisms

LF2(A)=L(F1 ◦ I)(A) ← LF1(LI(A)) → LF1(I(A)) = LF1(A). Since I sends projectives to projectives (see the proof of Proposition 2.6), the first arrow in this chain is an isomorphism. By the same proposition, the second arrow is also an ∼ ∼ isomorphism. Therefore, LF1(A) = LF2(A), as required. The isomorphism LF3(A) = LF2(A) is established similarly. Corollary 2.8. Let A be an R-⊗-algebra and X an A-⊗-bimodule. Then for any n ∈ Z, we have isomorphisms of vector spaces Ae,Re ∼ Ae,A⊗ Rop ∼ Ae,R⊗ Aop Tor n (X, A) = Tor n (X, A) = Tor n (X, A), n ∼ n ∼ n Ext Ae,Re (A, X) = Ext Ae,A⊗ Rop (A, X) = Ext Ae,R⊗ Aop (A, X). Remark 2.6. If we endow the Ext -spaces from Corollary 2.8 with the locally convex topology as in Remark 2.5, then the above isomorphisms will also be topological isomorphisms. Definition 2.1. Suppose A is an R-⊗-algebra and X is an A-⊗-bimodule. The vector spaces H Ae,Re Hn n n(A, R; X)=Torn (X, A), (A, R; X) = Ext Ae,Re (A, X) are called, respectively, the n-th relative Hochschild homology (cohomology) group of A with coefficients in X.

Definition 2.2. The number dbR A =dhAe,Re A is called the relative bidimension of the R-⊗-algebra A is. When R = C, it is denoted db A (see [25, III.5]). Remark 2.7. In view of Corollary 2.8, the above definitions of relative homology and cohomology and relative bidimension are equivalent to Hochschild’s original definitions in [28]. See also [19].

3. Arens–Michael envelopes Arens–Michael envelopes (under a diﬀerent name) were introduced in [66]. Here we adopt the terminology from [24]. Deﬁnition 3.1. Suppose A is a topological algebra. A pair (A, ιA) consisting of an Arens–Michael algebra A and a continuous homomorphism ιA : A → A is called the Arens–Michael envelope of A if for any Arens–Michael algebra B and any continuous homomorphism ϕ: A → B there exists a unique continuous homomorphism ϕ: A → B making the following diagram commute: _ ϕ_ _/ AO ?B (3.1) ιA ϕ A

We then say that ϕ extends ϕ (even though ιA is in general not injective; see Example 3.4 below). In other words, if AM denotes the full subcategory of Topalg consisting of Arens– Michael algebras and Set denotes the category of sets, then the Arens–Michael envelope of a topological algebra A is a representing object for the functor Hom Topalg (A, · ): AM → Set. The Arens–Michael envelope is clearly unique in the sense that if (A, ιA)and(A, jA)

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are Arens–Michael envelopes of the same topological algebra A, then there exists a unique isomorphism of topological algebras ϕ: A → A making the following diagram commute: _ _ _ ϕ_ _ _ _/ A _> ?A >> ÐÐ >> ÐÐ ι > Ð A > ÐÐ jA A Remark 3.1. Notice that it suﬃces to check the universal property of the Arens–Michael envelope expressed by diagram (3.1) only for homomorphisms with values in Banach algebras. This follows easily from the fact that each Arens–Michael algebra is the inverse limit of a system of Banach algebras (see, for example, [24, Ch. V]). Indeed, suppose A is a topological algebra, A is an Arens–Michael algebra, and ιA : A → A is a homomorphism such that for any Banach algebra B and any homomorphism ϕ: A → B there is a unique homomorphism ϕ: A → B making diagram (3.1) commutative. Take an arbitrary Arens– Michael algebra C and represent it as the inverse limit lim←− Cν of a system of Banach algebras. Then

Hom Topalg (A, C)=HomTopalg (A, ←lim− Cν ) =← lim− Hom Topalg (A, Cν) =← lim− Hom AM(A, Cν )=HomAM(A, ←lim− Cν )=HomAM(A, C). Therefore, (A, ιA) is the Arens–Michael envelope of A. Every topological algebra A has an Arens–Michael envelope (see [66] or [24, Ch. V]). More precisely, (A, ιA) can be obtained as the completion of A with respect to a new topology deﬁned by the family of all continuous submultiplicative prenorms on A.In particular, this implies that the image of the canonical homomorphism ιA : A → A is dense. It is easy to see that the correspondence A → A is a functor from Topalg to AM and that it is the left adjoint of the embedding functor from AM to Topalg. Henceforth we shall call it the Arens–Michael functor. If A is an algebra without topology, then we deﬁne its Arens–Michael envelope as the one with respect to the strongest locally convex algebra As (see Section 2). We now give several basic examples of Arens–Michael envelopes. Other examples will be given in Section 5.

Example 3.1. Suppose the polynomial algebra A = C[z1,...,zn], is endowed with the strongest locally convex topology. Taylor [66] noticed that its Arens–Michael envelope coincides with the algebra of entire functions O(Cn) endowed with the compact-open topology and that the homomorphism ιA is the tautological embedding. Since this example serves as the motivation for the rest of the paper, we sketch the basic idea behind the proof of the last assertion. For simplicity, we consider the case n =1. The Cauchy inequality for the coeﬃcients of a power series implies that the compact- open topology on O(C) can be deﬁned by the family of norms i i fρ = |ci|ρ f = ciz ∈ O(C); 0 <ρ<∞ . i i For any Banach algebra B and homomorphismϕ: C[z] → B,setb = ϕ(z). Then for any i ∈ O C i entire function f = i ciz ( )theseries i cib converges absolutely in B to some f(b) ∈ B;moreover,f(b)≤fρ for any ρ ≥b.Themapf → f(b) is the desired homomorphism from O(C)toB extending ϕ. The case of an arbitrary n is argued similarly.

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Example 3.2. Let A = Fn be the free C-algebra with n generators ζ1,...,ζn. For any ··· set of integers α =(α1,...,αk)from[1,n], let ζα be the element ζα1 ζαk of Fn.It is convenient to assume that the identity of the algebra corresponds to the set of zero | | F length (k = 0). We also set α = k. The algebra n of noncommutative power series ∈ C consists of all formal expressions a = α λαζα (where λα ) such that |α| aρ = |λα|ρ < ∞ for all 0 <ρ<∞. α

The system of prenorms { · ρ :0<ρ<∞} makes Fn intoaFréchet–Arens–Michael algebra. Clearly, Fn contains Fn as a subalgebra. Taylor showed [67] (cf. also the previous example), that F is the Arens–Michael envelope of F . Notice that when n ∼ n n =1,wehaveF1 = C[z]andF1 = O(C). Example 3.3. Let g be a semisimple Lie algebra and A = U(g) its universal enveloping algebra. Taylor showed [67] that the Arens–Michael envelope of U(g) is topologically isomorphic to the direct product of a countable family of full matrix algebras. More precisely, each finite-dimensional irreducible representation πλ of g yields a homomorphism → C ϕλ : U(g) Mdλ ( )(heredλ is the dimension of πλ). Let g be the set of equivalence classes of irreducible finite-dimensional representations of g; we then have a homomorphism → C ϕλ : U(g) Mdλ ( ). λ∈g λ∈g C The algebra λ Mdλ ( ) endowed with the direct product topology together with the above homomorphism is the Arens–Michael envelope of U(g).

Notice that for the algebras in Examples 3.1–3.3 the canonical homomorphism ιA to the Arens–Michael envelope is injective. The reason for this is that those algebras have suﬃciently many Banach representations. On the other hand, it may happen that a given algebra A has no nonzero Banach representations, and in that case its Arens–Michael envelope is trivial. For illustration, consider the following example. Example 3.4. Suppose A is a Weyl algebra (endowed, as the algebras from the above examples, with the strongest locally convex topology). Recall that it is generated (as a unital C-algebra) by generators ∂ and x, subject to the relation [∂,x]=1.Itis well known (see, for example, [59, 13.6]) that a nonzero normed algebra cannot contain elements ∂ and x with such a property. Hence there are no nonzero submultiplicative prenorms on A and therefore A =0. Other examples of algebras with trivial Arens–Michael envelopes can be found in [24, 5.2.26]. Example 3.5. Choose any Banach algebra A such that all homomorphisms from it to any other Banach algebra are continuous. For example, this could be the algebra B(H) of bounded operators in a Hilbert space or, more generally, in a Banach space E such ∼ that E = E ⊕ E; see [10, 5.4.12]. Endowing A with the strongest locally convex topology ∼ we have the topological algebra As. Then, clearly, we have an isomorphism As = A. When describing Arens–Michael envelopes it is often convenient to use the fact that the Arens–Michael functor commutes with the quotient operation on algebras. The following proposition is proved in [48]. Proposition 3.1. Suppose A is a topological algebra and I is a two-sided ideal in A. Let J denote the closure of the set ιA(I) in A.ThenJ is a two-sided ideal in A,and

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the homomorphism A/I → A/J induced by the homomorphism ιA : A → A extends to an isomorphism of topological algebras ∼ ∼ A/I = (A/J) . Since the quotient space of any Fréchet space by a closed subspace is complete, we have the following. Corollary 3.2. Under the assumptions of Proposition 3.1, assume in addition that A is ∼ aFréchet algebra. Then A/I = A/J. Corollary 3.3. If A is a finitely generated algebra, then As is a trace class Fréchet algebra. Proof. Since A is finitely generated, it is isomorphic to a quotient algebra of the free algebra Fn with finitely many generators. Now apply Example 3.2 and Corollary 3.2, as well as the fact that the Fréchet algebra Fn is trace class (see [37]). Example 3.6. Suppose V is an affine complex algebraic variety embedded in CN .Let Oalg(V ) denote the algebra of regular (polynomial) functions on V. Generalizing Exam- ple 3.1, we shall show that its Arens–Michael envelope is the Fréchet algebra of holomorphic functions O(V ). To this end, notice that Oalg(V )=Oalg(CN )/I,whereI is the ideal alg N alg N in O (C ) of all functions vanishing on V. Since the algebra O (C )=C[z1,...,zN ] is Noetherian, there is an exact sequence r (3.2) Oalg(CN ) → Oalg(CN ) → Oalg(V ) → 0 of Oalg(CN )-modules. It gives rise to an exact sequence Oalg r → Oalg → Oalg → (3.3) ( CN ) CN i∗ V 0 Oalg CN of sheaves of CN -modules, where i denotes the embedding of V in . Applying the analytization functor to (3.3) (see, for example, [69, 13.4] or [62]), we have an exact sequence r (OCN ) → OCN → i∗OV → 0

of OCN -modules. Finally, applying the global section functor we have, by Cartan’s The- orem B, an exact sequence r (3.4) O(CN ) → O(CN ) → O(V ) → 0 of O(CN )-modules. Let J ⊂ O(CN ) denote the ideal of all functions vanishing on V. Comparing (3.2) and (3.4), we see that if I ⊂ Oalg(CN ) is generated by r functions, then the same functions generate J as an ideal in O(CN ). Therefore, J is the smallest ideal in O(CN ) containing I.Inparticular,J ⊂ I. Since the inverse inclusion is obvious by the closedness of J,wehaveJ = I. Now it remains to apply Corollary 3.2, according to which Oalg(V ) = Oalg(CN )/I = O(CN )/J = O(V ). Next, we shall establish several auxiliary results about modules over Arens–Michael algebras. Suppose A is an algebra and X is a left A-module. We shall say that a prenorm ·X on X is m-compatible if there is a submultiplicative prenorm ·A on A such that a · xX ≤aAxX for all a ∈ A and x ∈ X. A similar deﬁnition makes sense when X is a right module or a bimodule. In the latter case we shall simultaneously require that a · xX ≤aAxX and x · aX ≤xX aA for all a ∈ A and x ∈ X. If A is a topological algebra and X is a topological A-module, then we shall also require that both prenorms ·X and ·A be continuous. Notice that if Y is a submodule of X, then the quotient prenorm on X/Y induced by an m-compatible prenorm on X is also m-compatible.

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Proposition 3.4. Suppose A is an Arens–Michael algebra and X is a left A-⊗-module. Then the topology on X canbedefinedbysomefamilyofm-compatible prenorms. Proof. Since every left A-⊗-module is a quotient of a free module, it suffices to prove the assertion for free modules. Thus let X = A ⊗ E be a free A-module. For each continuous submultiplicative prenorm on A and each continuous prenorm on E their projective tensor product is clearly an m-compatible prenorm on X. On the other hand, the prenorms of this form do define the topology on X.

Remark 3.2. Suppose {·λ : λ ∈ Λ1} is a directed system of submultiplicative prenorms defining the topology on A and {·µ : µ ∈ Λ2} is a directed system of prenorms defining the topology on X.Foreachλ ∈ Λ1 and each µ ∈ Λ2 we define a new prenorm on X by setting n n xλ,µ =inf aiλxiµ : x = aixi,ai ∈ A, xi ∈ X, n ∈ N . i=1 i=1 It is easy to check that this prenorm is m-compatible and that the system of all prenorms on X of this form is equivalent to the original one. Remark 3.3. Notice that the completeness of the algebras and modules in Proposition 3.4 is not essential. Proposition 3.5. Suppose that A is an Arens–Michael algebra. Then any left A-⊗- module X is isomorphic to the inverse limit of some system of Banach A-⊗-modules.

Proof. For each m-compatible prenorm ·X on X the set of x ∈ X such that xX =0 is a submodule of X. With the aid of Proposition 3.4, the rest of the proof is argued similarly to the proof in the case A = C (see, for example, [60, II.5.4]). Remark 3.4. Assertions similar to Propositions 3.4 and 3.5 for right modules and for bimodules also hold. Suppose E is a locally convex space whose topology is deﬁned by a system of prenorms { · λ : λ ∈ Λ}. We can make Λ a partially ordered set by deﬁning λ ≺ µ if xλ ≤xµ for all x ∈ E. Henceforth we shall assume that the indexing set for prenorms on a locally convex space is ordered this way. Proposition 3.6. Suppose A is an Arens–Michael algebra and X is an A-⊗-bimodule. Then the trivial extension A × X is also an Arens–Michael algebra.

Proof. Let { · λ : λ ∈ Λ1} be the system of all continuous submultiplicative prenorms on A and { · µ : µ ∈ Λ2} the system of all continuous m-compatible prenorms on X. In view of Proposition 3.4, these systems define the topologies on A and X, respectively. For each λ ∈ Λ1 and each µ ∈ Λ2 we define a prenorm on A × X by setting (a, x)λ,µ = aλ +xµ. Clearly, the system of all such prenorms defines the direct product topology on A × X. Using the definition of an m-compatible prenorm, for any µ ∈ Λ2 we choose an element ϕ(µ) ∈ Λ1 such that a · xµ ≤aϕ(µ)xµ and x · aµ ≤xµaϕ(µ) for any a ∈ A and any x ∈ X.SetΛ={(λ, µ) ∈ Λ1 × Λ2 : ϕ(µ) ≺ λ}. Clearly, Λ is cofinal in Λ1 × Λ2 and therefore the topology on A × X is defined by the system of prenorms { · λ,µ :(λ, µ) ∈ Λ}. For any elements (a, x)and(b, y)inA × X and any (λ, µ) ∈ Λwe have

(a, x)(b, y)λ,µ = (ab, a · y + x · b)λ.µ

= abλ + a · y + x · bµ ≤aλbλ + aλyµ + xµbλ

≤ (aλ + xµ)(bλ + yµ)=(a, x)λ,µ(b, y)λ,µ.

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Therefore, each prenorm ·λ,µ for (λ, µ) ∈ Λ is submultiplicative and A × X is an Arens–Michael algebra.

Using the preceding result it is not diﬃcult to show that the Arens–Michael envelope has the extension property not just for homomorphisms but also for derivations deﬁned on the original algebra. Before giving a rigorous statement and a proof of this assertion we want to make some observations. Suppose A is a topological R-algebra, B is a topological S-algebra, and (f,g)isanR-S-homomorphism from A to B (see (2.2)). It is easy to see that for any topological B-bimodule X and any S-derivation D : B → X the composition D ◦ f : A → X is an R-derivation. Thus we have a map

(3.5) Der S(B,X) → Der R(A, X),D→ D ◦ f, obviously giving rise to a morphism of functors of X acting from the category of topological B-bimodules to the category of vector spaces. Recall that a morphism f : X → Y in an arbitrary category A is said to be an epimorphism if for any Z ∈ Ob (A)andanyg, h: Y → Z we have g = h whenever gf = hf. If this condition is only satisﬁed for those Z which are objects of a certain full subcategory C ⊂ A, then we shall say that f is an epimorphism relative to C.Forease of reference we state the following lemma, the proof of which is obvious. Lemma 3.7. Suppose that in some category A we are given a diagram

f / h / AO BO ~?C ~~ (3.6) α β ~~ ~~ γ g / R S with a commutative left square. Suppose further that the diagram obtained by removing β is also commutative. If g is an epimorphism relative to some full subcategory C ⊂ A containing C, then the whole diagram (3.6) is commutative. Proposition 3.8. Suppose A, R, S are topological algebras and g : R → S is an epimorphism in Topalg relative to a subcategory of the category of Arens–Michael algebras AM. Suppose that A is a topological R-algebra and its Arens–Michael envelope A is a topological S-algebra such that the pair (ιA,g) is an R-S-homomorphism from A to A. Then for any A-⊗-bimodule X the canonical map (3.7) Der S (A, X) → Der R(A, X),D→ D ◦ ιA is a bijection. In other words, any R-derivation from A to X uniquely extends to an S-derivation from A to X. Proof. The injectivity of (3.7) is obvious since Im ιA is dense in A. To establish the sur- jectivity, take an arbitrary R-derivation D : A → X and construct an R-homomorphism ϕ: A → A × X, ϕ(a)=(a, D(a)) (see (2.3)). Composing it with the R-S-homomorphism ιA × 1X : A × X → A × X we have an R-S-homomorphism ψ : A → A × X.Since A × X is an Arens–Michael algebra (Proposition 3.6), there exists a unique continuous homomorphism ψ: A → A × X extending ψ. Thus for any a ∈ A we have (3.8) ψ(ιA(a)) = ψ(a)=(ιA(a),D(a)). Since Im ιA is dense in A it follows that ψ(b)=(b, D(b)) for some uniquely determined map D : A → X. This map is a derivation by virtue of (2.3) and we also have D ◦ ιA = D

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because of (3.8). It remains to show that D is an S-derivation, or, equivalently, that ψ is an S-homomorphism. To this end, consider the diagram ιA / ψ / A × O AO Ay< X yy (3.9) η η y A A yy y η × yy A X g / R S Its left square is commutative by assumption. The diagram obtained from it by removing ηA is also commutative, because ψιA = ψ is an R-S-homomorphism. Thus we are under the assumptions of Lemma 3.7 and therefore the right triangle in (3.9) is commutative, i.e., ψ is an S-homomorphism. Remark 3.5. Clearly, any homomorphism between topological algebras with a dense image is an epimorphism relative to AM. Further results about epimorphisms between topological algebras will be mentioned in Section 6. Corollary 3.9. For any topological R-algebra A and any A-⊗-bimodule X the canonical → → map Der R (A, X) Der R(A, X), D DιA is a bijection. Later on we will need a notion of the Arens–Michael envelope for modules. For def- initeness, we only consider left modules; the cases of right modules and bimodules are dealt with similarly. Definition 3.2. Let A be a topological algebra, A its Arens–Michael envelope, and X a left topological A-module. A pair (X,ιX ) consisting of a left A-⊗-module X and a continuous morphism ιX : X → X of A-modules is called the Arens–Michael envelope of X if for any left A-⊗-module Y and any continuous morphism α: X → Y of A-modules, there is a unique continuous morphism α : X → Y of A-⊗-modules making the following diagram commute: _ α_ _/ XO ?Y (3.10) ιX α X We shall also say that α extends α. To construct the Arens–Michael envelope of an A-module X, we define a new topology on X using the system of all continuous m-compatible prenorms. The LCS thus obtained will be denoted Xm and its completion, X. We obviously have a continuous linear map ιX : X → X. Furthermore, let Am denote the algebra A with topology defined by the system of all continuous submultiplicative prenorms. Recall that its completion is the Arens–Michael envelope A of A. It easily follows from the definition of an m-compatible prenorm that Xm is a left topological Am-module and that the action of Am on Xm is jointly continuous. We conclude that there is a unique structure of left A-⊗-module on X such that the canonical map Xm → X is a morphism of Am-modules. Therefore, ιX : X → X is a morphism of A-modules. Proposition 3.10. The pair (X,ιX ) constructed above is the Arens–Michael envelope of X. Proof. Suppose Y is a left A-⊗-module and α: X → Y is a morphism of A-modules. Then for any m-compatible prenorm · on Y the prenorm x →α(x) on X is also, as is easy

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to check, m-compatible. Since the topology on Y is defined by m-compatible prenorms (Proposition 3.4), we see that α is continuous as a map from Xm to Y. Therefore, there is a unique continuous linear map α : X → Y extending α.Thefactthatα is a morphism of A-modules is obvious since the images of ιA in A and of ιX in X are dense. Remark 3.6. The Arens–Michael envelope of a left topological A-module can also be obtained using the construction mentioned in the proof of Proposition 3.5. As in the case of algebras, it is easy to see that the correspondence X → X is a functor from the category of left topological A-modules to the category of left A-⊗-modules, which is left adjoint to the restriction-of-scalars functor along the homomorphism ιA.In other words, for any left topological A-module X and any left A-⊗-module Y ,wehave a natural (in X and Y ) isomorphism ∼ (3.11) Ah(X, Y ) = Ah(X,Y ). Notice that the functor X → X is additive. From (3.11) we immediately have the following. Proposition 3.11. If A is a ⊗-algebra and X aleftA-⊗-module, then there is a natural isomorphism of left A-⊗-modules ∼ (3.12) X = A ⊗A X. Remark 3.7. It is trivial to check that if A is a topological algebra, then the Arens– Michael envelope (A, ιA) is also the Arens–Michael envelope of A as a left A-module. Therefore, for any n ∈ N , the Arens–Michael envelope of the free A-module An is the n n pair (A ,ιA). The next proposition is a “module-theoretic analog” of Proposition 3.1. Proposition 3.12. Suppose A is a topological algebra, X is a left topological A-module, and Y ⊂ X is a submodule. Let Z denote the closure of ιX (Y ) in X.ThenZ is an A- submodule of X and the morphism X/Y → X/Z induced by the morphism ιX : X → X extends to an isomorphism of A-⊗-modules ∼ ∼ X/Y = (X/Z) . ∼ In particular, if X is metrizable, then X/Y = X/Z. Proof. That Z is an A-submodule of X follows from the fact that Im ιX is dense in X. For any left A-⊗-module W we have natural isomorphisms ∼ Ah(X/Y, W ) = ϕ ∈ Ah(X, W):ϕ =0 Y ∼ ∈ ∈ = ψ Ah(X,W):ψ =0 = ψ Ah(X,W):ψ =0 ιX (Y ) Z ∼ ∼ ∼ = Ah(X/Z,W) = Ah (X/Z) ,W . Moreover, for W =(X/Z )∼, the morphism X/Y → W mentioned in the statement of the proposition corresponds to the identity morphism 1W . The rest is clear. Remark 3.8. The definition and construction of the Arens–Michael envelope of left modules carries over, with obvious modifications, to the case of right modules or bimodules. In particular, in the case of bimodules, the natural isomorphism (3.12) becomes ∼ (3.13) X = A ⊗A X ⊗A A.

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4. Analytic analogs of classical ring-theoretic constructions 4.1. Analytic Ore extensions. We recall some standard facts about Ore extensions (see, for example, [31, 1.7]). Suppose R is an arbitrary associative algebra (without topology) and α: R → R is an endomorphism. A linear map δ : R → R is called an α- derivation if δ(ab)=δ(a)b+α(a)δ(b) for any a, b ∈ R.Inotherwords,δ is an α-derivation if and only if it is a derivation with values in the R-bimodule αR (see Section 2). The Ore extension (or the algebra of skew polynomials) R[t; α, δ]asaleftR-module consists of all n i polynomials i=0 rit in t with coefficients in R. Multiplication on R[t; α, δ] is uniquely determined by the relation tr = α(r)t + δ(r)forr ∈ R and the requirements that the natural embeddings R→ R[t; α, δ]andC[t] → R[t; α, δ] be algebra homomorphisms. Notice that the former embedding makes R[t; α, δ]anR-algebra. The algebra R[t; α, δ] and its element t have the following universal property, which completely characterizes them: for any R-algebra A and any x ∈ A such that x · r = α(r) · x + ηA(δ(r)) (r ∈ R), there exists a unique R-homomorphism f : R[t; α, δ] → A such that f(t)=x.When α = 1R, the algebra R[t; α, δ] is denoted R[t; δ], and when δ = 0 it is denoted R[t; α]. Next, we recall a useful formula for the multiplication on R[t; α, δ]. For each k, n ∈ Z+ ≤ → and each k n,letSn,k : R R denote the operator which is the sum of all compositions − n of k copies of δ and n k copies of α (with a total of k summands). Then for any r ∈ R we have n n n−k (4.1) t r = Sn,k(r)t . k=0 n n n In particular, when δ = 0 the last formula turns into t r = α (r)t ,andwhenα = 1R it becomes n n tnr = δk(r)tn−k. k k=0 Later on we shall construct a locally convex analog of R[t; α, δ], called an analytic Ore extension O(C,R; α, δ). In Section 5 we shall show that, under some additional assumptions, the Arens–Michael envelope of an Ore extension is an analytic Ore extension. Suppose E is a vector space and T is a family of linear operators in E. Definition 4.1. A prenorm ·on E will be called T-stable if for any T ∈ T there is a C>0 such that Tx≤Cx for all x ∈ X. Clearly, a scalar multiple of a T-stable prenorm and the sum of two T-stable prenorms is T-stable. It is also easy to see that if a subfamily S ⊂ T is such that T is contained in the subalgebra generated by S,thenanyS-stable prenorm is also T-stable. Suppose now that E is a locally convex space. Definition 4.2. A family T ⊂ L(E)issaidtobelocalizable if the topology on E can be defined by a family of T-stable prenorms. An operator T ∈ L(E)issaidtobelocalizable3 if {T } is. If E is complete, then it is not difficult to see that T is localizable if and only if E is ∈ ∈ T the inverse limit lim←− Eν of Banach spaces Eν (ν Λ) such that each operator T is { ∈ L ∈ } of the form T =← lim− Tν for some compatible family of operators Tν (Eν):ν Λ . This was the motivation for the term “localizable family”. Notice that the notion of localizable operator is closely related to the notion of tame operator (see [22]). More precisely, it is easy to see that an operator T in a Fréchet

3Operators with this property were independently introduced by W. Zelazko [73] under the name of m-topologizable operators.

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space E is localizable if and only if E admits a grading (in the sense of [22], i.e., an increasing defining sequence of prenorms) such that T is tame of order 0. Proposition 4.1. Suppose E is a locally convex space and T ⊂ L(E) is a finite set of operators. Then the following conditions are equivalent:4 (i) T is localizable; (ii) each operator T ∈ T is localizable; (iii) if { · ν : ν ∈ Λ} is a directed system of prenorms defining the topology on E, then for any T ∈ T and any ν ∈ Λ there are µ ∈ Λ and C>0 such that n n T xν ≤ C xµ for any x ∈ E and n ∈ Z+; (iv) if { · ν : ν ∈ Λ} is a directed system of prenorms defining the topology on E, n n then for any ν ∈ Λ there are µ ∈ Λ and C>0 such that T xν ≤ C xµ for any T ∈ T, x ∈ E,andn ∈ Z+. Proof. (i)=⇒(ii). This is obvious. (ii)=⇒(iii). Fix an arbitrary ν ∈ Λ. As T is localizable, it follows that there are T - stable prenorms ·on E,anelementµ ∈ Λ, and a constant C1 ≥ 1 such that ·ν ≤ ·≤C1·µ.ChooseaC2 > 0 such that Tx≤C2x for all x ∈ E. Then for any ∈ ∈ Z n ≤ n ≤ n ≤ n x E and any n + we have T x ν T x C2 x C1C2 x µ.Itremainsto set C = C1C2. (iii)=⇒(iv). This easily follows from the finiteness of T. (iv)=⇒(i). Let T = {T1,...,Tk}. For each finite set α =(α1,...,αs) of an arbitrary ∈{ } | | length, where αi 1,...,k for each i,set α = s and Tα = Tα1 ...Tαs . We also assume that the identity operator corresponds to the set of zero length (s = 0) (cf. the notation of Example 3.2). Fix an arbitrary ν ∈ ΛandchooseC>0andµ ∈ Λ satisfying (iii). |α| Then Tαxν ≤ C xµ for any set α of the above form and any x ∈ E. Therefore, for · the prenorm ν on E defined by the formula T x (4.2) x =sup α ν , ν |α| α C · ≤· ≤· we have ν ν µ. It now follows that the topology on E is defined by the { · ∈ } system of prenorms ν : ν Λ . On the other hand, it easily follows from (4.2) that ≤ ∈ T Tix ν C x ν for any x E and any i =1,...,k. Therefore, is localizable. Remark 4.1. Recall (see, for example, [72] or [57]) that a linear operator T in an LCS E is n called regular if for some λ>0 the set of operators {(λT ) : n ∈ Z+} is equicontinuous. If the topology of E is defined by a system of prenorms { · ν : ν ∈ Λ},thenT is regular if and only if there is a C>0 such that for any ν ∈ Λthereisaµ ∈ Λ n n such that T xν ≤ C xµ for any x ∈ E and any n ∈ Z+. Comparing this with condition (iii) of the preceding proposition, we see that any regular operator is localizable. The converse is however not true; for example, it is not difficult to check that the operator of multiplication by an independent variable in the space O(C) of entire functions is localizable (because of the local m-convexity of O(C)) but not regular (since there is no n λ>0 such that the family of functions {z → (λz) : n ∈ Z+} is bounded in O(C)). Suppose now that R is a ⊗-algebra, α: R → R is a (continuous) endomorphism, and δ : R → R is a (continuous) derivation. Assume in addition that α and δ are localizable. Our immediate goal is to show that on the space O(C,R)ofR-valued entire functions there is a jointly continuous multiplication which coincides with multiplication on R[z; α, δ] when it is restricted to polynomials. To this end, we first recall that O(C,R) is isomorphic (as an LCS and as a left R-⊗-module) to the projective tensor product

4In the case of one operator this proposition was independently proved by W. Zelazko [73].

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R ⊗ O(C) and that for any system of prenorms { · ν : ν ∈ Λ} deﬁning the topology on R the topology on O(C,R) is deﬁned by all prenorms of the form ∞ n (4.3) fν, ρ = cnν ρ (ν ∈ Λ,ρ>0), n=0 n ∈ O C where f(z)= n cnz is the Taylor expansion of f ( ,R). Lemma 4.2. Let R be a ⊗-algebra, α: R → R a localizable endomorphism, and δ : R → R a localizable derivation. Then there is a unique continuous linear map τ : O(C) ⊗ R → R ⊗ O(C), such that n n n−k (4.4) τ(z ⊗ r)= Sn,k(r) ⊗ z k=0

for all r ∈ R and n ∈ Z+. Proof. The uniqueness is obvious. To prove the existence, we first define the map on the elements zn ⊗ r by (4.4) and extend it by linearity to polynomials. Since polynomials form a dense subspace of O(C,R), it remains to check the continuity of the constructed map. By Proposition 4.1 (implication (ii)=⇒(i)), there is a system of {α, δ}-stable prenorms {·ν : ν ∈ Λ} defining the topology on R. Fix an arbitrary ν ∈ Λ and choose a constant C>0 such that α(r)ν ≤ Crν and δ(r)ν ≤ Crν for any r ∈ R. Then for any k, n ∈ Z+, k ≤ n,wehave n (4.5) S (r) ≤ Cnr . n,k ν k ν For any ρ>0, let ·ρ,ν be the prenorm on O(C) ⊗ R corresponding to the prenorm ∼ ·ν, ρ on R ⊗ O(C) under the permutation isomorphism O(C) ⊗ R = R ⊗ O(C). Then (4.4), (4.3), and (4.5) imply that n n n n−k n−k τ(z ⊗ r)ν, ρ = Sn,k(r) ⊗ z = Sn,k(r)νρ k=0 ν, ρ k=0 n n ≤ Cn ρn−k r = Cn(ρ +1)nr = zn ⊗ r . k µ µ C(ρ+1),µ k=0 ≤ This, together with (4.3), imply that τ(f) ν, ρ f C(ρ+1),µ for any polynomial f = n ⊗ n z rn, and therefore τ is continuous on the subspace of polynomials, which is dense in R ⊗ O(C). Extending τ by continuity to O(C,R), we have the desired assertion.

Remark 4.2. If α = 1R, then the map τ from the preceding lemma can also be deﬁned as follows. As δ is localizable, the formula ∞ ∞ n n (4.6) f · r = cnδ (r) f ∈ O(C),f(z)= cnz ,r∈ R n=0 n=0 deﬁnes a structure of a left O(C)-⊗-module on R. Consider the maps mR : O(C) ⊗ R → R, f ⊗ r → f · r, ∼ ∆: O(C) → O(C) ⊗ O(C) = O(C × C), (∆f)(z,w)=f(z + w).

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It is not diﬃcult to check that the composition ⊗ O(C) ⊗ R −∆−−−1→R O(C) ⊗ O(C) ⊗ R (4.7) m ⊗1O C −c−23→ O(C) ⊗ R ⊗ O(C) −−−−−−−R ( →) R ⊗ O(C),

where c23 is the transposition of the 2-nd and 3-rd tensor factors, coincides with the map τ from the previous lemma. Remark 4.3. Suppose now that the endomorphism α is arbitrary and δ = 0. Replacing δ by α in (4.6), we have another structure of a left O(C)-⊗-module on R. Consider the map ∼ ∆ : O(C) → O(C) ⊗ O(C) = O(C × C), (∆ f)(z,w)=f(zw). Now to get the map τ from the preceding lemma it suﬃces to replace ∆ by ∆ in (4.7). We now return to the general case and set A = R ⊗ O(C). Deﬁne a multiplication operator mA : A ⊗ A → A as the composition A ⊗ A = R ⊗ O(C) ⊗ R ⊗ O(C)

(4.8) 1 ⊗τ⊗1O C m ⊗mO C −−−−−−−−R ( →) R ⊗ R ⊗ O(C) ⊗ O(C) −−−−−−−R ( →) R ⊗ O(C)=A. Proposition 4.3. The operator mA : A ⊗ A → A introduced above makes A a ⊗-algebra. Moreover the tautological embedding i: R[z; α, δ] → A is a homomorphism. Proof. Combining (4.1), (4.4), and (4.8), we see that the diagram R[z; α, δ] ⊗ R[z; α, δ] m /R[z; α, δ]

(4.9) i⊗i i / A ⊗ A m A is commutative. This, together with the fact that the image of i is dense in A, implies that the multiplication on A is associative and that i is a homomorphism. Deﬁnition 4.3. The algebra A = R ⊗ O(C) with the above multiplication will be denoted O(C,R; α, δ) and called the analytic Ore extension of R.Whenα = 1R this algebra will be denoted O(C,R; δ)andwhenδ = 0 it will be denoted O(C,R; α). Notice that O(C,R; α, δ)containsR as a closed subalgebra and is therefore an R-⊗- algebra. Remark 4.4. It follows from Remarks 4.2 and 4.3 that the algebras O(C,R; δ)and O(C,R; α) are particular cases of the analytic smash product of R and the ⊗-bialgebra H acting on it (see [49]), the same way as R[t; δ]andR[t; α] are particular cases of the algebraic smash product (see [65]). In the case of O(C,R; δ)(resp.,R[t; δ]) for H one takes the bialgebra (O(C), ∆) (resp., (C[t], ∆)), and in the case of O(C,R; α)(resp.,R[t; α]) one takes the bialgebra (O(C), ∆) (resp., the bialgebra (C[t], ∆) isomorphic to the semigroup bialgebra CZ+). Remark 4.5. Notice that α and δ being localizable is not only a suﬃcient but also a necessary condition for the multiplication on R[z; α, δ]toextendtoA = R ⊗ O(C). Indeed, if m: A ⊗ A → A is a continuous linear operator making diagram (4.9) commute, then the map τ : O(C) ⊗ R → R ⊗ O(C),τ(a ⊗ r)=m(1 ⊗ a ⊗ r ⊗ 1)

must satisfy (4.4). Suppose {·ν : ν ∈ Λ} is a system of prenorms deﬁning the topology on R. Fix an arbitrary ν ∈ Λ. It follows from the continuity of τ that there are µ ∈ Λ

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and constants C, ρ > 0 such that τ(u)ν, 1 ≤ Cuρ,µ for all u ∈ O(C) ⊗ R. Therefore for any r ∈ R and any n ∈ Z+ we have an estimate n n n n−k δ (r)ν ≤ Sn,k(r)ν = Sn,k(r) ⊗ z k=0 k=0 ν, 1 n n n = τ(z ⊗ r)ν, 1 ≤ Cz ⊗ rρ,µ = Cρ rµ. Taking account of Proposition 4.1 (implication (iii)=⇒(ii)) we see that δ is localizable. A similar argument shows that α is localizable. Proposition 4.4. Suppose R is a ⊗-algebra, α: R → R is a localizable endomorphism, and δ : R → R is a localizable α-derivation. Suppose that B is an R-⊗-algebra which is also Arens–Michael and that x ∈ B is such that x · r = α(r) · x + ηB(δ(r)) for any r ∈ R. Then there exists a unique continuous R-homomorphism f : O(C,R; α, δ) → B such that f(z)=x. n ∈ O C · n Proof. For an arbitrary element a = n rnz ( ,R; α, δ)wesetf(a)= n rn x . Since B is an Arens–Michael algebra, it is easy to see that the latter series converges absolutely in B and that f : O(C,R; α, δ) → B is a continuous linear map. The fact that f is an R-homomorphism (and that f is a unique continuous R-homomorphism with the required properties) easily follows from the universal property of the algebra R[z; α, δ](seeabove)andthedensityofR[z; α, δ]inO(C,R; α, δ). Next, we want to determine under what conditions the algebra O(C,R; δ)isArens– Michael. Deﬁnition 4.4. Suppose R is an Arens–Michael algebra. We shall say that a family T ⊂ L(R)ism-localizable if the topology on R can be deﬁned by a family of T-stable submultiplicative prenorms. We shall say that an operator T ∈ L(R)ism-localizable if the family {T } is m-localizable. Example 4.1. It is easy to see that for any r in the Arens–Michael algebra R the multiplication operators Lr : s → rs and Rr : s → sr are m-localizable. Of course, the same is true for their compositions and linear combinations and, in particular, for inner derivations and inner automorphisms of R. Proposition 4.5. Let R be an Arens–Michael algebra, α: R → R an endomorphism, and δ : R → R an α-derivation. Suppose that the family {α, δ} is m-localizable. Then O(C,R; α, δ) is an Arens–Michael algebra.

Proof. Suppose { · ν : ν ∈ Λ} is a system of submultiplicative {α, δ}-stable prenorms deﬁning the topology on R. Fix an arbitrary ν ∈ Λ and choose a constant C>0such that α(r)ν ≤ Crν and δ(r)ν ≤ Crν for any r ∈ R. Then for any k, n ∈ Z+ and k ≤ n, inequality (4.5) holds. We shall show that for any a, b ∈ A = O(C,R; α, δ)andanyρ>0wehave

(4.10) abν, ρ ≤aν, C (ρ+1)bν, ρ . If a = s ⊗ zn and b = r ⊗ zm,then n n n m n+m−k n+m−k s ⊗ z · r ⊗ z ν, ρ = sSn,k(r) ⊗ z = sSn,k(r)νρ k=0 ν, ρ k=0 n n ≤s Cn ρn−k r ρm = s Cn(ρ +1)nr ρm ν k ν ν ν k=0 n m = s ⊗ z ν, C (ρ+1)r ⊗ z ν, ρ .

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Thus, (4.10) holds for monomials. Ifa and b are polynomials, then they can be written as ﬁnite sums a = i ai and b = j bj,whereai and bj are monomials; moreover, ∈ a ν, ρ = i ai ν, ρ and b ν, ρ = j bj ν, ρ for any ν Λandanyρ>0. This implies that abν, ρ = aibj ≤ aiν, C (ρ+1)bjν, ρ ν, ρ i,j i,j = aiν, C (ρ+1) bjν, ρ = aν, C (ρ+1)bν, ρ . i j Therefore, (4.10) holds for all polynomials and therefore for arbitrary a, b ∈ A.Using Corollary 2.2, we conclude that A is Arens–Michael.

Remark 4.6. Suppose R, α, δ are the same as in the preceding proposition. Then the property of A = O(C,R; α, δ) established in Proposition 4.4 is obviously universal and completely characterizes the pair (A, z ∈ A). In other words, if R-AM denotes the category of R-⊗-algebras which are Arens–Michael, then we have a bijection ∼ Hom R-AM(O(C,R; α, δ),B) = x ∈ B : x · r = α(r) · x + ηB(δ(r)) ∀r ∈ R , giving rise to an isomorphism of functors of B. Remark 4.7. The author does not know if an analog of Proposition 4.1 holds for m- localizable families and if there are localizable families of the form {α, δ} which are not m-localizable. We remark that the prenorms (4.3) defining the topology on O(C,R; α, δ)arenotnec- essarily submultiplicative, even in the case when O(C,R; α, δ) is Arens–Michael (cf. (4.10)). Thus it is convenient to single out a special case when the norms are submultiplicative. Definition 4.5. Suppose R is an Arens–Michael algebra. An operator T ∈ L(R)is said to be m-contracting (resp., m-isometric) if the topology on R canbedefinedbya family { · ν : ν ∈ Λ} of submultiplicative prenorms such that Trν ≤rν (resp., Trν = rν ) for all r ∈ R. Proposition 4.6. Suppose R is an Arens–Michael algebra and α: R → R is an m- contracting endomorphism. Fix a system of submultiplicative prenorms { · ν : ν ∈ Λ} defining the topology on R and such that α(r)ν ≤rν for all r ∈ R and ν ∈ Λ.Then each of the prenorms (4.3) on O(C,R; α) is submultiplicative. Proof. For any s, r ∈ R we have n m n n+m s ⊗ z · r ⊗ z ν, ρ = sα (r) ⊗ z ν, ρ n+m n m ≤sν rν ρ = s ⊗ z ν, ρ r ⊗ z ν, ρ .

This implies, as in the proof of Proposition 4.5, that abν, ρ ≤aν, ρ bν, ρ for any a, b ∈ O(C,R; α).

Remark 4.8. If α = 1R, then it is not diﬃcult to show that the conditions of Propo- sition 4.5 are not only suﬃcient but also necessary for O(C,R; δ) to be Arens–Michael. Indeed, if O(C,R; δ) is Arens–Michael, then so is its closed subalgebra R.Furthermore, the derivation δ coincides with the restriction to R of the inner derivation adz and therefore is m-localizable (see Example 4.1). Similarly, if δ =0,α is an automorphism, and O(C,R; α) is Arens–Michael, then α, being the restriction to R of the inner automorphism Adz,ism-localizable.

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Finally, if δ =0,1ν =1forallν ∈ Λ, and each of the prenorms (4.3) on O(C,R; α) is submultiplicative, then α is necessarily m-contracting. This follows at once from the chain of inequalities

α(r)ν = α(r)zν, 1 = zrν, 1 ≤zν, 1rν, 1 = rν . 4.2. Analytic tensor algebras. Suppose R is an arbitrary associative algebra (without topology) and M is an R-bimodule. For any n ∈ N set ⊗n ⊗ ···⊗ (4.11) M = M R R M . n We also set M ⊗0 = R. Recall that the tensor algebra of M is deﬁned as the R-bimodule ∞ ⊗n TR(M)= M n=0 together with the multiplication uniquely determined by the rule ab = a ⊗ b for a ∈ M ⊗m,b∈ M ⊗n

⊗m ⊗n ∼ ⊗(m+n) (here we used the canonical isomorphisms M ⊗R M = M that hold for any m, n ∈ Z+). ⊗n For any n ∈ Z+ let in : M → TR(M) denote the canonical embedding. Clearly, in is a morphism of R-bimodules and also i0 is an algebra homomorphism making TR(M) an R-algebra. The tensor algebra can also be deﬁned as a pair (TR(M),i1) consisting of an R-algebra TR(M)andanR-bimodule morphism i1 : M → TR(M) such that for any R-algebra A and any R-bimodule morphism α: M → A there exists a unique R-algebra homomorphism ψ : TR(M) → A making the following diagram commute: _ψ_ _/ TR(OM) x;A xx xx i1 xx xx α M Example 4.2. Suppose R is an algebra and α: R → R is an endomorphism. Then the Ore extension R[t; α] is canonically isomorphic to the tensor algebra TR(M), where M = Rα (see Section 2); under this isomorphism 1 ∈ M corresponds to t ∈ R[t; α]. Later we shall describe an “analytic analog” of the tensor algebra, the analytic tensor algebra TR(M), which is a generalization of Taylor’s algebra of power series in several free variables (see Example 3.2) and also of Cuntz’s “smooth tensor algebra” [8, 9]. We shall then show that the Arens–Michael envelope of the tensor algebra is an analytic tensor algebra. Suppose R is a ⊗-algebra and M is an R-⊗-bimodule. Replacing the algebraic tensor product ⊗R in (4.11) by the projective tensor product ⊗R (see Section 1) we have a ⊗ n family of R-⊗-bimodules M (n ∈ Z+). Suppose { · ν : ν ∈ Λ} is a directed system ∈ ∈ N ·⊗n of prenorms deﬁning the topology on M. For any ν Λandanyn ,let ν ⊗ n · denote the prenorm on M which is the projective tensor product of n copies of ν . + ∈ ∞ ⊗ n Consider the space TR(M) consisting of all x =(xn) n=1 M such that ∞ ⊗n n ∞∀ x ν, ρ = xn ν ρ < ρ>0. n=1

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+ The system of prenorms { · ν, ρ : ν ∈ Λ,ρ>0} makes TR(M) an LCS which is, as + is easily seen, complete. It is also easy to show that the topology on TR(M) does not depend on the deﬁning system { · ν : ν ∈ Λ} of prenorms on M. We now set + TR(M)=R ⊕ TR(M) . ⊗ n The elements of TR(M) which belong to M for some n ∈ Z+ will be called homogeneous. We shall use a similar convention for TR(M). Consider the canonical map i: TR(M) → TR(M) sending each elementary tensor ⊗n ⊗ n x1 ⊗···⊗xn ∈ M to x1 ⊗···⊗xn ∈ M . It is easy to see that its image is dense in TR(M). Proposition 4.7. There is a unique multiplication on TR(M) making it a ⊗-algebra and such that i is a homomorphism. On the homogeneous elements this multiplication coincides with the canonical map (4.12) M ⊗m × M ⊗n → M ⊗(m+n), (a, b) → a ⊗ b.

+ Moreover TR(M) is an ideal of TR(M) and a (nonunital) Arens–Michael algebra. If in addition R is an Arens–Michael algebra, then so is TR(M). Proof. Let TR(M)fin denote the subspace of TR(M) consisting of compactly supported sequences. Clearly, there exists a unique bilinear operator (4.13) TR(M)fin × TR(M)fin → TR(M)fin, (x, y) → xy ,

coinciding on homogeneous elements with canonical map (4.12). Letx ¯ =(xn)n≥1 and + y¯ =(yn)n≥1 be elements of TR(M)ﬁn ∩ TR(M) . Then for any ν ∈ Λandanyρ>0we have: ⊗n n x¯y¯ ν, ρ = xiyj ν ρ n i+j=n ≤ ⊗n n ≤ ⊗i ⊗j n (4.14) xiyj ν ρ xi ν yj ν ρ n i+j=n n i+j=n ⊗i i ⊗j j = xi ν ρ yj ν ρ = x¯ ν, ρ y¯ ν, ρ . i j Furthermore, as the action of R on M is jointly continuous, we have that for any ν ∈ Λ there are µ ∈ Λ and a continuous prenorm ·on R such that

a · xν ≤axµ and x · aν ≤xµa (a ∈ R, x ∈ M).

We may assume that ·µ majorizes ·ν . This, together with the properties of the projective tensor product, imply that · ⊗n ≤ ⊗n · ⊗n ≤ ⊗n a x ν a x µ and x a ν x µ a (a ∈ R, x ∈ M ⊗n,n∈ N ), whence

+ (4.15) a · xν, ρ ≤axµ,ρ and x · aν, ρ ≤xµ,ρa (a ∈ R, x ∈ TR(M) ). As the multiplication on R is jointly continuous, (4.14) and (4.15) show that the bilinear operator (4.13) is also jointly continuous. Therefore, it extends to a jointly continuous

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× → bilinear operator mT : TR(M) TR(M) TR(M). Let mT denote the multiplication on TR(M); then it follows easily from the construction that the diagram

mT / TR(M) × TR(M) TR(M)

i×i i m T / TR(M) × TR(M) TR(M) is commutative. Since the image of i is dense in TR(M), we see that mT is associative (hence TR(M)isa⊗-algebra), and that i is a homomorphism. Thus we have a multiplication on TR(M) with the required properties. Its uniqueness is obvious because the + image of i is dense. That TR(M) is a closed ideal of TR(M) is clear from the construc- + tion. Finally, it follows from (4.14) that TR(M) is Arens–Michael. This proves the ﬁrst part of the proposition. Assume now that R is an Arens–Michael algebra, and choose a directed system of submultiplicative prenorms { · λ : λ ∈ Λ1} deﬁning the topology on R.ByProposi- tion 3.4 (see also Remark 3.4), without loss of generality we may assume that for any ν ∈ Λ the prenorm ·ν on M is m-compatible. As in the proof of Proposition 3.6, with each ν ∈ Λ we associate ϕ(ν) ∈ Λ1 such that

a · xν ≤aϕ(ν)xν and x · aν ≤xν aϕ(ν) (a ∈ R, x ∈ M), and set Λ = {(λ, ν) ∈ Λ1 × Λ:ϕ(ν) ≺ λ}. As in (4.15), it follows from the last formula that + (4.16) a · xν, ρ ≤aλxν, ρ and x · aν, ρ ≤xν, ρ aλ (a ∈ R, x ∈ TR(M) ) for any (λ, ν) ∈ Λ and any ρ>0. Now we write an arbitrary element x =(xn) ∈ TR(M) + in the form x =(x0, x¯), wherex ¯ =(xn)n≥1 ∈ TR(M) ,andforany(λ, ν) ∈ Λ and any ρ>0 deﬁne a prenorm ·λ, ν,ρ on TR(M) by the formula

(4.17) xλ, ν,ρ = x0λ + x¯ν, ρ . As in the proof of Proposition 3.6, we have that the topology on TR(M) is deﬁned by the system of prenorms of the form (4.17). On the other hand, it follows from (4.14) and (4.16) that

xyλ, ν,ρ = x0y0λ + x0y¯ +¯xy0 +¯xy¯ν, ρ

≤x0λy0λ + x0λy¯ν, ρ + x¯ν, ρ y0λ + x¯ν, ρ y¯ν, ρ

=(x0λ + x¯ν, ρ )(y0λ + y¯ν, ρ )=xλ, ν,ρyλ, ν,ρ for any x, y ∈ TR(M), any (λ, ν) ∈ Λ ,andanyρ>0. Therefore, all prenorms {·λ, ν,ρ : (λ, ν) ∈ Λ ,ρ>0} are submultiplicative and TR(M) is Arens–Michael. Deﬁnition 4.6. The ⊗-algebra TR(M) thus constructed will be called the analytic tensor algebra of M. Remark 4.9. When A = C the algebra TC(M) coincides with Cuntz’s “smooth tensor algebra” [8]. If in addition n =dimM<∞, we recover Taylor’s algebra Fn of power series in n free variables (see Example 3.2). Now suppose, as before, that R is a ⊗-algebra and M is an R-⊗-bimodule. As in the ⊗ n case of the usual tensor algebra TR(M), we have the canonical morphisms in : M → TR(M)(n ∈ Z+)ofR-⊗-bimodules; moreover i0 : R → TR(M) is an algebra homomorphism making TR(M)anR-⊗-algebra.

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Proposition 4.8. Suppose R is a ⊗-algebra and M is an R-⊗-bimodule. Then for any R-⊗-algebra A which is Arens–Michael, and any morphism of R-⊗-bimodules α: M → A there exists a unique continuous R-homomorphism ψ : TR(M) → A making the following diagram commute:

_ψ_ _/ TR(OM) y

Proof. We fix a directed system of prenorms { · ν : ν ∈ Λ} defining the topology on M and also choose a system of submultiplicative prenorms { · µ : µ ∈ Λ2} defining the topology on A. For any n ∈ N consider the morphism n ⊗ n → ⊗···⊗ → ··· mA : A A, a1 an a1 an.

As the prenorms ·µ are submultiplicative, we have n ≤ ⊗n ∈ ⊗ n ∈ N ∈ (4.19) mA(u) µ u µ for all u A ,n ,µ Λ2. Deﬁne a map ψ : TR(M) → A by the formula ∞ n ◦ ⊗n ψ(x)=ηA(x0)+ (mA α )(xn). n=1

To show that ψ is well-deﬁned and continuous, we ﬁx an arbitrary µ ∈ Λ2 and, using the continuity of α,chooseν ∈ ΛandC>0 such that α(x)µ ≤ Cxν for all x ∈ M. Then, obviously, ⊗n ⊗n ≤ n ⊗n ∈ ⊗ n ∈ N (4.20) α (xn) µ C xn ν for all xn M ,n . Using (4.19) and (4.20), for each x ∈ TR(M)wehave ∞ ≤ n ◦ ⊗n ψ(x) µ ηA(x0) µ + (mA α )(xn) µ n=1 ∞ ≤ n ⊗n ηA(x0) µ + C xn ν = ηA(x0) µ + x¯ ν, C . n=1

Since ηA is continuous, the latter estimate implies that ψ is well-deﬁned and continuous. The commutativity of diagram (4.18) follows directly from the construction. A simple calculation on elementary tensors (which we omit) shows that ψ is a homomorphism of R-algebras. Finally, the uniqueness of ψ with the required properties follows easily from the fact that the images of i0 and i1 generate a dense subalgebra of TR(M).

Suppose R is a ⊗-algebra and α is a (continuous) endomorphism of it. We shall show that the analytic tensor algebra TR(Rα) can be realized as some algebra of “skew power series”. For this we fix a directed system of prenorms { · ν : ν ∈ Λ} defining the (n) topology on R,andforeachν ∈ Λandeachn ∈ N define a prenorm ·ν on R by the formula k (n) ··· (4.21) r ν =inf r0,i ν rn−1,i ν , i=1

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where inf is taken over all representations of r in the form k 2 n−1 (4.22) r = r0,i α(r1,i)α (r2,i) ···α (rn−1,i)(rj,i ∈ R, k ∈ N ). i=1 (0) We also set ·ν = ·ν . It is not diﬃcult to see that for a ﬁxed n the system of { · (n) ∈ } prenorms ν : ν Λ on R is equivalent to the original one. Consider the space { } ∞ n R t; α consisting of all formal expressions a = n=0 rnt such that ∞ (n) n ∞∀ a ν, ρ = rn ν ρ < ρ>0. n=0

The system of prenorms { · ν, ρ : ν ∈ Λ,ρ>0} makes R{t; α} an LCS, which, as is easy to see, is complete. Proposition 4.9. There is a unique structure of a ⊗-algebra on R{t; α} such that the embedding R[t; α] → R{t; α} is a homomorphism. Moreover, there exists a unique R-⊗- algebra isomorphism R{t; α}→TR(Rα) sending t to 1 ∈ Rα. Consequently, if R is an Arens–Michael algebra, then so is R{t; α}. Proof. It is easy to see that for any n ∈ N we have an isomorphism of R-⊗-bimodules

⊗ n n−1 ϕn :(Rα) → Rαn ,r0 ⊗···⊗rn−1 → r0 α(r1) ···α (rn−1). Moreover, (n) ⊗n ϕn(u) ν = u ν ⊗ n for any ν ∈ Λandanyu ∈ (Rα) . Therefore, there is an isomorphism of LCS, ∞ { }→ n → ∞ R t; α TR(Rα), rnt x =(xn)n=0, n=0 ⊗ ⊗···⊗ ∈ ⊗ n where xn = rn 1 1 (Rα) . n−1 It remains to carry the structure of a ⊗-algebra from TR(Rα)toR{t; α}. Under this construction, if R[t; α]isidentiﬁedwithTR(Rα) (see Example 4.2), then the homomorphism i: TR(Rα) → TR(Rα) corresponds to R[t; α] → R{t; α} (see Proposition 4.7). Therefore, the above embedding is an algebra homomorphism. The assertion about uniqueness obviously follows from the density of R[t; α]inR{t; α}. Finally, if R is an Arens–Michael algebra, then so is TR(Rα) (see Proposition 4.7) and therefore R{t; α}.

Combining the preceding proposition with Proposition 4.8, we have the following property of R{t; α}. Corollary 4.10. Suppose R is a ⊗-algebra and α: R → R is an endomorphism. Then for any R-⊗-algebra A which is Arens–Michael and any x ∈ A such that x · r = α(r) · x for all r ∈ R there is a unique continuous R-homomorphism ψ : R{t; α}→A such that ψ(t)=x. Remark 4.10. If R is an Arens–Michael algebra and M is an R-⊗-bimodule, then the property of TR(M) established in Proposition 4.8 is obviously universal and completely characterizes the pair (TR(M),i1). In other words, there is a bijection ∼ Hom R-AM(TR(M),A) = RhR(M,A),

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giving rise to an isomorphism of functors of A. In particular, if α is an endomorphism of R,then ∼ Hom R-AM(R{t; α},A) = x ∈ A : x · r = α(r) · x ∀r ∈ R (cf. Remark 4.6). Proposition 4.11. Assume, in addition to the conditions of Corollary 4.10 that R is an Arens–Michael algebra and α is m-localizable. Then there exists a unique R-isomorphism R{t; α}→O(C,R; α),t→ (z → z · 1). Proof. This is a consequence of the universal properties of the above algebras (see Re- marks 4.6 and 4.10).

Example 4.3. We shall now give an example of an endomorphism α of an Arens– Michael algebra R such that the underlying LCS of R{t; α} coincides with the space of formal power series R[[t]]. Of course, in view of the preceding proposition, such an endomorphism cannot be m-localizable. Set R = O(C) and define α: R → R by setting α(f)(z)=f(z − 1) (z ∈ C). For the defining system of prenorms on R take { · ρ : ρ>0},wherefρ =max|z|≤ρ |f(z)|. (n) We shall show that for any ρ>0thereisanN ∈ N such that ·ρ ≡ 0 for all n>N. Henceforth, for z ∈ C and r>0, the open disc of radius r and center z will be denoted Ur(z). Fix an arbitrary ρ>0andchooseanr>ρsuch that 2r<[2ρ]+1. Then for any integer n ≥ [2ρ]+1wehaveUr(−n) ∩ Ur(0) = ∅.SetU = Ur(−n) ∪ Ur(0), take an arbitrary f ∈ O(C)andε>0, and define a function h ∈ O(U) by setting h(z)=0forz ∈ Ur(0) and h(z)=f(z + n)forz ∈ Ur(−n). By Runge’s theorem (see, for example, [42, Ch. 4, Section 2]), the restriction homomorphism O(C) → O(U) has a dense image. Hence there is a g ∈ O(C) such that |g(z) − h(z)| <ε/2 for any z ∈ Uρ(−n) ∪ Uρ(0). It follows n n (n) from the equality f = α (g)+(f − α (g)) and the definition of the prenorms ·ρ (see (4.21) and (4.22)) that (n+1) ≤ − n (4.23) f ρ g ρ + f α (g) ρ.

Bythechoiceofh and g,wehavegρ <ε/2. Furthermore, n f − α (g)ρ =max |f(z) − g(z − n)| : z ∈ Uρ(0) =max |f(z + n) − g(z)| : z ∈ Uρ(−n) ε =max |h(z) − g(z)| : z ∈ U (−n) < . ρ 2 (n+1) Comparing this with (4.23), we have an estimate fρ <ε.Asε>0andf ∈ O(C) (n+1) were arbitrary, this means that ·ρ ≡ 0 for all n ≥ [2ρ] + 1. Therefore it follows from the deﬁnition of R{t; α} that, as a locally convex space, it coincides with the space of formal power series R[[t]]. 4.3. Analytic skew Laurent extensions. Suppose R is an arbitrary associative algebra (without topology) and α: R → R is an automorphism. Recall that the skew Laurent −1 n i extension R[t, t ; α], as a left R-module, consists of all Laurent polynomials i=−m rit in t with coeﬃcients in R; the multiplication on R[t, t−1; α] is uniquely determined by the relation tr = α(r)t for r ∈ R and the requirement that the natural embeddings R→ R[t, t−1; α]andC[t, t−1] → R[t, t−1; α] be algebra homomorphisms. Notice that the former embedding makes R[t, t−1; α]anR-algebra. The algebra R[t, t−1; α]andits

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element t have the following universal property, which completely characterizes them: For any R-algebra A and any invertible element x ∈ A such that x · r = α(r) · x (r ∈ R), there exists a unique R-homomorphism f : R[t, t−1; α] → A such that f(t)=x. Clearly, R[t, t−1; α]containsR[t; α] as a subalgebra. Now we shall construct a locally convex analog of the algebra R[t, t−1; α]. Suppose R is × a ⊗-algebra. Set C = C\{0} and recall that for any system of prenorms {·ν : ν ∈ Λ} defining the topology on R, the topology on O(C×,R) is defined by all prenorms of the form ∞ n (4.24) fν, ρ = cnν ρ (ν ∈ Λ,ρ>0), n=−∞ n ∈ O C× where f(z)= n cnz is the Laurent expansion of f ( ,R). Lemma 4.12. Suppose R is a ⊗-algebra and α: R → R is an automorphism. If α and α−1 are localizable, then there is a unique continuous linear map τ : O(C×) ⊗ R → R ⊗ O(C×), such that (4.25) τ(zn ⊗ r)=αn(r) ⊗ zn for all r ∈ R and n ∈ Z. The proof of this lemma is similar and in fact simpler than that of Lemma 4.2 and is therefore omitted. Notice that the description of τ given in Remark 4.3 carries over in an obvious way to the case in question. × Now let A = R ⊗ O(C ) and define a multiplication mA : A ⊗ A → A by the formula obtained from (4.8) by replacing C with C×. It is easy to see that we have the following analog of Proposition 4.3. Proposition 4.13. The operator mA : A ⊗ A → A thus defined makes A a ⊗-algebra. Moreover the tautological embedding i: R[z,z−1; α] → A is a homomorphism. Definition 4.7. The algebra A = R ⊗ O(C×) with the above multiplication will be denoted O(C×,R; α) and called the analytic Laurent extension of R. Notice that the algebra O(C×,R; α)containsR as a closed subalgebra and is therefore an R-⊗-algebra. As in the case of analytic Ore extensions (see Remark 4.4), this algebra can be described as the analytic smash product of R and the ⊗-bialgebra (O(C×), ∆) acting on it (the latter, as is not difficult to see, is the Arens–Michael envelope of the group bialgebra CZ). Finally, similarly to the argument in Remark 4.5, one shows that α and α−1 being localizable is not only a sufficient but also a necessary condition for the multiplication on R[z,z−1; α]toextendtoA = R ⊗ O(C). The next three propositions are proved similarly to Propositions 4.4, 4.5, and 4.6. We omit their proofs. Proposition 4.14. Let R be a ⊗-algebra and α: R → R an automorphism such that α and α−1 are localizable. Suppose that B is an R-⊗-algebra which is Arens–Michael and x ∈ B is an invertible element such that x · r = α(r) · x for any r ∈ R. Then there exists a unique continuous R-homomorphism f : O(C×,R; α) → B, such that f(z)=x.

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Proposition 4.15. Let R be an Arens–Michael algebra and α: R → R an automorphism. Suppose that the family {α, α−1} is m-localizable. Then O(C×,R; α) is an Arens–Michael algebra. Proposition 4.16. Let R be an Arens–Michael algebra and α: R → R an m-isometric automorphism. Fix a system of submultiplicative prenorms { · ν : ν ∈ Λ} deﬁning the topology on R and such that α(r)ν = rν for all r ∈ R and ν ∈ Λ. Then each of the prenorms (4.24) on O(C×,R; α) is submultiplicative.

±1 −1 Notice that the equality α =Adz±1 immediately implies that {α, α } being m- localizable is not only suﬃcient but also necessary for O(C×,R; α) to be Arens–Michael. × If 1ν =1forallν and each of the prenorms (4.24) on O(C ,R; α) is submultiplicative, then α must be m-isometric (cf. Remark 4.8). In the case of the analytic Laurent extension, the universal property of the analytic Ore extension mentioned in Remark 4.6 becomes × ∼ (4.26) Hom R-AM(O(C ,R; α),B) = x ∈ B : x is invertible, x · r = α(r) · x ∀r ∈ R .

5. Examples of Arens–Michael envelopes

5.1. Ore extensions: the case α = 1R. Universal enveloping algebra. Suppose R is an algebra (without topology) and δ : R → R is a derivation. Let Rδ be the Arens– Michael algebra obtained by completing R with respect to the system of all δ-stable submultiplicative prenorms. The canonical homomorphism R → Rδ will be denoted j. ˆ Clearly, δ uniquely defines an m-localizable derivation δ of the algebra Rδ such that δˆ ◦ j = j ◦ δ. Moreover we have homomorphisms ˆ ˆ R[z; δ] → Rδ[z; δ] → O(C,Rδ; δ) , where the former coincides with j on R and sends z to z, and the latter is the canonical embedding (see above). The composition of these homomorphisms will be denoted ιR[z;δ]. ˆ Theorem 5.1. The pair (O(C,Rδ; δ),ι) is the Arens–Michael envelope of R[z; δ]. Proof. Let ϕ: R[z; δ] → B be a homomorphism from R[z; δ] to an Arens–Michael algebra B.Setf = ϕ|R and x = ϕ(z). Fix a submultiplicative prenorm · on B and define a prenorm ·f on R by the formula rf = f(r). Clearly, this prenorm is submultiplicative and it is not difficult to check that it is δ-stable. Indeed, δ(r)f = f(δ(r)) = ϕ([z,r]) = [x, f(r)] ≤ 2x·f(r) =2x·rf for any r ∈ R. It now follows that f is continuous in the topology on R defined by the system of all δ-stable submultiplicative prenorms. Therefore, there exists a unique continuous homomorphism ηB : Rδ → B such that ηBj = f.ThusB becomes an Rδ- ⊗-algebra and ϕ an R-Rδ-homomorphism. This implies that the relation x · r = r · x + ˆ ηB(δ(r)) holds for any r ∈ Im j and hence for any r ∈ Rδ. Therefore, by Proposition 4.4, ˆ there exists a unique continuous Rδ-homomorphism ϕ: O(C,Rδ; δ) → B such that ϕ(z)= x. Therefore, the composition ϕ ◦ ιR[z;δ] : R[z; δ] → B is an R-homomorphism and sends z to x. The uniqueness of an R-homomorphism with this property implies that ϕ ◦ ιR[z;δ] = ϕ, i.e., ϕ extends ϕ in the sense of Definition 3.1. The uniqueness of such ˆ an extension is clear because the image of ιR[z;δ] is dense in O(C,Rδ; δ). The rest is clear.

Remark 5.1. A similar result can be proved for more general smash products (see [49], Theorem 2.2).

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The above construction of the Arens–Michael envelope of R[t; δ] can be used, in particular, for an inductive description of the Arens–Michael envelopes of universal enveloping algebras of solvable Lie algebras. If g is a solvable finite-dimensional Lie algebra, then it has an ideal h of codimension 1. Choose an arbitrary x ∈ g \ h and set R = U(h). It is ∼ easy to see that U(g) = R[x; δ], where δ =adx. Therefore to describe the Arens–Michael envelope of U(g) it suffices to know the system of all submultiplicative δ-stable prenorms on U(h). We shall illustrate this approach by two very simple examples. Example 5.1. Suppose g is a two-dimensional Lie algebra with basis {x, y} and commu- C C d tator [x, y]=y.Seth = y and R = U(h)= [y]. Consider the derivation δ =adx = y dy of R;bytheabove,U(g)=R[x; δ] and our goal is to describe all submultiplicative δ- stable prenorms on R. ∈ Z i ∈ n | | For any n + and any r = i ciy R we set r n = i=0 ci . A standard · calculation (which we omit) shows that n is a submultiplicative prenorm on R.Since i ≤ ∈ · δ(r)= i iciy ,wehave δ(r) n n r n for any r R,sothat n is δ-stable. Moreover, any δ-stable prenorm on R is majorized by some ·n. Indeed, suppose · is a δ-stable prenorm and let a constant C>0 be such that δ(r)≤Cr for any r ∈ R. Then for r = yn we have δ(yn)=nyn, whence nyn≤Cyn. Therefore, yn =0for sufficiently large n, which easily implies that ·≤M·n for some n ∈ Z+ and some constant M>0. Using the notation introduced above, we have that Rδ coincides with the completion of R with respect to the system of prenorms { · n : n ∈ Z+}, i.e., with the algebra of formal power series C[[y]]. Applying Theorem 5.1, we have the following { } Proposition 5.2. Suppose g is a Lie algebra with basis x, y and commutator[x, y]=y. The Arens–Michael envelope U(g) of U(g) consists of all power series a = c yixj i,j ij | | j ∞ ∈ Z such that j cij ρ < for any i + and any ρ>0. The topology on U(g) is defined by the system of prenorms ∞ n j an,ρ = |cij|ρ (n ∈ Z+,ρ>0). j=0 i=0 In other words, the underlying LCS of U(g) coincides with O(C)[[y]] = C[[y]] ⊗ O(C), and the multiplication is uniquely determined by the relation [x, y]=y (where x ∈ O(C) denotes the identity map on C) and by the condition that the embeddings C[[y]] → U(g) and O(C) → U(g) be homomorphisms. Example 5.2. Suppose g is a three-dimensional Lie algebra with basis {x, y, z} and commutator [x, y]=z,[x, z]=[y, z] = 0 (i.e., the Heisenberg algebra). Set h =span{y, z} C ∂ and R = U(h)= [y, z]. Consider the derivation δ =adx = z ∂y of R.Asinthe preceding example, we need to describe all submultiplicative δ-stable prenorms on R. i j ∈ For any ρ>0andanya = i,j aijy z R we set i! (5.1) a = |a | ρi+j. ρ ij (i + j)! i,j

Clearly, ·ρ is a norm on R. Lemma 5.3. The norms (5.1) are submultiplicative and δ-stable. Moreover, any submultiplicative δ-stable prenorm on R is majorized by some norm (5.1).

i! i+j Proof. Fix an arbitrary ρ>0 and for each pair of indices (i, j)setpij = (i+j)! ρ . Obviously, to prove that ·ρ is submultiplicative it suﬃces to show that

pi+k,j+l ≤ pijpkl for any i, j, k, l ∈ Z+.

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After canceling ρi+j+k+l, the last inequality becomes

(i + k)! ≤ i!k! (i + k + j + l)! (i + j)!(k + l)!

or, equivalently, i + k i + k + j + l ≤ . i i + j This formula can easily be deduced by induction on j and l from the obvious inequalities n n +1 n n +1 ≤ and ≤ . m m +1 m m Thus the norms (5.1) are submultiplicative. i j ∈ Furthermore, for any a = i,j aijy z R we have − i−1 j+1 (i 1)! i+j i! i+j δ(a)ρ = aijiy z = |aij|i ρ ≤ |aij| ρ = aρ. (i + j)! (i + j)! i,j ρ i,j i,j i≥1 i≥1 Therefore, the norms (5.1) are δ-stable. Now suppose that ·is a submultiplicative δ-stable prenorm on R. Fix a constant C ≥ 1 such that δ(a)≤Ca for any a ∈ R and notice that for any i, j ∈ Z+, (i + j)! δj(yi+j)= yizj. i! Therefore, substituting a = yi+j in δj(a)≤Cja,wehave (i + j)! yizj ≤Cjyi+j≤Cjri+j, i! where r = y. Hence i! i! yizj ≤Cj ri+j ≤ (Cr)i+j = yizj . (i + j)! (i + j)! Cr i j Therefore, setting ρ = Cr, for any a = i,j aijy z we have an estimate i j i j i j a = aijy z ≤ |aij|y z ≤ |aij|y z ρ = aρ. i,j i,j i,j

Thus a≤aρ for any a ∈ R.

Combining the last lemma with Theorem 5.1, we have Proposition 5.4. Suppose g is a Lie algebra with basis {x, y, z} and commutator [x, y]= z, [x, z]=[y, z]=0(i.e., the Heisenberg algebra). The Arens–Michael envelope U(g) of U(g) consists of all power series i j k a = aijky z x , i,j,k such that for any ρ>0, i! a = |a | ρi+j+k < ∞. ρ ijk (i + j)! i,j,k The topology on U(g) is deﬁned by the prenorms { · ρ : ρ>0}.

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5.2. Tensor algebras. Suppose R is an algebra and M is an R-bimodule. Using Propo- sition 4.8, it is not diﬃcult to describe the Arens–Michael envelope of the tensor algebra −ι→R −i→0 TR(M). For this, we ﬁrst notice that the homomorphism R R TR (M)andtheR- −ι−M→ −i→1 bimodule morphism M M TR (M) uniquely determine (by the universal property → of TR(M)) an R-homomorphism ιT : TR(M) TR (M). Proposition 5.5. Let R be an algebra and M an R-bimodule. Then the pair (TR (M),ιT ) is the Arens–Michael envelope of TR(M).

Proof. Suppose ϕ: TR(M) → A is a homomorphism with values in an Arens–Michael algebra A. The homomorphism θ = ϕi0 : R → A extends to a homomorphism ηA = θ: R → A,whichmakesA an R-⊗-algebra and, in particular, an R-⊗-bimodule. There- fore the R-bimodule morphism α = ϕi1 : M → A extends to an R-bimodule morphism → → α: M A, which, by Proposition 4.8, yields an R-homomorphism ϕ: TR (M) A coinciding with α on M. It is now easy to see that ϕι T = ϕ, because both R-homomorphisms coincide with α on M.Thusϕ is a homomorphism extending ϕ in the sense of Defini- tion 3.1. Its uniqueness is obvious because the image of ιT is dense in TR (M). 5.3. Ore extensions: the case δ =0. Quantum affine spaces. As a consequence of Proposition 5.5, we can easily describe the Arens–Michael envelope of the algebra R[t; α], where R is an algebra and α is an endomorphism of it. Notice that the definition of the Arens–Michael envelope implies that there exists a unique continuous endomorphism α of R such that αι R = ιRα. Moreover, we have homomorphisms R[t; α] → R[t; α] → R{t; α},

where the former coincides with ιR on R and sends t to t, and the latter is the canonical embedding (see Proposition 4.9). The composition of these homomorphisms will be denoted ιR[t;α]. Corollary 5.6. Suppose R is an algebra and α is an endomorphism of it. Then the pair (R{t; α},ιR[t;α]) is the Arens–Michael envelope of R[t; α]. Proof. We shall show that the pair (Rα ,ιR) is the Arens–Michael envelope of the bimodule R in the sense of Definition 3.2 (see also Remark 3.8). Indeed, that ιR : Rα → Rα is an R-bimodule morphism easily follows from the relation ιRα = αι R. Suppose X is an R-⊗-bimodule and ϕ: Rα → X a morphism of R-bimodules. Set x = ϕ(1); then, clearly, x · r = α(r) · x for any r ∈ R, and therefore x · r = α(r) · x for any r ∈ R.This,inturn, implies that ϕ: Rα → X, ϕ(r)=r · x is a morphism of R-bimodules extending ϕ.The uniqueness of such extension is clear because the image of Im ιR is dense in R.Thus, Rα = Rα as R-⊗-bimodules. It remains to combine this fact with Propositions 5.5 and 4.9. Example 5.3. Set R = C[x] and define an endomorphism α: R → R by the formula α(x)=x−1. Then it is easy to see that the algebra R[y; α] is isomorphic to the universal enveloping algebra of U(g), where g is the two-dimensional Lie algebra with basis {x, y} and commutator [x, y]=y (see Example 5.1). By Corollary 5.6, we have an isomor- ∼ phism of topological algebras U(g) = R{y; α}. On the other hand, R = O(C), and it is not difficult to see that the endomorphism α: R → R acts by the formula α(f)(z)= f(z−1). Therefore, by Example 4.3, the underlying LCS of U(g)=R{y; α} is isomorphic to O(C)[[y]]; under this isomorphism the coordinate function z → z from O(C) corresponds to x ∈ g. Thus we have a second proof of Proposition 5.2, which is independent of the results of 5.1.

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Example 5.4. Fix a number q ∈ C, q = 0. Recall (see [40]) that the quantum plane is an algebra with two generators x, y, subject to the relation xy = qyx. This algebra will Oalg C2 5 be denoted q ( ). n × More generally, let q =(qij)i,j=1 be a complex (n n)-matrix such that qii =1and −1 qij = qji for all i, j (such matrices are usually called multiplicatively antisymmetric). A Oalg Cn multiparameter quantum aﬃne space (see, for example, [1, 20]) is an algebra q ( ) with generators x1,...,xn and relations xixj = qij xjxi (i, j =1,...,n). In the one- Oalg Cn parameter case, i.e., when qij = q for all i

C[x1][x2; α2] ···[xn; αn],

where for each i =2,...,n, the endomorphism αi is uniquely determined by the condition Oalg Cn αi(xj)=qijxj (jj Oalg Cn Then in q ( )wehave (5.3) xαxβ = c(α, β)xα+β (see, for example, [1, 21]). Zn → R ·w Let w : + + be an arbitrary function. For each ρ>0 deﬁne a prenorm ρ Oalg Cn on q ( ) by the formula w | | |α| α (5.4) a ρ = cα w(α)ρ for a = cαx . α α ·w Lemma 5.7. The prenorm ρ is submultiplicative if and only if | | ≤ ∈ Zn (5.5) c(α, β) w(α + β) w(α)w(β)(α, β +). Proof. This follows easily from relations (5.3).

ConsiderthefreealgebraFn with generators ξ1,...,ξn. Obviously, there exists a → Oalg Cn → − unique homomorphism Fn q ( ) such that ξi xi for each 1,...,n. The image of an arbitrary element u ∈ Fn under the action of this homomorphism will be denoted u¯. Fix an N ∈ Z+ and let XN ⊂ Fn denote the set of all words in ξ1,...,ξn of length N. The symmetric group SN acts on XN in an obvious way. Moreover, it follows from the Oalg Cn ∈ ∈ ∈ C× deﬁnition of q ( )thatforanyu XN and σ SN there is a unique λ(σ, u) such thatu ¯ = λ(σ, u)σ(u). × Lemma 5.8. The function λ: SN × XN → C satisﬁes the relation

λ(στ, u)=λ(σ, τ (u))λ(τ,u)(σ, τ ∈ SN ,u∈ XN ).

5 2 In the literature this algebra is usually denoted Oq(C )orsimplyAq; we have added the superscript 2 “alg” in order to use the notation Oq(C ) for an analytic analog of this algebra.

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Proof. It follows from the deﬁnition of λ that u¯ = λ(τ,u)τ(u)andτ(u)=λ(σ, τ (u))στ(u), whence (5.6)u ¯ = λ(τ,u)λ(σ, τ (u))στ(u). On the other hand, (5.7)u ¯ = λ(στ, u)στ(u). It remains to combine (5.6) and (5.7).

Fix an α ∈ Zn and consider the element ξα = ξα1 ···ξαn ∈ X ,whereN = |α|.Set + 1 n N α wq(α)=min |λ(σ, ξ )| : σ ∈ SN . ≤ ∈ Z+ Since λ(e, u)=1,wheree is the identity permutation, we have wq(α) 1 for any α n . Zn → R Lemma 5.9. The function wq : + + satisﬁes (5.5).

Proof. Set N1 = |α|, N2 = |β|,andN = N1 + N2. Clearly, there is a τ ∈ SN such that τ(ξαξβ)=ξα+β.Moreover,xαxβ = λ(τ,ξαξβ)xα+β, whence λ(τ,ξαξβ)=c(α, β). Therefore, in view of Lemma 5.8, α β α β min |λ(σ, ξ ξ )| : σ ∈ SN =min |λ(στ, ξ ξ )| : σ ∈ SN (5.8) α+β α β =min |λ(σ, ξ )|·|λ(τ,ξ ξ )| : σ ∈ SN = wq(α + β)|c(α, β)|. ∈ ∈ Suppose now that σ1 SN1 and σ2 SN2 are such that α β |λ(σ1,ξ )| = wq(α)and|λ(σ2,ξ )| = wq(β). Let σ × σ ∈ S denote the permutation 1 2 N σ1(i),i≤ N1, (σ1 × σ2)(i)= N1 + σ2(i − N1),i>N1.

In other words, σ1 ×σ2 acts on the ﬁrst N1 elements as σ1 and on the remaining elements α β α β as σ2. Clearly, (σ1 × σ2)(ξ ξ )=σ1(ξ )σ2(ξ ), and therefore α β α α β β α β α β x x = λ(σ1,ξ )σ1(ξ )λ(σ2,ξ )σ2(ξ )=λ(σ1,ξ )λ(σ2,ξ )(σ1 × σ2)(ξ ξ ). On the other hand, α β α β α β x x = λ(σ1 × σ2,ξ ξ )(σ1 × σ2)(ξ ξ ), whence λ(σ × σ ,ξαξβ)=λ(σ ,ξα)λ(σ ,ξβ). Taking account of (5.8) we see that 1 2 1 2 w (α)w (β)=λ(σ × σ ,ξαξβ) q q 1 2 α β ≥ min |λ(σ, ξ ξ )| : σ ∈ SN = wq(α + β)|c(α, β)|,

·wq ·q Henceforth the prenorm ρ will be denoted ρ . ·q Lemma 5.10. The prenorm ρ is submultiplicative and is the largest of all the sub- · Oalg Cn ≤ multiplicative prenorms on q ( ) such that xi ρ for all i =1,...,n. ·q Proof. First, ρ is submultiplicative by Lemmas 5.9 and 5.7. It is also clear that q · Oalg Cn xi ρ = ρ for all i =1,...,n. Suppose is some other prenorm on q ( ) satisfying N the assumptions of the lemma. Clearly, u¯≤ρ for any u ∈ XN and any N ∈ Z+. ∈ Zn | | ∈ | α | Take an arbitrary α +,setN = α ,andchooseσ SN such that wq(α)= λ(σ, ξ ) . Since xα = λ(σ, ξα)σ(ξα), we have α α |α| x = wq(α)σ(ξ )≤wq(α)ρ .

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·q ≤q This, together with the explicit description of ρ , implies that a a ρ for any ∈ Oalg Cn a q ( ).

Suppose, as before, that q =(qij) is a multiplicatively antisymmetric complex (n × n)-matrix. Set O Cn α q | | |α| ∞∀ q( )= a = cαx : a ρ = cα wq(α)ρ < ρ>0 . ∈Zn ∈Zn α + α + n If qij = q for all i0 . Notice that assigning to each a = α cαx q( ) the func- Cn → C α O Cn tion f : ,f(z)= α cαwq(α)z we have an LCS isomorphism between q( ) and the space of entire functions O(Cn).

n Theorem 5.11. There is a unique multiplication on Oq(C ) making it a topological algebra and such that xixj = qijxjxi for all i, j =1,...,n. Algebra thus obtained, Oalg Cn together with the tautological embedding of q ( ) in it, is the Arens–Michael envelope Oalg Cn { · q } of q ( ). Moreover, each prenorm of the system ρ : ρ>0 deﬁning the topology n on Oq(C ) is submultiplicative. ·q Oalg Cn O Cn Proof. By Lemma 5.10, the prenorm ρ is submultiplicative on q ( ). Since q( ) Oalg Cn { · q } is, obviously, a completion of q ( ) with respect to the system ρ : ρ>0 ,we Oalg Cn see that the multiplication on q ( ) extends uniquely to a continuous multiplication n on Oq(C ) and makes the latter an Arens–Michael algebra. Oalg Cn → Suppose now that A is an Arens–Michael algebra and ϕ: q ( ) A is a homomorphism. Fix a continuous submultiplicative prenorm · on A and set ρ = ≤ q ∈ Oalg Cn max1≤i≤n ϕ(xi) . Applying Lemma 5.10, we have ϕ(a) a ρ for any a q ( ). { · q } Therefore ϕ is continuous in the topology deﬁned by the system ρ : ρ>0 .The rest is clear.

We shall now examine the two simplest cases when wq can be computed explicitly. | |≥ ∈ Zn Proposition 5.12. (i) If qij 1 for all i

Deﬁne a permutation τ ∈ SN by the formula τ(i)=N − i (i =1,...,N). Clearly, α αˇ Oalg Cn τ(ξ )=η ,whereˇα =(αn,...,α1). Furthermore, the multiplication law on q ( ) shows that α α − α1 ··· αn i j αn ··· α1 1 αn ··· α1 (5.10) x1 xn = qij xn x1 = c(α, α) xn x1 , i

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or, equivalently, λ(τ,ξα)=c(α, α)−1. Therefore, in view of Lemma 5.8 and equality (5.9), we have w (α)=min |λ(σ, ξα)| : σ ∈ S =min |λ(στ, ξα)| : σ ∈ S q N N αˇ α −1 =min |λ(σ, η )| : σ ∈ SN ·|λ(τ,ξ )| = wqˇ(ˇα)|c(α, α)| . −1 But wqˇ ≡ 1 by (i) and therefore wq(α)=|c(α, α)| .

Corollary 5.13. (i) If |qij |≥1 for i0 . ∈Zn ∈Zn α + α +

(ii) If |qij|≤1 for i0 . ∈Zn ∈Zn α + α + n In both cases the topology on Oq(C ) is deﬁned by the system of submultiplicative { · q } prenorms ρ : ρ>0 and the multiplication is uniquely determined by the relations

xixj = qijxjxi (i, j =1,...,n). Remark 5.2. Notice that assertion (i) of the preceding corollary can be deduced by induction from Corollary 5.6, Proposition 4.11, and Proposition 4.6. In the case of the quantum plane, Corollary 5.13 becomes especially simple. Corollary 5.14. Suppose q ∈ C \{0}. (i) If |q|≥1,then ∞ ∞ O C2 i j q | | i+j ∞∀ q( )= a = cijx y : a ρ = cij ρ < ρ>0 . i,j=0 i,j=0 (ii) If |q|≤1,then ∞ ∞ O C2 i j q | || |ij i+j ∞∀ q( )= a = cijx y : a ρ = cij q ρ < ρ>0 . i,j=0 i,j=0 2 In both cases the topology on Oq(C ) is deﬁned by the system of submultiplicative prenorms { · q } ρ : ρ>0 and the multiplication is uniquely determined by the relation xy = qyx. 5.4. Connections with the algebra of holomorphic functions on the quantum n n ball. Let Bρ ⊂ C be the closed ball in C of radius ρ>0 and center at the origin. Let A(Bρ) denote the Banach algebra consisting of all continuous functions on Bρ which are holomorphic on its interior. Fix an arbitrary R>1 and consider the inverse system of Banach algebras

(5.12) A(B1) ← A(BR) ← A(BR2 ) ← A(BR3 ) ←··· , where the arrows denote the restriction homomorphisms. It is easy to see that the inverse limit of (5.12) is, up to isomorphism of topological algebras, the algebra O(Cn)ofentire functions. n Let z1,...,zn be the coordinates on C . Notice that (5.12) is isomorphic to the inverse system γ γ γ γ (5.13) A(B) ←−r A(B) ←−r A(B) ←−r A(B) ←−···r ,

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−1 where B = B1, r = R ,andγr denotes the unique continuous endomorphism of A(B) such that γr(zi)=rzi for all i =1,...,n.Thus O Cn ∼ B (5.14) ( ) = ←lim− A( ),γr . Below we shall show that (5.14) has a quantum analog, in which O(Cn) is replaced n by Oq(C ) (see above) and the role of the algebra A(B) is played by the algebra of holomorphic functions on the quantum ball, introduced by Vaksman [70]. First, we recall some deﬁnitions from [55] and [70]. Fix q ∈ (0, 1) and let Aq denote + + the algebra with generators a1,...,an,a1 ,...,an and relations + + + + ai aj = qaj ai (ii This algebra was introduced by W. Pusz and S. Woronowicz [55]; this is a q-analog of the algebra of canonical commutation relations. There is a unique involution a → a∗ A ∗ + on q such that ai = ai for all i =1,...,n. Notice that Aq is a particular case of the quantum Weyl algebra (see [14, 29]; in the simplest case of n = 1 the quantum Weyl algebra is discussed below in Example 5.5). This algebra can be viewed as an iterated Ore extension of the base ﬁeld C; as a consequence, ± ± A the ordered monomials in a1 ,...,an form a basis of q (ibid.). This, in turn, implies + + A Oalg Cn that a1 ,...,an generate a subalgebra of q isomorphic to q ( ). { ∈ Zn } Let H be a Hilbert space with orthonormal basis eα : α + .Foreachα = ∈ Zn | ∈ Z (α1,...,αn) + the basis element eα will also be denoted α1,...,αn . For any k + we set 1 − q2k [k]= , [k]! = [1] · [2] ···[k]. 1 − q2 ∈ Zn Finally, for α =(α1,...,αn) + set

[α]! = [α1]! ···[αn]! .

By [55], there is an irreducible ∗-representation π : Aq → B(H), uniquely determined by the formulas αk (5.15) π(aj)|α1,...,αn = [αj] q k>j |α1,...,αj − 1,...,αn, + | k>j αk | (5.16) π(aj ) α1,...,αn = [αj +1]q α1,...,αj +1,...,αn . This is a q-analog of the classical Fock representation. Notice that, unlike the classical ∈ A ± ≤ − 2 −1/2 case, all the operators π(a)witha q are bounded and π(aj ) (1 q ) for all j =1,...,n. (It is easy to check that the last inequality is in fact an equality, but we will not need this fact.) D. P. Proskurin and Yu. S. Samo˘ılenko [52] showed that π is faithful and that the ∗ ∗ C -algebra π(Aq) ⊂ B(H) is the universal enveloping C -algebra of Aq in the sense that ∗ any ∗-homomorphism from Aq to a C -algebra uniquely factors through π(Aq). As was noticed by L. L. Vaksman [70], the algebra C(B)q = π(Aq) can be viewed as a q-analog of the algebra C(B) of continuous functions on the closed ball B, whereas its closed B + subalgebra A( )q generated by the operators π(ak )(k =1,...,n) can be viewed as a q-analog of the algebra A(B).

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Henceforth we shall use the notation introduced in the preceding section. Recall, in particular, that for any α ∈ Z+ we have −1 αiαj (5.17) wq(α)=|c(α, α)| = q i0 (see Corollary 5.13 (ii)) will be denoted ·ρ. ≤ ∈ Oalg Cn Lemma 5.15. (i) a a 1 for any a q ( ). 2 (ii) For any r< 1 − q there is a Cr > 0 such that ar ≤ Cra for any a ∈ Oalg Cn q ( ). + ≤ − 2 −1/2 ≤ Proof. Since π(ai ) (1 q ) (see above), we have xi 1 for any i =1,...,n. In view of Lemma 5.10, this yields (i). To prove (ii), notice that by induction one easily deduces from (5.16) that + α1 ··· + αn i

γr γr γr γr (5.18) A(B)q ←− A(B)q ←− A(B)q ←− A(B)q ←−··· n is isomorphic to Oq(C ). Proof. For any s>0 consider a homomorphism Oalg Cn → Oalg Cn → γ˜s : q ( ) q ( ),xi sxi (i =1,...,n). It is easy to see that alg n (5.19) γ˜s(a)t = ast a ∈ O (C ),s,t>0 . q Fix an arbitrary r< 1 − q2. It follows from Lemma 5.15 and (5.19) that

γ˜r(a)≤γ˜r(a)1 = ar ≤ Cra

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∈ Oalg Cn Oalg Cn B for any a q ( ). This and the fact that q ( )isdenseinA( )q imply the existence and uniqueness of the homomorphism γr with the required properties. For the sake of convenience, label the terms of (5.18) by nonnegative integers. More ∈ Z B k+1 → precisely, for any k + set Ak = A( )q and define homomorphisms τk : Ak+1 Ak k+1 ∈ Z k+1 by setting τk = γr for all k +.SetA =← lim−(Ak,τk ), and let τk denote the canonical morphism from A to Ak. Define a prenorm ·k on A by setting ak = τk(a) for any a ∈ A. Notice that the sequence of prenorms {·k : k ∈ Z+} defines the standard inverse limit topology on A. −1 Set R = r and for any k ∈ Z+ define a homomorphism Oalg Cn → → k ϕk : q ( ) Ak,xi R xi (i =1,...,n). k+1 ∈ Z Clearly, ϕk = τk ϕk+1 for any k +. Therefore, there exists a unique homomorphism Oalg Cn → ∈ Z ϕ: q ( ) A such that ϕk = τkϕ for all k +. In turn, it gives rise to a n homomorphism ϕ: Oq(C ) → A. We shall show that ϕ is an isomorphism. For this, we first notice that for any k ∈ Z+ the image of ϕk is dense; hence the same is true for ϕ. Therefore, to prove that ϕ is an isomorphism, it suffices to check that ϕ is topologically injective. Using (5.19) and Lemma 5.15, we have

aRk = γ˜Rk+1 (a)r ≤ Crγ˜Rk+1 (a) = Crϕk+1(a) = Crτk+1ϕ(a) = Crϕ(a)k+1 ∈ Z ∈ Oalg Cn Oalg Cn for any k + and any a q ( ). Since the topology on q ( ) induced n from Oq(C ) is deﬁned by { · Rk : k ∈ Z+}, it follows from the last computation that ϕ is topologically injective. As we mentioned before, this implies that ϕ is an isomorphism.

5.5. Ore extensions: the “general” case. Quantum Weyl algebras and quantum matrices. In the general case, i.e., when α = 1R and δ = 0, there does not seem to be a satisfactory description of the Arens–Michael envelope of the Ore extension R[t; α, δ] as an algebra of “skew power series” (cf. Example 3.4). Below we shall describe a special but often encountered situation when such a description is in fact possible and we shall also illustrate it with several examples. Suppose R is an algebra, α: R → R is an endomorphism, and δ : R → R is an α- derivation. As before (see Section 5.3), α will denote the (unique) endomorphism of R such that

(5.20) αι R = ιRα. Clearly, ιR can be viewed as an R-bimodule morphism αR → α R. Therefore, ιRδ : R → α R is a derivation. It follows from Corollary 3.9 that there is a unique derivation δ : R → α R, i.e., a unique α-derivation of R such that (5.21) διR = ιRδ. In the next theorem, as in Section 4.1, z ∈ O(C, R) denotes the function sending λ ∈ C ∈ to λ1R R. Theorem 5.17. Suppose that the family {α, δ} is m-localizable. Then there exists a unique R-homomorphism (5.22) ιR[t;α,δ] : R[t; α, δ] → O(C, R; α, δ) such that t → z. The algebra O(C, R; α, δ) with the homomorphism (5.22) is the Arens–Michael envelope of R[t; α, δ].

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Proof. The existence and uniqueness of the R-homomorphism (5.22) directly follows from the universal property of Ore extensions and relations (5.20) and (5.21). Further- more, the algebra O(C, R; α, δ) is Arens–Michael by Proposition 4.5. Suppose now that ϕ: R[t; α, δ] → A is a homomorphism to some Arens–Michael algebra A.Setψ = ϕ|R and x = ϕ(t). The homomorphism ψ: R → A extending ψ makes A an R-⊗-algebra; moreover, since ϕ is a homomorphism, it follows from (5.20) and (5.21) that x · r = α(r) · x + ψ(δ(r)) for any r ∈ Im ιR.SinceImιR is dense in R, the latter equality holds for any r ∈ R. Therefore, by Proposition 4.4 (see also Remark 4.6), there exists a unique R- homomorphism (or, equivalently, R-homomorphism) ϕ: O(C, R; α, δ) → A such that ϕ(z)=x. This equality and the fact that ϕ is an R-homomorphism is equivalent to the relation ϕ ◦ ιR[t;α,δ] = ϕ. The rest is clear. To apply the preceding theorem in concrete situations it is useful to have a purely algebraic condition guaranteeing that the family {α, δ} is m-localizable. The next theorem gives one such possible condition of that kind.

Theorem 5.18. Suppose A is an algebra with generators x1,...,xn+1 and relations ⎧ ≤ ⎨⎪ xixj = qijxjxi (i, j n), (5.23) xn+1xi = λixixn+1 (2 ≤ i ≤ n), ⎩⎪ γ1 ··· γn xn+1x1 = λ1x1xn+1 + µx1 xn , n × ∈ C where (qij)i,j=1 = q is a multiplicatively antisymmetric (n n)-matrix, λ1,...,λn,µ , ∈ Zn ∈ Zn and γ =(γ1,...,γn) +. For any j =1,...,n let ej + denote the element with 1 Zn × Zn → C× at the j-th place and 0 elsewhere. Deﬁne a function c: + + by (5.2) and assume that

c(γ,ej) (5.24) λ1 = c(γ,e1),λj = (j ≥ 2). q1jc(ej,γ) Moreover, assume that

(5.25) |λi|≤1(i =1,...,n), |qij|≥1(i0 , ∈Zn+1 ∈Zn+1 α + α + and the multiplication on it is uniquely determined by (5.23).

Proof. Suppose F is the free algebra with generators ξ1,...,ξn,andletI ⊂ F be the two-sided ideal generated by the relations ξiξj = qijξjξi (i, j =1,...,n). Set R = F/I, let π denote the quotient homomorphism from F to R,andsetyi = π(ξi)(i =1,...,n). Oalg Cn As R = q ( ), the multiplication law (5.3) (with x replaced by y) holds on R. It is easy to see that there are a unique endomorphismσ ˜ and a uniqueσ ˜-derivation δ˜ of F such that ˜ γ ˜ σ˜(ξi)=λiξi (1 ≤ i ≤ n), δ(ξ1)=µξ , δ(ξk)=0 (k ≥ 2). Clearly,σ ˜(I) ⊂ I.Weshallshowthatδ˜(I) ⊂ I. Indeed,

δ˜(ξiξj − qijξjξi)=0 when i, j ≥ 2,

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and πδ˜(ξ ξ − q ξ ξ )=π δ˜(ξ )ξ − q σ˜(ξ )δ˜(ξ ) = µπ ξγξ − q λ ξ ξγ 1 j 1j j 1 1 j 1j j 1 j 1j j j ej +γ = µ c(γ,ej) − q1jλjc(ej,γ) y for any j ≥ 2. But the latter expression is zero by (5.24). In view of the relation −1 ˜ ⊂ qj1 = q1j , this yields δ(I) I. Thusσ ˜ and δ˜ ﬁx I and therefore give rise to linear maps σ, δ : R → R. Moreover, it is easy to see that σ is an endomorphism and δ is a σ-derivation of R. Consider the Ore extension B = R[yn+1; σ, δ]. The constructions of R, σ,andδ show that y1,...,yn+1 satisfy the same relations (5.23) as the xi. Therefore, there exists a unique homomorphism A → B such that xi → yi (i =1,...,n+ 1). On the other hand, the universal property of Ore extensions yields a unique homomorphism B → A such that yi → xi (i =1,...,n+ 1). This implies that A and B are isomorphic. Henceforth we shall assume that A = B and xi = yi (i =1,...,n+1). Oalg Cn We shall show that σ and δ satisfy the conditions of Theorem 5.17. Since R = q ( ) and |qij|≥1fori0 , ∈Zn ∈Zn β + β +

and all the prenorms ·ρ (ρ>0) are submultiplicative. Since σ(xi)=λixi and |λi|≤1 for all i =1,...,n,wehave

(5.26) σ(r)ρ ≤rρ for all r ∈ R and all ρ>0. It is easily checked by induction that c(γ,e )m − λm m 1 1 (m−1)e1+γ δ(x1 )=µ x c(γ,e1) − λ1

for all m ∈ Z+. This and the fact that δ(xk)=0fork>1implythat

β1 β1 c(γ,e1) − λ − β β1 β2 ··· βn 1 (β1 1)e1+γ β2 ··· βn (5.27) δ(x )=δ(x1 )x2 xn = µ x x2 xn c(γ,e1) − λ1 ∈ Zn for all β +, β1 > 0, and β (5.28) δ(x )=0 when β1 =0.

Furthermore, the condition |qij|≥1fori>jand (5.2) imply that |c(γ,e1)|≤1. Since |λ1|≤1, it follows from (5.27) and (5.28) that

|γ|−1 β |β| 2µρ |β| δ(x )ρ ≤ Cρρ , where Cρ = ρ , |c(γ,e1) − λ1| ∈ Zn · for all β + and all ρ>0. This and the explicit description of ρ imply that δ(r)ρ ≤ Cρrρ for all r ∈ R and all ρ>0. Comparing this with (5.26), we have that the family {σ, δ} is m-localizable. To ﬁnish the proof, apply Theorem 5.17 and the canonical isomorphism of locally convex spaces

∼ n ∼ n n+1 O(C, R) = O(C, O(C )) = O(C × C )=O(C ).

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Example 5.5 (quantum Weyl algebras). Recall (see, for example, [20]) that the quantum Weyl algebra (with parameter q ∈ C \{0}) is the algebra A1(q) with generators x, ∂ and relation ∂x − qx∂ =1.Whenq = 1 we have the usual Weyl algebra (see Example 3.4). Suppose that |q|≤1andq = 1. Then it is not difficult to see that A1(q) satisfies the conditions of Theorem 5.18; moreover, n = 1 (and therefore c ≡ 1), λ1 = q, µ =1,and γ =0.If|q|≥1 but still q = 1, then it is convenient for our purposes to represent A1(q) as the algebra with generators ∂,x and relation x∂ − q−1∂x = −q−1. It also satisfies −1 −1 the conditions of Theorem 5.18 with n =1,λ1 = q , µ = −q ,andγ = 0. Applying this theorem, we have the following description of the Arens–Michael envelope of the quantum Weyl algebra. Corollary 5.19. Suppose q ∈ C \{0} and q =1 . (i) If |q|≤1,then ∞ ∞ i j i+j A1(q) = a = cijx ∂ : aρ = |cij|ρ < ∞∀ρ>0 . i,j=0 i,j=0 (ii) If |q|≥1,then ∞ ∞ i j i+j A1(q) = a = cij∂ x : aρ = |cij|ρ < ∞∀ρ>0 . i,j=0 i,j=0

In both cases the topology on A1(q) is deﬁned by the system of prenorms { · ρ : ρ>0}, and the multiplication is uniquely determined by the relation ∂x − qx∂ =1. Remark 5.3. Thus we see that for all q = 1, the Arens–Michael envelopes of the corresponding quantum Weyl algebras are similar. In particular, their underlying LCS are 2 isomorphic to the space of entire functions O(C ), and the canonical morphism from A1(q) to A1(q) is injective. The case q = 1 is thus degenerate: recall that A1(1) = {0} (see Example 3.4).

Remark 5.4. It is not difficult to check that if q is not a root of unity, then A1(q)canbe realized as the algebra of q-differentiable operators on C[x] with polynomial coefficients: N n ∈ C ∈ Z A1(q)= a = an(x)Dq : an [x],N + , i=0

where the q-derivative Dq is given by f(qx) − f(x) (D f)(x)= (f ∈ C[x]). q qx − x

On the other hand, it is easy to check that when |q|≤1,q= 1, the operator Dq can be viewed as a bounded operator on the Hardy space H2. Therefore, if |q|≤1andis not a root of unity, then the quantum Weyl algebra A1(q) has a faithful representation 2 π : A1(q) → B(H ), which is uniquely determined by the conditions 2 (π(x)f)(z)=zf(z), (π(∂)f)(z)=(Dqf)(z)(f ∈ H , |z| < 1). In particular, for the above values of q we have a simple proof that the Arens–Michael envelope of A1(q) is not trivial, which is not based on Theorem 5.18. Example 5.6 (quantum matrices). Suppose q ∈ C \{0}. Recall (see, for example, [31]), that the algebra of quantum (2 × 2)-matrices Mq(2) has generators a, b, c, d and relations ab = qba, ac = qca, bc = cb, (5.29) bd = qdb, cd = qdc, ad − da =(q − q−1)bc.

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Set x1 = a, x2 = b, x3 = c,andx4 = d. Then it is not diﬃcult to see that Mq(2) belongs to the type of algebras considered in Theorem 5.18. Moreover, n =3,λ1 =1, λ = λ = q−1, µ = q−1 − q, γ =(0, 1, 1), and 2 3 ⎛ ⎞ 1 qq q = ⎝q−1 11⎠ . q−1 11 It is easy to see that −2 c(γ,e1)=q ,c(γ,e2)=c(e2,γ)=c(γ,e3)=c(e3,γ)=1, so that conditions (5.24) hold if and only if q = ±1. It is also clear that conditions (5.25) hold if and only if |q|≥1. To cover the case |q|≤1, consider the algebra Mq−1 (2). To distinguish its standard generators from the generators a, b, c, d of Mq(2), denote them a ,b,c,d, respectively. Then it is easy to see that there is an isomorphism Mq(2) → Mq−1 (2) such that a → d , d → a, b → b, c → c. Corollary 5.20. Suppose q ∈ C \{0}. (i) If |q|≥1,then ∞ ∞ i j k l i+j+k+l Mq(2) = u = cijkla b c d : uρ = |cijkl|ρ < ∞∀ρ>0 . i, j, k, l=0 i, j, k, l=0 (ii) If |q|≤1,then ∞ ∞ i j k l i+j+k+l Mq(2) = u = cijkld b c a : uρ = |cijkl|ρ < ∞∀ρ>0 . i, j, k, l=0 i, j, k, l=0

In both cases the topology on Mq(2) is deﬁned by the system of prenorms {·ρ : ρ>0}, and the multiplication is uniquely determined by relations (5.29). Proof. In view of the above, in the case q = ±1 the proof reduces to applying Theorem 5.18. If q =1,thenMq(2) = C[a, b, c, d], and there is nothing to prove (see Example 3.1). − Oalg C4 If q = 1, then it is not diﬃcult to see that Mq(2) = q ( ), where ⎛ ⎞ 1 −1 −11 ⎜ ⎟ −111−1 (5.30) q = ⎜ ⎟ . ⎝−111−1⎠ 1 −1 −11 Thus in the case q = −1 the proof reduces to applying Corollary 5.13. 5.6. Skew Laurent extensions. Quantum tori. Let R be an algebra, α: R → R an automorphism, and α its canonical extension to R (see (5.20)). Clearly, α is an automorphism of R and α−1 =(α−1). Theorem 5.21. Suppose that the family {α, α−1} is m-localizable. Then there exists a unique R-homomorphism −1 × (5.31) ιR[t,t−1;α] : R[t, t ; α] → O(C , R; α) such that t → z. The algebra O(C×, R; α) with the homomorphism (5.31) is the Arens–Michael envelope of R[t, t−1; α]. In view of Propositions 4.15 and 4.14, the proof of this theorem is similar to the proof of Theorem 5.17.

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n × Example 5.7. Suppose q =(qij)i,j=1 is a multiplicatively antisymmetric (n n)-matrix. Recall that themultiparameter quantum torus (see, for example, [41, 20]) is the alge- Oalg C× n −1 −1 bra q ( ) with generators x1,x1 ,...,xn,xn and relations xixj = qijxjxi and −1 −1 xixi = xi xi =1(i, j =1,...,n). n × Corollary 5.22. Let q =(qij)i,j=1 be a multiplicatively antisymmetric (n n)-matrix | | Cn such that qij =1for all i, j.Letx1,...,xn denote the coordinates on .Then × n O (C ) admits a unique multiplication making it a ⊗-algebra such that xixj = qijxjxi −1 −1 and xixi = xi xi =1for all i, j =1 ,...,n . The algebra thus obtained, together Oalg C× n with the tautological embedding of q ( ) in it, is the Arens–Michael envelope of Oalg C× n q ( ) . Its topology is defined by the family of submultiplicative prenorms |α| (5.32) aρ = |cα|ρ (ρ>0), α∈Zn α ∈ O C× n where a = α cαx is the Laurent expansion of a ( ) . Proof. We induct on n.Whenn = 0, the assertion is obvious. Suppose that it is true − n−1 Oalg C× n−1 for matrices of order n 1. Set q¯ =(qij)i,j=1 and R = q¯ ( ) . It is not difficult Oalg C× n −1 → to see that q ( ) = R[xn,xn ; α], where the automorphism α: R R is uniquely determined by the conditions α(xi)=qnixi (i =1,...,n−1). The induction assumption and the condition |qni| = 1 imply that α(a)ρ = aρ for any a ∈ R and any ρ>0. It remains to apply Theorem 5.21, Proposition 4.16 and the canonical isomorphism of locally convex spaces × ∼ × × n−1 ∼ × × n−1 × n O(C , R) = O C , O (C ) = O C × (C ) = O (C ) . Oalg C× n Definition 5.1. The Arens–Michael envelope of q ( ) described in the preceding × n proposition will be denoted Oq (C ) .

When the condition |qij | = 1 fails for at least one pair of indices i, j,theArens–Michael envelope of the quantum torus degenerates: | | Oalg C× n Proposition 5.23. Suppose that qij =1for some i, j.Then q ( ) =0. Oalg C× n Proof. It suﬃces to show that there are no homomorphisms from q ( ) to nonzero Oalg C× n → Banach algebras. Suppose that ϕ: q ( ) A is such a homomorphism. Set u = ϕ(xi),v= ϕ(xj), and q = qij. The spectrum of an arbitrary element a ∈ A will be denoted σ(a). We have that u, v ∈ A are invertible and uv = qvu. Therefore, setting K = σ(uv) and using the fact that σ(uv)=σ(vu) (see, for example, [24, 2.1.8]), we have that K = qK.SinceK is a nonempty compact and |q| = 1, the last relation is only possible when K = {0}. But this is also impossible since uv is invertible. The contradiction obtained shows that A =0.

6. Homological epimorphisms Let A, B be categories and F : A → B a covariant functor. Recall that F is said to be fully faithful if it is full and faithful (see [39] for terminology), i.e., if it gives rise to a bijection between Hom A(X, Y )andHomB(F (X),F(Y )) for any X, Y ∈ Ob (A). In other words, F is fully faithful if and only if it gives rise to an equivalence between A and a full subcategory of B. Proposition 6.1. Let f : A → B be a homomorphism of ⊗-algebras. The following conditions are equivalent: (i) f is an epimorphism in the category of ⊗-algebras;

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(ii) the functor f • : B-mod → A-mod is fully faithful; (iii) the canonical morphism B ⊗A B → B is an isomorphism of LCS; (iv) for any X ∈ Ob (mod-B) and any Y ∈ Ob (B-mod) the canonical morphism X ⊗A Y → X ⊗B Y is an isomorphism of LCS. Proof. The proof that (i)⇐⇒ (ii)⇐⇒ (iii) is a verbatim repetition of the proof given in [64, XI.1] for the category of rings. The implication (iv)=⇒(iii) is obvious, and (iv) can be obtained from (iii) by tensoring B with X on the left and with Y on the right.

Remark 6.1. In [67], ⊗-algebra epimorphisms are called pseudo-quotient homomorphisms. Lemma 6.2. Suppose A, B are exact categories and F : A → B is an exact additive covariant functor. The following conditions are equivalent: (i) RF : Db(A) → Db(B) is fully faithful; n (ii) for any X, Y ∈ Ob (A) and any n ∈ Z+ the canonical morphism Ext A(X, Y ) → n Ext B(F (X),F(Y )) is a bijection. The proof of this lemma given in [17] for abelian categories carries over verbatim to exact categories. Theorem 6.3. Suppose f : A → B is an R-S-homomorphism from an R-⊗-algebra A to an S-⊗-algebra B. The following conditions are equivalent: (1) Rf • : Db((B,S)-mod) → Db((A, R)-mod) is fully faithful; (2) for any X, Y ∈ Ob ((B,S)-mod) and any n ∈ Z+ the canonical morphism n → n Ext B,S(X, Y ) Ext A,R(X, Y ) is bijective; (3) for any X ∈ Ob (Db((B,S)-mod)) the canonical morphism ⊗ L • → B A(Rf (X)) X is an isomorphism in Db((B,S)-mod); (4) f is an epimorphism of ⊗-algebras and, moreover, any X ∈ Ob ((B,S)-mod), viewed as an object of (A, R)-mod, is acyclic relative to the functor B ⊗A ( · ): (A, R)-mod → (B,S)-mod . Proof. (1)⇐⇒ (2): This is a particular case of Lemma 6.2. • (1)⇐⇒ (3): Since B ⊗A ( · )isleftadjointtof :(B,S)-mod → (A, R)-mod,thesame is true for their derived functors [32, Lemma 15.6]. It remains to apply Theorem 4.3.1 from [39]. (3)=⇒(4): Taking into account that f • is exact, it follows from (3) that for any ∈ ⊗ L → X Ob ((B,S)-mod), the canonical morphism B A X X is an isomorphism in Db((B,S)-mod) and therefore in Db(LCS). Applying to it the functor H0,wehavethat the composition of the canonical morphisms A,R → ⊗ → Tor 0 (B,X) B A X X is an isomorphism in LCS. Since the ﬁrst morphism in this composition is the canonical map from an LCS to its completion (see Section 2), we conclude that both morphisms are isomorphisms in LCS. Therefore, B ⊗A X → X is an isomorphism in B-mod.The rest is clear. (4)=⇒(2): Suppose P• is a projective resolution of X in (A, R)-mod.By(4),B ⊗A P• is a projective resolution of X in (B,S)-mod. Therefore n ∼ n ∼ n ⊗ ∼ n Ext A,R(X, Y ) = H (Ah(P•,Y)) = H Bh(B A P•,Y) = Ext B,S(X, Y ).

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Deﬁnition 6.1. If the homomorphism f : A → B satisﬁes the conditions of the preceding theorem, then we shall call it a left relative homological epimorphism. The homomorphism f : A → B is called a right relative homological epimorphism if f : Aop → Bop is a left relative homological epimorphism. Lemma 6.4. Suppose f : A → B is an R-S-homomorphism from an R-⊗-algebra A to an S-⊗-algebra B which is an epimorphism of ⊗-algebras. Let A denote any of the exact categories (A, R)-mod-(A, R), (A, A)-mod-(A, R), (A, R)-mod-(A, A), and B any of the exact categories (B,S)-mod-(B,S), (B,B)-mod-(B,S), (B,S)-mod-(B,B). Consider the functor (6.1) T = B ⊗A ( · ) ⊗A B : A → B. Then, if A is T -acyclic for some choice of the exact categories Aand B,thenitisalso T -acyclic for any other choice of those categories. Proof. Suppose that (2.5) is a projective resolution of A in (A, R)-mod-(A, R). In view of Proposition 2.6, it is also a projective resolution of A in (A, R)-mod-(A, A)andin ∼ (A, A)-mod-(A, R). Since f is an epimorphism, it is easy to see that T (A) = B.Moreover, objects of T (P•) are projective in B for any choice of B. Therefore, again applying Proposition 2.6, we have that the admissibility of the complex

0 ← B = T (B) ← T (P•) in B does not depend on the choice of B. It remains to notice that, in view of the above, the latter complex is admissible in B if and only if A is T -acyclic. Deﬁnition 6.2. An R-S-homomorphism f : A → B from an R-⊗-algebra A to an S-⊗- algebra B is called a two-sided relative homological epimorphism if f is an epimorphism of ⊗-algebras and A is acyclic relative to functor (6.1) from Lemma 6.4. Proposition 6.5. Any two-sided relative homological epimorphism f : A → B from an R-⊗-algebra A to an S-⊗-algebra B is a left and a right relative homological epimorphism. Proof. Fix a projective resolution

(6.2) 0 ← A ← P• in (A, R)-mod-(A, R). In view of Proposition 2.6, the augmented complex (6.2) is split in R-mod-A. Therefore, for any X ∈ Ob ((B,S)-mod) (6.3) 0 ← X ← P• ⊗A X is a projective resolution of X in (A, R)-mod. On the other hand, it follows from the assumption that the complex 0 ← B ← Q• = B ⊗A P• ⊗A B is split in S-mod-B, and therefore the complex 0 ← X ← Q• ⊗B X ∼ is split in S-mod. Taking account of the isomorphism B ⊗A X = X,wehavethat the latter complex is isomorphic to the complex B ⊗A (6.3). Thus B ⊗A (6.3) is split in S-mod, and therefore the conditions of Theorem 6.3, (4) hold. This shows that f is a left relative homological epimorphism. A similar argument shows that f is a right relative homological epimorphism.

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Example 6.1. Fix a ⊗-algebra A and set B = S = A, R = C,andf = 1A.Thenitis easy to see that f is a left (resp., right) relative homological epimorphism if and only if dg A =0(resp.,dgAop = 0). On the other hand, f is a two-sided relative homological epimorphism if and only if db A =0. The next proposition shows that one-sided relative homological epimorphisms do in- herit some of the properties of two-sided ones. Proposition 6.6. Suppose f : A → B is an R-S-homomorphism from an R-⊗-algebra A to an S-⊗-algebra B and that f is a left or right relative homological epimorphism. Then A is acyclic relative to the functors T = B ⊗A ( · ): (A, R)-mod-(A, R) → LCSspl, Tr =(· ) ⊗A B :(A, R)-mod-(A, R) → LCSspl, T = B ⊗A ( · ) ⊗A B :(A, R)-mod-(A, R) → LCSspl.

Proof. Fix a projective resolution P• of A in (A, R)-mod-(A, R). By Proposition 2.6, augmented complex (2.5) is split in R-mod-A. Therefore the complex (6.4) 0 ← B ← P• ⊗A B

is split in R-mod-B and, in particular, in LCS. This proves that A is Tr-acyclic. A similar argument shows that the complex (6.5) 0 ← B ← B ⊗A P•

is split in B-mod-R and, in particular, in LCS, whence the T-acyclicity of A.Further- more, notice that (6.4) (resp., (6.5)) without the term on the left is a projective resolution of B in (A, R)-mod (resp., in mod-(A, R)). Therefore if f is a left relative homological epimorphism, then, applying the functor B ⊗A ( · )to(6.4),wehaveacomplex (6.6) 0 ← B ← B ⊗A P• ⊗A B, which is split in S-mod.Iff is a right relative homological epimorphism, then, applying the functor ( · ) ⊗A B to (6.5), we have that (6.6) is split in mod-S.Thusinboth cases (6.6) is split in LCS, which means that A is T -acyclic. Remark 6.2. In fact, the proof of the preceding proposition reduces to the analysis of the diagram

T T (A, A)-mod-(A, R) /mod-(A, R) r /mod-(B,S) O (exact) (A-acyclic) (exact) T / (A, R)-mod-(A, R) LCSOspl

(A-acyclic) (exact) T T (A, R)-mod-(A, A) r /(A, R)-mod /(B,S)-mod (exact) Here the vertical arrows are functors acting identically on objects and morphisms. Start- ing with an object A ∈ (A, R)-mod-(A, R) and moving along the arrows of the diagram, at each step we have an object which is acyclic relative to the preceding arrow. Moreover, the right half of the top row corresponds to the case when f is a right relative homological epimorphism, and the right half of the bottom row, to the case when f is a left relative homological epimorphism. The next theorem shows that in the case R = S = C the possible distinctions between left, right, and two-sided homological epimorphisms disappear.

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Theorem 6.7. item Suppose f : A → B is a homomorphism of ⊗-algebras. The following conditions are equivalent: (1) Rf • : Db(B-mod) → Db(A-mod) is fully faithful; (1)Rf • : Db(mod-B) → Db(mod-A) is fully faithful; (2) for any X, Y ∈ Ob (B-mod) and any n ∈ Z+ the canonical morphism n → n Ext B(X, Y ) Ext A(X, Y ) is bijective; (2 ) for any X, Y ∈ Ob (mod-B) and any n ∈ Z+ the canonical morphism n → n Ext B(X, Y ) Ext A(X, Y ) is bijective; (3) for any Y ∈ Ob (B-mod) the canonical morphism Y → Ah(B,Y ) is bijective, n ≥ and Ext A(B,Y )=0for all n 1; (3 ) for any Y ∈ Ob (mod-B) the canonical morphism Y → hA(B,Y ) is bijective, n ≥ and Ext A(B,Y )=0for all n 1; (4) for any X ∈ Ob (Db(B-mod)) the canonical morphism ⊗ L • → B A(Rf (X)) X is an isomorphism in Db(B-mod); (4) for any X ∈ Ob (Db(mod-B)) the canonical morphism • ⊗ L → (Rf (X)) A B X is an isomorphism in Db(mod-B); (5) f is an epimorphism of ⊗-algebras and, furthermore, any X ∈ Ob (B-mod),as an object of A-mod, is acyclic relative to the functor B ⊗A ( · ): A-mod → B-mod; (5) f is an epimorphism of ⊗-algebras and, furthermore, any X ∈ Ob (mod-B),as an object of mod-A, is acyclic relative to the functor ( · ) ⊗A B : mod-A → mod-B; (6) f is an epimorphism of ⊗-algebras and, furthermore, B,asanobjectofA-mod, is acyclic relative to the functor B ⊗A ( · ): A-mod → B-mod; (6) f is an epimorphism of ⊗-algebras and, furthermore, B,asanobjectofmod-A, is acyclic relative to the functor ( · ) ⊗A B : mod-A → mod-B; (7) f is an epimorphism of ⊗-algebras and, furthermore, A,asanobjectofA-mod-A, is acyclic relative to the functor B ⊗A ( · ) ⊗A B : A-mod-A → B-mod-B. Proof. The implications (1) ⇐⇒ (2) ⇐⇒ (4) ⇐⇒ (5) and (1) ⇐⇒ (2) ⇐⇒ (4) ⇐⇒ (5) follow from Theorem 6.3. For (5) ⇐⇒ (5) ⇐⇒ (6) ⇐⇒ (6) ⇐⇒ (7), see [48, 3.2]. (2) =⇒ (3) and (2)=⇒ (3) are obvious. • (3) =⇒ (6). As the functors B ⊗A ( · )andf are adjoint, it follows from (3), that for any Y ∈ Ob (B-mod) we have natural isomorphisms ∼ ∼ ∼ Bh(B,Y ) = Y = Ah(B,Y ) = Bh(B ⊗A B,Y ).

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It is easy to see that the composition of these isomorphisms is induced by the canonical morphism B ⊗A B → B. Therefore it is an isomorphism in B-mod, i.e., f is an epimorphism of ⊗-algebras. Now we ﬁx a projective resolution

0 ← B ← P• in A-mod. It follows from (3) that for any Y ∈ Ob (B-mod)thecomplex

0 → Ah(B,Y ) → Ah(P•,Y) is exact. On the other hand, it is isomorphic to the complex 0 → Bh B ⊗A B,Y → Bh B ⊗A P•,Y , which is exact for any Y ∈ Ob (B-mod) if and only if the complex 0 ← B ⊗A B ← B ⊗A P• is split in B-mod. Therefore, (6) holds. The implication (3) =⇒ (6) is proved similarly. Definition 6.3. A ⊗-algebra homomorphism f : A → B is called a homological epimorphism if it satisfies the equivalent conditions of Theorem 6.7. In this case we shall say (following [46]) that B is stably flat over A. Remark 6.3. Theorem 6.7 is a “locally convex analog” of a purely algebraic Theorem 4.4 from [17]. We decided to give a detailed proof, since the proof in [17] does not carry over verbatim to the case of ⊗-algebras. Moreover, [17] does not mention conditions (4), (4) and (7), and conditions (5), (5), (6), and (6) are stated in terms of the functor Tor , which is not enough in the ⊗-case. Remark 6.4. If we state condition (7) of Theorem 6.7 in an explicit form, then we arrive at the following definition: a homomorphism A → B is a homological epimorphism if and only if for some (or, equivalently, for any) projective resolution

0 ← A ← P0 ← P1 ←··· of the bimodule A in A-mod-A the complex 0 ← B ← B ⊗A P0 ⊗A B ← B ⊗A P1 ⊗A B ←··· is split in LCS. Thus a homological epimorphism is the same thing as the absolute localization in Taylor’s terminology [67]. For examples of homological epimorphisms the reader is referred to [67, 68, 37, 26, 17, 46, 16, 15, 48, 44] (see also the introduction to the present paper). A basic example, obtained by Taylor [67, 4.3], is the embedding of the polynomial algebra C[x1,...,xn]with the strongest locally convex topology in the Fr´echet algebra O(Cn) of entire functions. Other examples will be given in Section 9 of this paper (see Theorem 9.12).

7. Relatively quasi-free algebras Throughout this section R is a ﬁxed ⊗-algebra. Suppose A is an R-⊗-algebra. By an R-extension of A we shall understand an arbitrary open homomorphism of R-⊗-algebras σ : A → A,whereA is an R-⊗-algebra. As usual, an extension will written as a sequence (7.1) 0 → I −→i A −→σ A → 0, where I =Kerσ and i is the tautological embedding. Notice that (7.1) is a topologically exact (see the Appendix, Example 10.5) sequence of R-⊗-bimodules. Exten- sion (7.1) is said to be R-admissible (henceforth, simply admissible) if this sequence is split in R-mod-R, i.e., if σ has a right inverse morphism of R-⊗-bimodules. If σ has a right

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inverse morphism of R-⊗-algebras, then extension (7.1) is said to be R-split (henceforth, simply split). Recall that extension (7.1) is said to be singular if I2 =0.InthiscaseI becomes an A-⊗-bimodule when, for a ∈ A and x ∈ I,weseta · x = ux and x · a = xu for arbitrary u ∈ σ−1(a). If X is an A-⊗-bimodule, then an R-extension of A by X is, by deﬁnition, a topologically exact sequence κ (7.2) 0 → X −→ A −→σ A → 0, in which A is an R-⊗-algebra, σ is a homomorphism of R-⊗-algebras, and κ is a morphism of A-⊗-bimodules. Clearly, each R-extension of A by X gives rise to a singular R- extension of the form (7.1), and conversely, as follows from the above, any singular extension can be obtained this way; moreover, κ establishes an isomorphism between X and I in A-mod-A. Two R-extensions 0 → X → A → A → 0and0→ X → A → A → 0ofA by X are said to be equivalent if there is an isomorphism of R-⊗-algebras ϕ: A → A such that the diagram q8A MM qqq MM qq MM& X M ϕ 8A MMM qqq MM&qqq A is commutative. As in the case R = C, it is not diﬃcult to check that the set of equivalence classes of admissible R-extensions of A by X is in one-to-one correspondence with H2(A, R; X); moreover the zero element of H2(A, R; X) corresponds to the equivalence class of a split extension. Proposition 7.1. The following properties of the R-⊗-algebra A are equivalent:

(i) dbR A ≤ 1; (ii) H2(A, R; X)=0for any X ∈ A-mod-A; (iii) any admissible singular R-extension of A is split; (iv) for any R-⊗-algebra B, any admissible singular R-extension 0 → I → A → B → 0,andanyR-homomorphism ϕ: A → B there is an R-homomorphism ψ : A → A making the diagram

~ A ψ ~ ~ ϕ ~ / / / / 0 I A B 0 commute. Proof. Clearly, (i)⇐⇒ (ii) and (iv)=⇒(iii); furthermore, it follows from the above that (ii)⇐⇒ (iii). To prove the implication (iii)=⇒(iv), take an arbitrary admissible singular R-extension 0 → I −→i A −→σ B → 0 and consider its preimage under ϕ:

/ i / σ / / 0 I A ×B A A 0

π ϕ / i / σ / / 0 I A B 0 Here A ×B A = {(u, a) ∈ A × A : σ(u)=ϕ(a)} (it is easy to see that this is an R-⊗- algebra), i(a)=(i(a), 0), σ(u, a)=a and π(u, a)=u. The top row of this diagram is an admissible singular R-extension of A (its admissibility follows from Axiom Ex2 applied

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to the exact category (R-mod-R)spl, see the Appendix, but it is also easy to check it directly). By (iii), there is an R-homomorphism ψ : A → A×B A which is a right inverse of σ.Itremainstosetψ = πψ. Definition 7.1. An R-⊗-algebra A, satisfying the equivalent conditions of Proposi- tion 7.1 is said to be relatively quasi-free.IfR = C,thenA is said to be quasi-free. To check whether an algebra is quasi-free it is convenient to use the bimodule of noncommutative differential 1-forms [7]. It is defined as follows. Fix an R-⊗-algebra A and consider the functor Der R(A, · )fromA-mod-A to the category of sets. As was shown by Cuntz and Quillen [7] (in a purely algebraic situation), this functor is representable. 1 Its representing object is denoted ΩRA and is called the bimodule of relative differential 1-forms of A. In other words, the bimodule of relative differential 1-forms is a pair 1 ⊗ 1 → 1 (ΩRA, dA) consisting of an A- -bimodule ΩRA and an R-derivation dA : A ΩRA such that for any A-⊗-bimodule X and any R-derivation D : A → X there exists a unique ⊗ 1 → morphism of A- -bimodules ϕ:ΩRA X making the diagram 6Ω1 A dAmmm R mmm (7.3) A SSSS ϕ SSS) D X

commute. The derivation dA is called the universal R-derivation of A. 1 To construct ΩRA, denote the completion of the quotient space A/Im ηA by A; clearly, this is an R-⊗-bimodule. The image of a ∈ A under the canonical map A → A will be ⊗ 1 ⊗ denoteda ¯.AsaleftA- -module, ΩRA coincides with A R A. To deﬁne a structure of arightA-⊗-module on it, denote the elementary tensor a0 ⊗ a¯1 ∈ A ⊗R A by a0da1;the tensor 1da will be simply written as da. It is not diﬃcult to see that there is a unique 1 × → 1 → · continuous bilinear map ΩRA A ΩRA,(ω, a) ω a, such that

(7.4) a0da1 · a = a0d(a1a) − a0a1da. 1 ⊗ This map defines a right action of A on ΩRA and makes it an A- -bimodule. It follows → 1 → → from (7.4) that the map dA : A ΩRA, a da,isanR-derivation. If D : A X is an arbitrary R-derivation of A with values in a bimodule X, then the unique morphism ϕ making diagram (7.3) commute is given by ϕ(a0da1)=a0 · D(a1); it is straightforward 1 to check that ϕ is a morphism of bimodules. Thus, the pair (ΩRA, dA) is indeed the bimodule of relative differential 1-forms. Consider the sequence of A-⊗-bimodules ← ←m− ⊗ ←−j 1 ← (7.5) 0 A A R A ΩRA 0. Here m is the multiplication on A,andj is the morphism of bimodules corresponding to the inner derivation −ad1⊗1 : A → A ⊗R A, a → [1 ⊗ 1,a]. In other words, j is uniquely determined by the formula j(a0da1)=a0(1 ⊗ a1 − a1 ⊗ 1). Clearly, mj = 0, i.e., (7.5) is a complex of A-bimodules. Proposition 7.2. Sequence (7.5) is split in A-mod-R and R-mod-A. Proof. Consider the morphisms of A-R-⊗-bimodules, s: A → A ⊗R A, a → a ⊗ 1; ⊗ → 1 ⊗ → t: A R A ΩRA, a0 a1 a0da1. A direct calculation shows that s and t provide a contracting homotopy for (7.5) making it split in A-mod-R. Furthermore, define a morphism of R-A-⊗-bimodules s : A → A ⊗R A

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by setting s (a)=1⊗ a; then, since ms = 1A and j =kerm, we have that (7.5) is also split in R-mod-A. ⊗ 1 Proposition 7.3. The R- -algebra A is relatively quasi-free if and only if ΩRA is projective in some (equivalently, in any) of the exact categories (A, R)-mod-(A, R), (A, A)-mod-(A, R),and(A, R)-mod-(A, A). Proof. By Proposition 7.2, the sequence (7.5) is admissible in the exact categories (A, R)-mod-(A, R), (A, A)-mod-(A, R), and (A, R)-mod-(A, A). Moreover, its middle term is projective in each of these categories. Hence, taking into account Corollary 2.8, for any A-⊗-bimodule X we have isomorphisms H2 ∼ 1 1 ∼ 1 1 ∼ 1 1 (A, R; X) = Ext Ae,Re (ΩRA, X) = Ext Ae,A⊗ Rop (ΩRA, X) = Ext Ae,R⊗ Aop (ΩRA, X). The rest is clear. Remark 7.1. Notice that the notion of a relatively quasi-free algebra has an obvious purely algebraic analog (see [7] for R = C). Everything that was said above about relatively quasi-free ⊗-algebras and noncommutative differential forms carries over easily to algebras without topology. In that regard we remark that if R is an algebra and A is an R-algebra having a countable dimension as a vector space, then its bimodule of relative differential 1-forms, constructed in a purely algebraic situation and then endowed with the strongest locally convex topology, is isomorphic to the bimodule of relative differential forms of the strongest locally convex algebra As over Rs (see Proposition 2.3). Combining this with Remark 2.2, we have that As is relatively quasi-free over Rs as a ⊗-algebra if and only if A is relatively quasi-free over R in the purely algebraic sense. Remark 7.2. Quasi-free algebras admit some other equivalent definitions. In particular, they can be characterized in terms of connections. For details (when R = C), see [7, 9] or [43]. Next, we want to determine the action of the Arens–Michael functor on the bimod- 1 ule ΩRA (see Remark 3.8). Consider first a more general situation. Suppose A is an R-⊗-algebra, B is an S-⊗-algebra, and (f,g)isanR-S-homomorphism from A to B. Fix an arbitrary B-⊗-bimodule X and consider the commutative diagram / Der S(B,X) Der R(A, X)

1 / 1 ⊗ 1 ⊗ BhB(ΩSB,X) AhA(ΩRA, X) BhB(B A ΩRA A B,X) Here the equalities denote canonical isomorphisms, the top arrow was deﬁned in (3.5), and the bottom arrow is uniquely determined by the top one and the commutativity of 1 the diagram. Substituting ΩSB for X in the bottom row, we see that the morphism of ⊗ ⊗ 1 ⊗ → 1 B- -bimodules B A ΩRA A B ΩSB (this is a noncommutative analog of the pull- back operation of diﬀerential forms), denoted f∗, corresponds to the identity morphism 1 of ΩSB. Proposition 7.4. Let A, R, S be ⊗-algebras and g : R → S an epimorphism of ⊗- algebras. Assume that A is an R-⊗-algebra and that its Arens–Michael envelope A is an S-⊗-algebra such that the pair (ιA,g) is an R-S-homomorphism from A to A.Then the morphism ⊗ 1 ⊗ → 1 (ιA)∗ : A A ΩRA A A ΩSA is an isomorphism of A-⊗-bimodules.

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Proof. Use Proposition 3.8. Corollary 7.5. For any R-⊗-algebra A the morphism 1 1 (ι )∗ : A ⊗ Ω A ⊗ A → Ω A A A R A R is an isomorphism of A-⊗-bimodules. Theorem 7.6. Let A, R, S be ⊗-algebras and g : R → S an epimorphism of ⊗-algebras. Suppose that A is an R-⊗-algebra and its Arens–Michael envelope A is an S-⊗-algebra such that the pair (ιA,g) is an R-S-homomorphism from A to A. Assume also that A is relatively quasi-free over R.Then: (i) A is relatively quasi-free over S; (ii) ιA : A → A is a two-sided relative homological epimorphism. Proof. Assertion (i) follows from Propositions 7.3 and 7.4. The same propositions imply that (7.5) is a projective resolution of A in (A, R)-mod-(A, R). Applying to it the functor A ⊗A ( · ) ⊗A A and using Propositions 6.1 (iv) and 7.4, we have a sequence ← ←m− ⊗ ←−j 1 ← 0 A A S A ΩSA 0, which is admissible in (A, S)-mod-(A, S) by Proposition 7.2. The rest is clear. Corollary 7.7. Suppose A is a relatively quasi-free R-⊗-algebra. Then A is relatively quasi-free over R and ιA : A → A is a two-sided relative homological epimorphism.

Remark 7.3. Setting in the preceding corollary R = C and A = Fn we recover Taylor’s result [67] that the embedding of the free algebra Fn in its Arens–Michael envelope is a homological epimorphism. Now we want to give several examples of relatively quasi-free algebras. Proposition 7.8. Suppose R is an Arens–Michael algebra and A is any of the following two R-⊗-algebras: (i) A = O(C,R; α, δ),whereα: R → R is an endomorphism and δ : R → R is an α-derivation such that the family {α, δ} is m-localizable; (ii) A = O(C×,R; α),whereα: R → R is an automorphism such that the family {α, α−1} is m-localizable. 1 ⊗ Then the bimodule ΩRA is isomorphic to Aα R A, and the universal derivation d: A → Aα ⊗R A is uniquely determined by the formula d(z)=1⊗ 1. Consequently, A is relatively quasi- free. Proof. Consider ﬁrst case (i). In view of (2.3) and Remark 4.6, for any A-⊗-bimodule X we have the canonical bijections ∼ Der (A, X) = ϕ ∈ Hom (A, A × X): p ϕ = 1 R R-AM 1 A ∼ = b ∈ A × X : p (b)=z, b · r = α(r) · b + δ × (r) ∀r ∈ R 1 A X ∼ = x ∈ X :(zr,x · r)=(α(r)z,α(r) · x)+(δ(r), 0) ∀r ∈ R ∼ = x ∈ X : x · r = α(r) · x ∀r ∈ R = RhR(Rα,X) ∼ ∼ = AhA(A ⊗R Rα ⊗R A, X) = AhA(Aα ⊗R A, X). It is easy to see that these bijections give rise to an isomorphism of functors of X, whence the ﬁrst assertion. To prove the second assertion, we set X = Aα ⊗R A and notice that

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the R-derivation d corresponding to the identity morphism 1X under the above bijections sends z ∈ A to 1 ⊗ 1 ∈ X. To prove (ii), notice that for any A-⊗-bimodule X and any x ∈ X the element (z,x) ∈ A × X is invertible and its inverse is (z−1, −z−1 · x · z−1). In view of this and (4.26), we can ﬁnish the argument as in case (i). Proposition 7.9. Suppose R is an algebra of countable dimension and A is any of the following two R-algebras: (i) A = R[t; α, δ],whereα: R → R is an endomorphism and δ : R → R is an α-derivation; (ii) A = R[t, t−1; α],whereα: R → R is an automorphism. 1 Endow R and A with the strongest locally convex topology. Then the bimodule ΩRA is isomorphic to Aα ⊗R A (which, in turn, is isomorphic to Aα ⊗R A endowed with the strongest locally convex topology), and the universal derivation d: A → Aα ⊗R A is uniquely determined by the formula d(t)=1⊗1. Consequently, A is relatively quasi-free. The proof of this proposition is similar to that of Proposition 7.8 and is therefore omitted. We also remark that the assumption that R is of countable dimension was needed only to guarantee that the multiplications on R and A are jointly continuous. A purely algebraic analog of the preceding proposition is, of course, true without any restrictions on the dimension of R. The same applies to Proposition 7.11 below. The following two propositions are proved exactly the same way as Proposition 2.9 of [7]. We omit the proofs. Proposition 7.10. Suppose R is an Arens–Michael algebra, M is an R-⊗-bimodule, and 1 ⊗ ⊗ A = TR(M). Then the bimodule ΩRA is isomorphic to A R M R A, and the universal derivation d: A → A ⊗R M ⊗R A is uniquely determined by the formula d(m)=1⊗m⊗1 for m ∈ M. As a consequence, A is relatively quasi-free. Proposition 7.11. Suppose R is a countably dimensional algebra, M is a countably dimensional R-bimodule, and A = TR(M). Endow R, M,andA with the strongest locally 1 ⊗ ⊗ convex topology. Then the bimodule ΩRA is isomorphic to A R M R A (which, in turn, is isomorphic to A ⊗R M ⊗R A endowed with the strongest locally convex topology), and the universal derivation d: A → A ⊗R M ⊗R A is uniquely determined by the formula d(m)=1⊗ m ⊗ 1 for m ∈ M. Consequently, A is relatively quasi-free.

8. Finiteness conditions Definition 8.1. Suppose A is a topological algebra and X is a left topological A-module. We shall say that X is strictly finitely generated if for some n ∈ N there is an open morphism from An onto X. Remark 8.1. It is not difficult to see that if A is locally convex and X is finitely generated and endowed with the strongest locally convex topology, then it is strictly finitely generated. Remark 8.2. If A is a Fréchet algebra, then it follows easily from the Banach Open Mapping Theorem that a left topological A-module X is strictly finitely generated if and only if it is finitely generated and is Fréchet. Remark 8.3. If A → B is a homomorphism of Fréchet algebras and X is a finitely generated left Fréchet A-module, then the left Fréchet B-module B ⊗A X is also finitely generated. This follows from the preceding remark and the fact that the functor of taking projective tensor product with a Fréchet module sends open morphisms of Fréchet modules to open morphisms (see [25, II.4.12]).

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Proposition 8.1. Suppose A is a topological algebra and X is a strictly finitely generated left topological A-module. Suppose that A is metrizable. Then the A-module X is strictly finitely generated. Proof. We may assume that X = An/Y for some submodule Y ⊂ An. It follows from n n Proposition 3.12 and Remark 3.7 that X = A /ιA(Y ). The rest is clear. Suppose A is an R-⊗-algebra and P is an A-⊗-bimodule. Recall that P is projective in (A, R)-mod-(A, R) if and only if P is a retract of a bimodule A ⊗R M ⊗R A,where M ∈ R-mod-R. Definition 8.2. We shall say that a projective (A, R)-⊗-bimodule P satisfies the finiteness condition (f1) if it is a retract of a bimodule A ⊗R M ⊗R A,whereM ∈ R-mod-R is strictly finitely generated as a left R-module. Definition 8.3. We shall say that a projective (A, R)-⊗-bimodule P satisfies the finiteness condition (f2) if it is a retract of a bimodule A ⊗R M ⊗R A,whereM ∈ R-mod-R is isomorphic to the R-⊗-bimodule Rn for some n.

Remark 8.4. Clearly, condition (f2) implies condition (f1). If R = C, then both conditions are equivalent to P being a retract of (A ⊗ A)n for some n. This means precisely that P is strictly projective (see [27]) and finitely generated in A-mod-A. Definition 8.4. We shall say that an R-⊗-algebra A is an algebra of (f1)-finite type (resp., (f2)-finite type)ifin(A, R)-mod-(A, R) it has a finite projective resolution all of whose bimodules satisfy the finiteness condition (f1)(resp.,(f2)). When R = C, the preceding definition means that A has a finite projective resolution all of whose bimodules are strictly projective and finitely generated. These conditions are satisfied, for example, by the algebra of polynomials C[t1,...,tn] with the strongest locally convex topology, and also by the algebra O(U) of holomorphic functions on a polydomain U ⊂ Cn: in both cases the resolution can be chosen to be a Koszul complex [67]. Other examples will appear in the next section. Definition 8.5. We shall say that an R-⊗-algebra A is relatively (f1)-quasi-free (resp., 1 relatively (f2)-quasi-free)ifΩRA satisfies condition (f1)(resp.,(f2)).

It follows immediately from Propositions 7.2 and 7.3 that a relatively (f1)-quasi-free (resp., relatively (f2)-quasi-free) algebra is quasi-free in the sense of Definition 7.1 and is an algebra of (f1)-finite type (resp., (f2)-finite type). Remark 8.5. Notice that Definitions 8.3, 8.4, and 8.5 have obvious purely algebraic analogs (cf. Remark 7.1). For example, the R-algebra A is relatively (f1)-quasi-free if the 1 bimodule ΩRA constructed in a purely algebraic situation is a retract of the A-bimodule A ⊗R M ⊗R A,wheretheR-bimodule M is finitely generated as a left R-module. It is not difficult to see that if we restrict ourselves to countably dimensional algebras and modules, then the “algebraic” finiteness conditions (i.e., conditions (f1)and(f2)for bimodules, the property of being an algebra of (f1)- or (f2)-finite type, or (f1)-quasi-free, or (f2)-quasi-free) are equivalent to the corresponding “topological” finiteness conditions for the same algebras and bimodules endowed with the strongest locally convex topology. We skip the verification of all these assertions since this is not difficult and reduces to applying Proposition 2.3. Example 8.1. The algebras O(C,R; α, δ)andO(C×,R; α) of Proposition 7.8 are rel- −1 atively (f1)-quasi-free. The same is true for the algebras R[t; α, δ]andR[t, t ; α]of Proposition 7.9. If α = 1R, then the above algebras are relatively (f2)-quasi-free.

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Example 8.2. The algebras TR(M)andTR(M) of Propositions 7.10 and 7.11 are relatively (f1)-quasi-free if M is strictly ﬁnitely generated in R-mod, and relatively (f2)- quasi-free if M is a retract of Rn in R-mod-R for some n ∈ N . Lemma 8.2. Suppose B is an Arens–Michael algebra and N ∈ B-mod is a strictly ﬁnitely generated module. Then the algebra Bh(N) endowed with the topology of simple convergence is Arens–Michael.

n Proof. If N = B ,thenBh(N) is topologically isomorphic to Mn(B) and is therefore Arens–Michael (see Section 2). In the general case, for some n ∈ N there is a open n n morphism σ : B → N.SetK =Kerσ and C = {ϕ ∈ Bh(B ):ϕ(K) ⊂ K}. Clearly, C n is a closed subalgebra of Bh(B ) and is therefore Arens–Michael. It is also clear that to each ϕ ∈ C there corresponds a unique ψ ∈ Bh(N), acting by the rule ψ(σ(x)) = σ(ϕ(x)) n (x ∈ B ). Moreover, the map θ : C → Bh(N), ϕ → ψ, is a homomorphism of algebras. The morphism σ induces two continuous linear maps ∗ n σ : Bh(N,N) → Bh(B ,N),ψ→ ψ ◦ σ, n n n σ∗ : Bh(B ,B ) → Bh(B ,N),ϕ→ σ ◦ ϕ. Obviously, σ∗ is injective and ∗ n Im σ = ϕ ∈ Bh(B ,N): ϕ(K)=0 . Therefore Im σ∗ is closed. Moreover, a simple verification shows that for any finite set D ⊂ Bn and any neighborhood of zero U ⊂ N we have σ∗ M(σ(D),U) = M(D, U) ∩ Im σ∗ (see (2.1) for the notation). Since any finite subset in N is an image of some finite subset of Bn,wehavethatσ∗ is topologically injective, i.e., it is a topological isomorphism ∗ between Bh(N,N)andImσ . The inverse isomorphism will be denoted κ.Noticethat ∗ the space Bh(N,N) is complete, because it is isomorphic to a closed subspace Im σ of n ∼ n thecompletespaceBh(B ,N) = N . Furthermore, it is not difficult to see that σ∗ is open. Indeed, under the natural n ∼ n n isomorphism Bh(B ,X) = X (X ∈ B-mod)itissenttoσ , which is open as so is σ. −1 ∗ It follows from the definition of the algebra C that C = σ∗ (Im σ ). Since the restric- ∗ tion of an open map to the preimage of any set is open, we have that σ∗|C : C → Im σ is open. It follows from the definition of the homomorphism θ that θ = κ ◦ σ∗|C ; hence θ is continuous and open. Therefore, Bh(N) is isomorphic to the quotient algebra C/Ker θ and is therefore locally m-convex. In view of the aforementioned completeness of Bh(N), this concludes the proof. Lemma 8.3. Suppose R is a ⊗-algebra whose Arens–Michael envelope S is metrizable. Let B be an Arens–Michael algebra and N aFréchet B-R-⊗-bimodule which is strictly finitely generated in B-mod.ThenN admits a structure of a right S-module, which extends the original structure of a right R-module (i.e., such that n · ιR(r)=n · r for all n ∈ N and r ∈ R) and makes it a Fréchet B-S-⊗-bimodule.

Proof. Consider the algebra Bh(N) and endow it, as in the preceding lemma, with the topology of simple convergence. Clearly, the map op m: R → Bh(N) ,m(r)(n)=n · r is an algebra homomorphism and it is not diﬃcult to check that it is continuous. Indeed, for any neighborhood of zero V ⊂ N and any ﬁnite subset D ⊂ N there is a neighborhood of zero U ⊂ R such that D·U ⊂ V, which is equivalent to the inclusion m(U) ⊂ M(D, V ). Therefore m is continuous. By the preceding lemma, Bh(N) is Arens–Michael; hence

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op there exists a unique continuous homomorphism m : S → Bh(N) such that mι R = m. Setting n · s = m (s)(n)fors ∈ S, n ∈ N,wehaveastructureofarightS-module on N extending the original structure of a right R-module and making N a B-S-bimodule. Clearly, the constructed action N × S → N is separately continuous. But S and N are Fr´echet spaces; hence this action is jointly continuous. Therefore N is a Fr´echet B-S-⊗-bimodule.

Corollary 8.4. Suppose R is a ⊗-algebra whose Arens–Michael envelope S is metrizable, and a bimodule M ∈ R-mod-R is strictly finitely generated as a left R-module. Then the S-⊗-bimodule M = S ⊗R M ⊗R S is isomorphic to S ⊗R M in S-mod and, in particular, is strictly finitely generated in S-mod. Proof. Consider the S-R-⊗-bimodule N = S ⊗R M. It follows from Propositions 8.1 and 3.11 that it is strictly finitely generated in S-mod and, in particular, is a Fréchet module. On the other hand, it follows from Lemma 8.3 that on N there is a structure of a right S-⊗-module extending the structure of a right R-⊗-module and making it an S-⊗-bimodule. As the canonical image of R in S is dense, we have the isomorphisms of S-⊗-bimodules, ∼ ∼ M = N ⊗R S = N ⊗S S = N. But, as we mentioned before, N is strictly finitely generated in S-mod.

Corollary 8.5. Suppose A, R, S are ⊗-algebras and g : R → S is an epimorphism of ⊗-algebras. Suppose that on A wehaveastructureofanR-⊗-algebra, and on its Arens– Michael A envelope we have a structure of an S-⊗-algebra such that the pair (ιA,g) is an R-S-homomorphism from A to A. Assume also that A is relatively quasi-free over R. Then: (i) If A is relatively (f2)-quasi-free over R,thenA is relatively (f2)-quasi-free over S; (ii) If (S, g)=(R, ιR), S is metrizable and A is relatively (f1)-quasi-free over R, then A is relatively (f1)-quasi-free over S. 1 ⊗ ⊗ Proof. Suppose ΩRA is a retract of A R M R A in A-mod-A. By Proposition 7.4, ⊗ 1 ⊗ 1 ⊗ the A- -bimodule ΩSA is isomorphic to A A ΩRA A A and is therefore a retract of A ⊗R M ⊗R A, which, in turn, is isomorphic to A ⊗S M ⊗S A. ∼ n ∼ n ∼ n If M = R for some n ∈ N,thenM = (S ⊗R S) = S . In view of the above, this proves (i). To prove (ii), apply Corollary 8.4. Corollary 8.6. If A is a relatively (f2)-quasi-free R-⊗-algebra, then A is relatively (f2)- quasi-free over R.IfA is a relatively (f1)-quasi-free R-⊗-algebra and R is metrizable, then A is relatively (f1)-quasi-free over R.

9. Theorems on stable flatness of Arens–Michael envelopes In this section we shall show, for some finitely generated algebras A, that the canonical homomorphism ιA from A to its Arens–Michael envelope A is a homological epimorphism, i.e., A is stably flat over A (see Definition 6.3). Our main technical tool will be the following theorem, establishing a connection between the notions of relative and “absolute” homological epimorphism. Theorem 9.1. Suppose (f,g): (A, R) → (B,S) is an R-S-homomorphism from an R- ⊗-algebra A to an S-⊗-algebra B. Assume that g is a homological epimorphism and f is a left or right relative homological epimorphism. Also assume that A is projective

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in R-mod and B is projective in S-mod, and that one of the following two conditions is satisfied: (i) S and B are Fréchet–Arens–Michael algebras such that S = R, g = ιR,andA is an algebra of (f1)-finite type; (ii) A is an algebra of (f2)-finite type. Then f is a homological epimorphism. For the proof we need the following lemma. Lemma 9.2. Let A, B be exact categories and F : A → B an additive covariant functor having a left derived functor. Suppose X → Y → Z is an admissible pair in A and that F sends it to an admissible pair in B. Suppose also that two objects from X, Y, Z are F -acyclic. Then so is the third. Proof. The admissible pairs X → Y → Z and FX → FY → FZ give rise to distinguished triangles X → Y → Z → X[1] and FX → FY → FZ → (FX)[1] in D−(A)and D−(B), respectively (see [32, 11.6]). Applying the functor LF to the first one, we have a distinguished triangle in D−(B) which maps to the second one:

LF (X) /LF (Y ) /LF (Z) /LF (X)[1]

/ / / FX FY FZ (FX)[1]

By the assumptions, the ﬁrst two vertical arrows are isomorphisms in D−(B). Hence so is the third one (see Proposition 10.1).

Corollary 9.3. Suppose A, B and F are the same as in the preceding lemma. Assume furthermore that F sends admissible epimorphisms to epimorphisms. Let

0 ← X0 ← X1 ←···←Xn ← 0 be an admissible complex in A which is sent by F to an admissible complex in B and whose objects X1,...,Xn are F -acyclic. Then X0 is also F -acyclic. Proof. When n = 1, there is nothing to prove, and when n = 2 everything follows from Lemma 9.2. Suppose that the assertion is true for complexes of length at most n,where n ≥ 2. We shall show that it is true for complexes of length n + 1. By the assumption, the complexes

d0 dn−2 dn−1 dn 0 ← X0 ←−···←−−− Xn−1 ←−−− Xn ←− Xn+1 ← 0,

Fd0 Fdn−2 Fdn−1 Fdn 0 ← FX0 ←−−···←−−−− FXn−1 ←−−−− FXn ←−−− FXn+1 ← 0

are admissible. Setting K =Kerdn−2 and L =KerFdn−2, we have admissible complexes

d0 d1 dn−2 (9.1) 0 ← X0 ←− X1 ←−···←−−− Xn−1 ← K ← 0,

dn (9.2) 0 ← K ← Xn ←− Xn+1 ← 0,

Fd0 Fd1 Fdn−2 0 ← FX0 ←−− FX1 ←−−···←−−−− FXn−1 ← L ← 0,

Fdn 0 ← L ← FXn ←−−− FXn+1 ← 0.

Since they are admissible, we have, in particular, that K =Cokerdn,andL =CokerFdn.

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Consider the diagram

··· o Fdn−2 oFdn−1 oFdn ··· FXOn−cH1 FXn HH vv HγH δvv HvHv α vv HH β vv HH vv H o{v u L /FK v

Here α =kerFdn−2, β = F (coker dn), γ = F (ker dn−2), δ =cokerFdn,andu and v are uniquely determined by the conditions αu = γ and vδ = β. We have αuvδ = γβ = Fdn−1 = αδ.

Since α is a monomorphism and δ is an epimorphism, we also have uv = 1L.Furthermore, the assumption on F and the admissibility of (9.2) imply that β is an epimorphism. As vδ = β,wehavethatv is also an epimorphism. This and the equality uv = 1 show ∼ L that u and v are inverses of each other and thus FK = L. To ﬁnish the proof, apply the induction assumption to (9.1) and (9.2). Proof of Theorem 9.1. Consider the commutative diagram

· ⊗ ( ) A B / modO-A modO-B (9.3) ( · ) ⊗R A ( · ) ⊗S B / mod-R mod-S ( · ) ⊗R S We need to show that B ∈ mod-A is acyclic relative to ( · ) ⊗A B. Choose a projective resolution

(9.4) 0 ← A ← P0 ← P1 ←···←Pn ← 0

of A in (A, R)-mod-(A, R) such that all the bimodules Pi satisfy condition (f1)incase(i) (resp., condition (f2) in case (ii)). By Proposition 6.6, the complexes (9.5) 0 ← B ← B ⊗A P0 ← B ⊗A P1 ←···←B ⊗A Pn ← 0, (9.6) 0 ← B ← B ⊗A P0 ⊗A B ← B ⊗A P1 ⊗A B ←···←B ⊗A Pn ⊗A B ← 0 are split in LCS. Notice that (9.6) can be obtained from (9.5) by applying the functor ( · ) ⊗A B : mod-A → mod-B. Since the epimorphisms in these categories are exactly the morphisms with dense images, it is easy to see that this functor sends epimorphisms to epimorphisms. Therefore, in view of Corollary 9.3, to finish the proof it suffices to show that the modules B ⊗A Pi (i =0,...,n) are acyclic relative to this functor. Fix an arbitrary i ∈{0,...,n}. Without loss of generality we may assume that Pi = A ⊗R M ⊗R A,whereM ∈ R-mod-R and, in addition, that M is strictly finitely generated in R-mod in case (i), and that M is isomorphic to Rm in R-mod-R in case (ii). Now we shall show that the structure of a right R-⊗-module on the B-R-⊗-bimodule ∼ m N = B ⊗R M extends to a structure of a right S-⊗-module. Indeed, in case (ii), N = B in B-mod-R and the assertion is true for obvious reasons. In case (i), M is strictly finitely generated in R-mod; hence S ⊗R M is strictly finitely generated in S-mod (see Propo- sition 8.1) and therefore N = B ⊗R M = B ⊗S (S ⊗R M) is strictly finitely generated in B-mod (see Remark 8.3). Thus, applying Lemma 8.3, we have that the structure of a right R-⊗-module on N extends to a structure of a right S-⊗-module. Combining this result with the fact that g is a homological epimorphism, we see that N is acyclic relative to the bottom arrow in (9.3). On the other hand, since A ∈ R-mod

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and B ∈ S-mod are projective, the vertical arrows in that diagram are exact functors. Therefore, by Corollary 2.5, the right A-⊗-module N ⊗R A = B ⊗A Pi is acyclic relative to the top arrow in (9.3). In view of the above, this implies that f is a homological epimorphism.

Henceforth all finitely generated algebras and finitely generated modules over them will be assumed to be endowed with the strongest locally convex topology. Corollary 9.4. Let R be a finitely generated algebra and A a finitely generated relatively (f1)-quasi-free R-algebra which is projective as a left R-module. Suppose that A is projective in R-mod.IfιR : R → R is a homological epimorphism, then ιA : A → A is also a homological epimorphism. Proof. Combine Corollary 7.7 with Theorem 9.1.

Corollary 9.5. Let R be a ﬁnitely generated algebra and M an R-bimodule which is ﬁnitely generated and projective as a left R-module. Suppose that TR (M) is projective → → in R-mod.IfιR : R R is a homological epimorphism, then ιTR(M) : TR(M) TR(M) is also a homological epimorphism.

Proof. Notice that since M is projective as a left R-module, TR(M)isprojectiveasa left R-module. Now use Corollary 9.4 and Example 8.2.

Remark 9.1. When R = C, the preceding corollary is Taylor’s result [67] that the embedding of the free algebra Fn in its Arens–Michael envelope is a homological epimorphism (see Remark 4.9 and Remark 7.3).

Remark 9.2. In regard to the proof of the preceding corollary, we remark that the R- bimodule M = R ⊗R M ⊗R R is isomorphic to R ⊗R M in R-mod (see Corollary 8.4) and, therefore, is projective in R-mod. By induction, this easily implies that all its tensor powers M⊗n (see Section 4.2) are also projective in R-mod. Since the analytic tensor ⊗ n algebra TR (M) is the completion of the direct sum of the bimodules M , it is natural to ask whether it is projective in R-mod, i.e., whether one of the conditions of the preceding corollary is redundant. Unfortunately, the analytic tensor algebra has the following drawback, which is not present for the algebraic tensor algebra: if S is an Arens–Michael algebra and N is an S-⊗-bimodule, then the projectivity of N in S-mod (or in mod-S) does not, in general, imply that the analytic tensor algebra TS (N) is projective in S-mod (resp., in mod-S). In particular, if α is an m-localizable automorphism S, then the analytic Ore extension O(C,S; α)=S{t; α} = TS(Sα) (see Propositions 4.9 and 4.11), being a projective (and even free) left S-⊗-module need not be a projective right S-⊗- module. In fact, one can show that the algebra of holomorphic functions on the quantum 2 plane Oq(C ) (see Corollary 5.14) with |q| > 1 is not projective as a right module over its subalgebra generated by the element x (in contrast with a purely algebraic situation). We will not prove this assertion, since we do not need it for our purposes. Corollary 9.6. Let R be a ﬁnitely generated algebra, α: R → R an endomorphism, and δ : R → R an α-derivation. Suppose that their canonical extensions α, δ to R form an m- localizable family. If ιR : R → R is a homological epimorphism, then ιR[t;α,δ] : R[t; α, δ] → O(C, R; α, δ) is also a homological epimorphism. Proof. Set A = R[t; α, δ]. By Theorem 5.17, we have ∼ A = O(C, R; α, δ).

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This implies, in particular, that A is projective (and even free) in R-mod.Inviewof Example 8.1, to finish the proof it suffices to apply Corollary 9.4. Corollary 9.7. Let R be a finitely generated algebra, α: R → R an automorphism, and α its canonical extension to R. Suppose that the family {α, α−1} is m-localizable. If −1 × ιR : R → R is a homological epimorphism, then ιR[t,t−1;α] : R[t, t ; α] → O(C , R; α) is also a homological epimorphism. Proof. In view of Theorem 5.21, the proof is similar to that of the preceding corollary. Notice that in Corollaries 9.4–9.7 we dealt with the situation described in Theorem 9.1, (i). We shall now examine part (ii). Corollary 9.8. Let R be a finitely generated algebra and δ : R → R a derivation. If j : R → Rδ is a homological epimorphism, then ιR[t;δ] : R[t; δ] → O(C,Rδ; δ) is also a homological epimorphism. Proof. The proof is based on Theorem 5.1 and Theorem 9.1, (ii) and is similar to the proof of Corollary 9.6. To apply these results to the concrete algebras from Section 5, we will need some facts about homological epimorphisms of polynomial algebras. An augmented ⊗-algebra will be understood as a ⊗-algebra A with a homomorphism εA : A → C. Homomorphisms between augmented ⊗-algebras are defined in an obvious way. If A is an augmented ⊗-algebra, then we consider C as an A-⊗-module obtained by a change of scalars along εA. Definition 9.1. Suppose f : A → B is a homomorphism of augmented ⊗-algebras. We ⊗ L C → shall say that f is a weak homological epimorphism if the canonical morphism B A C is an isomorphism in Db(B-mod). Remark 9.3. Weak homological epimorphisms were introduced in [48] under the name of “weak localization”. Notice that f : A → B is a weak homological epimorphism if and only if C ∈ A-mod is acyclic relative to the functor B ⊗A ( · ): A-mod → B-mod and the canonical morphism B ⊗A C → C is an isomorphism in B-mod.Thisisprovedby considering the morphisms ⊗ L C → ⊗ C → C B A B A and using arguments similar to the ones used in the proof of the implication (3)=⇒(4) in Theorem 6.3. Remark 9.4. In the “expanded” form, Definition 9.1 means that for some (or, equivalently, for any) projective resolution 0 ← C ← P• in A-mod the complex 0 ← C ← B ⊗A P• is admissible (cf. Remark 6.4). It follows from Theorem 6.7 (4) (see also Proposition 3.6 of [48]) that any homological epimorphism of augmented ⊗-algebras is a weak homological epimorphism. Henceforth we shall use the notion of Hopf ⊗-algebra. Formally, a Hopf ⊗-algebra is defined as a Hopf algebra in a symmetric monoidal category of complete LCS endowed with the functor ⊗ of completed projective tensor product. For details, see [36, 5, 47, 48]; we just remark that the definition of Hopf ⊗-algebra can be obtained from the usual algebraic definition of Hopf algebra by replacing vector spaces with complete LCS, and the symbol ⊗,withthesymbol⊗. We also remark that the Arens–Michael envelope of aHopf⊗-algebra H has a canonical structure of a Hopf ⊗-algebra, which is uniquely determined by the requirement that ιH : H → H be a morphism of Hopf ⊗-algebras; see [48, 6.7]. The next result is Proposition 3.7 from [48].

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Proposition 9.9. Suppose f : U → H is a morphism of Hopf ⊗-algebras with invertible antipodes. Then f is a homological epimorphism if and only if f is a weak homological epimorphism. Next, we recall the deﬁnition of the Koszul complex (see, for example, [6]). Suppose A is a commutative algebra and x =(x1,...,xn)isasetofelementsofA. For any p n n p =0,...,n set Kp(x,A)=A ⊗ C . Choose a basis e1,...,en in C and for any p =1,...,n deﬁne a morphism dp : Kp(x,A) → Kp−1(x,A) by the formula p ⊗ ∧···∧ − k−1 ⊗ ∧···∧ ∧···∧ dp(a ei1 eip )= ( 1) xik a ei1 eik eip . k=1

The sequence of A-modules K•(x,A)=(Kp(x,A),dp)0≤p≤n forms a chain complex, called the Koszul complex of x ∈ An. If A is an augmented commutative algebra and xi ∈ Ker εA for all i =1,...,n,then εA we can also deﬁne the augmented Koszul complex 0 ← C ←−− K•(x,A). Proposition 9.10. Let A be an augmented ⊗-algebra containing the polynomial algebra C[x1,...,xn] as a dense augmented subalgebra. Suppose that for any p =0,...,n−1 there (p) (p) → exist continuous linear operators T1 ,...,Tn : A A such that for any homogeneous polynomial f of degree m we have

(p) 1 ∂f Ti (f)= (i =1,...,n). p + m ∂xi 6 Then the embedding C[x1,...,xn] → A is a weak homological epimorphism.

Proof. Set A0 = C[x1,...,xn], x =(x1,...,xn) and consider the augmented Koszul complex ε (9.7) 0 ← C ←− K•(x,A0). It is well known (see, for example, [6]) that complex (9.7) is exact and is thus a projective resolution of C in A0-mod. Notice that since A0 is dense in A, the canonical morphism ⊗ C → C ⊗ · A A0 is an isomorphism. Therefore, applying the functor A A0 ( ) to (9.7), we have an augmented Koszul complex ε (9.8) 0 ← C ←− K•(x,A). Notice that (9.7) is a dense subcomplex of (9.8). For each p =0,...,n−1 deﬁne an operator hp : Kp(x,A) → Kp+1(x,A) by the formula n ⊗ (p) ⊗ ∧ ∈ ∈ pCn hp(a ω)= Ti (a) ei ω (a A, ω ). i=1

We also deﬁne an operator h−1 : C → A,1C → 1A. It is known (see, for example, [31, 18.7.6 (3)]) that the restrictions of the operators hp to the subcomplex (9.7) yield a contracting homotopy of the latter. As hp is continuous and (9.7) is dense in (9.8), it follows that {hp} is a contracting homotopy of the complex (9.8). Therefore, (9.8) is split in LCS, which means that A0 → A is a weak homological epimorphism (see Remark 9.4). Let g be a ﬁnite-dimensional Lie algebra and h ⊂ g an ideal of codimension 1. Fix some ∼ x ∈ g \ h and set R = U(h). It is easy to see that there is an algebra isomorphism U(g) = R[x; δ], where δ =adx (see the discussion preceding Example 5.1). Identifying U(g)

6 Of course, C[x1,...,xn]isviewedhereasa⊗-algebra relative to the strongest locally convex topology.

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ˆ with R[x; δ], and its Arens–Michael envelope U(g), with O(C,Rδ; δ) (see Theorem 5.1), ˆ we have Hopf ⊗-algebra structures on R[x; δ]andO(C,Rδ; δ); moreover, the canonical ˆ morphism ιR[x;δ] : R[x; δ] → O(C,Rδ; δ) becomes a morphism of Hopf ⊗-algebras (see the discussion preceding Proposition 9.9). ˆ We shall consider Rδ as the closed subalgebra of H = O(C,Rδ; δ) consisting of constant functions. Clearly, Rδ is complemented in H as a subspace; therefore Rδ ⊗ Rδ is canonically identiﬁed with a closed subalgebra of H ⊗ H.Let∆H and SH denote, respectively, the comultiplication and the antipode of H.Noticethat∆H (Rδ) ⊂ Rδ ⊗ Rδ and SH (Rδ) ⊂ Rδ (this immediately follows from the facts that R is a Hopf subalgebra of R[x; δ], ι: R[x; δ] → H is a morphism of Hopf ⊗-algebras, and ι(R)=Rδ). Thus Rδ is a Hopf ⊗-algebra and the canonical homomorphism j = ι|R : R → Rδ is a morphism ⊗ 2 of Hopf -algebras. Notice also that SH = 1H immediately implies that Rδ is a Hopf ⊗-algebra with an invertible antipode. Corollary 9.11. Let g be a ﬁnite-dimensional Lie algebra, h ⊂ g acodimension1 ideal, and R = U(h).Fixsomex ∈ g\h and set δ =adx : R → R. Suppose that j : R → Rδ is a weak homological epimorphism. Then ιU(g) : U(g) → U(g) is a homological epimorphism. Proof. Taking account of Proposition 9.9 and the above discussion, this is a particular case of Corollary 9.8.

Theorem 9.12. Suppose A is any of the following algebras: Oalg Cn n (i) A = q ( ), the multiparameter quantum aﬃne space (where q =(qij)i,j=1 is a multiplicatively antisymmetric (n × n)-matrix such that either |qij|≥1 for all | |≤ i

whereq ˇij = qn−i,n−j, sending xi to xn−i (cf. the proof of Proposition 5.12 (ii)). Therefore, without loss of generality we may assume that |qij |≥1 for all i0. Therefore, α is m-localizable, and to complete the proof, apply Corollary 9.6. (ii) If |qij| = 1 at least for one pair i, j,thenA = 0 (see Proposition 5.23), and there is nothing to prove. If |qij| =1foralli, j, then, using Corollaries 5.22 and 9.7, we can argue as in the proof of (i).

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Oalg Cn → (iii) Set R = q ( ); then A = R[xn+1,σ,δ], where the endomorphism σ : R R and the σ-derivation δ : R → R are uniquely determined by the conditions γ σ(xi)=λixi (1 ≤ i ≤ n),δ(x1)=µx ,δ(xk)=0 (k ≥ 2) (see the proof of Theorem 5.18). Furthermore, the canonical extension of σ and δ to R form an m-localizable family {σ, δ} (ibid.). Now use (i) and Corollary 9.6. (iv) If q =1,thenA = 0 (see Example 3.4), and there is nothing to prove. If q =1, then A satisﬁes the conditions of Theorem 5.18 (see Example 5.5), and everything reduces to (iii). (v) If q = ±1, then A satisﬁes the conditions of Theorem 5.18 (see Example 5.6) and 4 the desired assertion follows from (iii). If q = −1, then A = Oq (C ) (see (5.30)) and everything reduces to (i). Finally, when q =1wehaveA = C[a, b, c, d]. Now apply Proposition 4.3 from [67] (or use (i) with the matrix q consisting of ones). (vi) Set h = Cy, R = U(h)=C[y], and δ =adx : R → R.ThenRδ = C[[y]] (see Example 5.1). Consider the linear operator (0) C → C n → n−1 T1 : [[y]] [[y]], cny cny . n n≥1

(0) Clearly, T1 is continuous and satisfies the conditions of Proposition 9.10. There- fore, j : R → Rδ is a weak homological epimorphism, and the desired assertion follows from Corollary 9.11. (vii) Set h =span{y, z}, R = U(h)=C[y, z], and δ =adx : R → R.Then ∞ ∞ i! R = a = c yizj : a = |c | ρi+j < ∞∀ρ>0 δ ij ρ ij (i + j)! i,j=0 i,j=0 (see Example 5.2). For p =0, 1 consider the linear operators i T (p) : R → R , c yizj → c yi−1zj, 1 δ δ ij ij i + j + p i,j≥0 i≥1,j≥0 j T (p) : R → R , c yizj → c yizj−1. 2 δ δ ij ij i + j + p i,j≥0 i≥0,j≥1 It is not difficult to check that these operators are well-defined and continuous. Indeed, i j ∈ for any a = i,j cijy z Rδ and any ρ>0wehave i (i − 1)! i! |c | ρi+j−1 ≤ |c | ρi+j−1 ≤ ρ−1a , ij i + j + p (i + j − 1)! ij (i + j)! ρ i≥1,j≥0 i≥1,j≥0

(p) ≤ −1 whence T1 (a) ρ ρ a ρ.Furthermore, j i! 2j−1i! |c | ρi+j−1 ≤ |c | ρi+j−1 ij i + j + p (i + j − 1)! ij (i + j)! i≥0,j≥1 i≥0,j≥1 i! ≤ |c | (2ρ)i+j−1 ≤ (2ρ)−1a , ij (i + j)! 2ρ i≥0,j≥1 (p) ≤ −1 (p) (p) and therefore T2 (a) ρ (2ρ) a 2ρ. Thus the operators T1 and T2 are well- deﬁned, continuous, and, clearly, satisfy the conditions of Proposition 9.10. Therefore j : R → Rδ is a weak homological epimorphism, and the desired assertion follows from Corollary 9.11.

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10. Appendix: Derived categories For the convenience of the reader, in this appendix we mention some facts from the theory of derived categories. Notice that most of the standard references [23, 18, 30] only consider the derived categories of abelian categories, whereas the typical categories of functional analysis are not abelian. We shall consider the more general case of exact categories in the sense of Quillen [56]. We shall mostly follow the survey paper by B. Keller [32]. Let A be an additive category. We shall always assume that idempotents split in A; i.e., each morphism p: X → X in A such that p2 = p has a kernel. An exact pair p in A is any sequence X −→i Y −→ Z, such that i =kerp and p =cokeri. Suppose E is some class of exact pairs in A, closed under isomorphisms. Elements of E will be called admissible pairs. A morphism i: X → Y (resp., p: Y → Z)issaidtobean admissible monomorphism (resp., admissible epimorphism)ifitispartofanadmissible p pair X −→i Y −→ Z. An exact category is an additive category A endowed with a class of exact pairs E satisfying the following axioms: Ex0. The identity morphism of a zero object is an admissible epimorphism. Ex1. The composition of two admissible epimorphisms is an admissible epimorphism. Ex1◦. The composition of two admissible monomorphisms is an admissible monomorphism. Ex2. Each diagram Y

p / X Y where p is an admissible epimorphism, can be extended to a pull-back square

p / X Y

p / X Y where p is an admissible epimorphism. Ex2◦. Each diagram i / X Y

X where i is an admissible monomorphism, can be extended to a push-out square i / X Y

i / X Y where i is an admissible monomorphism.

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If A is an exact category, then the dual category A◦ is also exact, provided the p p admissible pairs in A◦ are deﬁned as the pairs X −→i Y −→ Z such that the pair Z −→ Y −→i X in A is admissible.

Example 10.1. Each additive category A canbemadeexactifE is deﬁned as the class of all split exact sequences. The corresponding exact category will be denoted Aspl.We shall then say that A is endowed with the split exact structure.

Example 10.2 (quasi-abelian and abelian categories). Suppose that A is an additive category with kernels and cokernels, and let E denote the class of all exact pairs in A.If, under these assumptions, A satisfies axioms Ex2 and Ex2◦,thenA is said to be quasi- abelian [53, 61] (or, in the original terminology, semi-abelian [58]). Since axioms Ex0, Ex1, and Ex1◦ are automatically satisfied, any quasi-abelian category endowed with a class of all exact pairs becomes an exact category. Notice that many standard categories of functional analysis are quasi-abelian [54]. The examples include: the category LCS of locally convex spaces (LCS), its full subcategory consisting of Hausdorff LCS, the category of Fréchet spaces, the categories of all normed spaces and of all Banach spaces, and others. Notice, however, that the category of all complete LCS is not quasi-abelian [54]. Recall that A is said to be abelian if any monomorphism in it is a kernel, and any epimorphism is a cokernel (see, for example, [39]). Any abelian category is quasi-abelian and therefore is an exact category.

Example 10.3. Suppose A is an additive category with kernels and cokernels, B is an exact category, and : A → B is an additive functor preserving kernels and cokernels. Then it is easy to check that A becomes an exact category if one deﬁnes admissible pairs in A as the pairs whose images under are admissible in B. An important particular case of this construction is when B = Bspl. In this case we denote the exact category obtained by AB.

For a given additive category A the category of all chain complexes in A is denoted C(A). Its full subcategory consisting of complexes bounded on the right (resp., bounded on the left, bounded), i.e., complexes X ∈ Ob (C(A)) such that Xn =0foralln>N (resp., for all nN) holds for some N ∈ Z, is denoted C−(A)(resp., C+(A), Cb(A)). The differential in a chain complex X mapping Xn to Xn+1 will be n A denoted dX . Recall that the homotopy category H( ) is defined as follows: it has the same objects as C(A), and the group of morphisms Hom H(A)(X, Y ) is defined as the quotient of Hom C(A)(X, Y ) by the subgroup of all null-homotopic morphisms. The categories H−(A), H+(A), and Hb(A) are defined similarly. Following the convention, we identify chain complexes with cochain complexes by n n n viewing a chain complex (Xn,dn) as a cochain complex (X ,d ), where X = X−n and n ◦ ∼ ◦ d = d−n. Notice also that there is an isomorphism C(A ) = C(A) , assigning to a complex (Xn,dn)inA◦ acomplex(X¯ n, d¯n)inA, such that X¯ n = X−n and d¯n = d−n−1. − ◦ ∼ ◦ − ◦ ∼ This isomorphism, in turn, gives rise to isomorphisms C (A ) = C+(A) and H (A ) = H+(A)◦. For any cochain complex X in A,letX[1] be the complex such that X[1]n = Xn+1 n − n+1 ∈ Z → A and dX[1] = dX for all n .Iff : X Y is a morphism in C( ), then the morphism f[1]: X[1] → Y [1] is defined by the formula f[1]n = f n+1.Thuswehave a functor X → X[1] from C(A)inC(A), called the suspension functor (or the shift functor). Similarly, one defines the suspension functor from H(A)toH(A).

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Recall that for any morphism f : X → Y in C(A)itscone M(f) is the complex deﬁned as follows: dn 0 n n+1 ⊕ n n X[1] M(f) = X Y ,dM(f) = n n . f dY Each morphism f : X → Y is part of a sequence

f α(f) β(f) (10.1) X −→ Y −−−→ M(f) −−−→ X[1], where α(f): y → (0,y)andβ(f): (x, y) → x. A triangle in H(A) is any sequence of morphisms of the form X → Y → Z → X[1]. A morphism from this triangle to a triangle X → Y → Z → X[1] is a triple of morphisms (u, v, w) which is part of a commutative diagram / / / X Y Z X[1]

u v w u[1] / / / X Y Z X [1] A triangle in H(A)issaidtobedistinguished if it is isomorphic to a triangle of the form (10.1). The category H(A) endowed with the functor X → X[1] and the class of all distinguished triangles is a triangulated category. We do not give here the precise definition of a triangulated category (see, for example, [18] or [30]); we just remark that a triangulated category (C,T) is an additive category C endowed with an automorphism T : C → C and a class of sequences X → Y → Z → T (X), called distinguished triangles, satisfying certain axioms. For the proof of the following standard fact, see, for example, [18] or [30]. Proposition 10.1. Suppose (u, v, w) is a morphism of distinguished triangles in a triangulated category (C,T) such that two of the morphisms u, v, w are isomorphisms. Then so is the third. We continue to assume that A is an exact category. A complex X in A is said to be admissible if for any n ∈ Z the differential dn has a kernel, and the canonical sequence Ker dn → Xn → Ker dn+1 is an admissible pair. A morphism f : X → Y in C(A)is called a quasi-isomorphism if its cone M(f) is admissible. Example 10.4. If A is abelian, then a complex in A is admissible if and only if it is exact, and a morphism f : X → Y in C(A) is a quasi-isomorphism if and only if it induces an isomorphism in cohomology. Example 10.5. Suppose that A is an additive category with kernels and cokernels. Recall that a morphism f : X → Y is said to be strict if the induced morphism Coim f → Im f is an isomorphism. Notice that A is abelian if and only if each morphism in it is strict. If A is quasi-abelian (see Example 10.2), then a complex in A is admissible if and only if it is strictly exact, i.e., when it is exact and each differential dn is a strict morphism [53, 61]. In particular, a complex in the quasi-abelian category LCS is admissible if and only if it is topologically exact [25], i.e., when each dn is an open map from Xn to Ker dn+1 [54]. Example 10.6. For an arbitrary additive category B acomplexX ∈ Ob (C(B)) is admissible in the exact category Bspl if and only if it is split (i.e., null-homotopic). In this case quasi-isomorphisms are precisely homotopy equivalences.

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Example 10.7. As an immediate consequence of the preceding example, we have that in the exact category AB (see Example 10.3) a complex X is admissible if and only if X is split in B, and a morphism of complexes f : X → Y is a quasi-isomorphism if and only if f : X → Y is a homotopy equivalence.

Suppose (C,T)and(C,T) are triangulated categories. A triangular functor from (C,T)to(C,T)isapair(F, α), where F : C → C is an additive functor and α: FT → f g T F is a natural isomorphism such that for any distinguished triangle X −→ Y −→ Z −→h T (X)inC the triangle

Ff Fg αFT(X)◦Fh FX −−→ FY −−→ FZ −−−−−−−→ T (FX)

is distinguished in C. A morphism of triangular functors and a triangular equivalence of triangulated categories are defined in an obvious way. The derived category of an exact category A is defined as the localization of the homotopy category H(A) by the system of all quasi-isomorphisms. In other words, the derived category is a pair (D(A),QA) consisting of a category D(A) and a functor QA : H(A) → D(A) sending quasi-isomorphisms to isomorphisms; also, for any category C and any functor F : H(A) → C sending quasi-isomorphisms to isomorphisms there exists a unique functor G: D(A) → C such that F = G ◦ QA.Moreover,D(A) has a canonical structure of a triangulated category and the functor QA is triangular. The derived category always exists and is unique up to a triangular equivalence. If C is a triangulated category and F : H(A) → C is a triangular functor sending admissible objects to zero, then it sends quasi-isomorphisms to isomorphisms, and the functor G: D(A) → C,whose existence is postulated above, is also triangular. In a similar way (replacing H(A)byH−(A), H+(A), or Hb(A)) one defines derived − − ◦ ∼ categories D (A), D+(A), and Db(A). Notice that there are isomorphisms D (A ) = ◦ ◦ ∼ − ◦ − D+(A) and D+(A ) = D (A) .Moreover,D (A), D+(A), and Db(A)canbeidentified with full subcategories of D(A). For more details, see [32] or (for the abelian case) [18, 30]. Associating with each X ∈ Ob (A) the complex whose degree zero term is X and all other terms are zero, we have a fully faithful additive functor A → C(A). Its compositions with the canonical functors C(A) → H(A)andC(A) → D(A) are also fully faithful functors; for this reason A is customarily identified with the full subcategories of C(A), H(A), and D(A) consisting of complexes concentrated in degree zero. A similar convention is used for C∗(A), H∗(A), and D∗(A), where ∗∈{+, −,b}. Let A, B be exact categories and F : A → B an additive covariant functor. In an obvious way, F gives rise to a triangular functor H−(F ): H−(A) → H−(B). The left derived functor of F is defined as a pair (LF, s), consisting of a triangular functor

LF : D−(A) → D−(B)

and a morphism of triangular functors

− (10.2) sF :LF ◦ QA → QB ◦ H (F ), such that for any other such pair (G, s) there exists a unique morphism of triangular functors α: G → LF with s = sF ◦ (αQA). Right derived functors, as well as derived functors of contravariant functors can be deﬁned by passing to the dual categories. For each functor F : A → B we have the ◦ ◦ ◦ ◦ ◦ ◦ functors F : A → B, F◦ : A → B ,andF◦ : A → B ,actingthesamewayasF ; ◦ ◦ moreover, if F is covariant (resp., contravariant), then so is F◦ ,whereasF and F◦ are contravariant (resp., covariant).

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If F : A → B is an additive covariant functor, then its right derived functor is defined ◦ ◦ + + as the functor RF =L(F◦ )◦ : D (A) → D (B). If F is contravariant, then we set − + ◦ ◦ + − RF =L(F◦)◦ : D (A) → D (B)andLF =L(F ) : D (A) → D (B). A functor F : A → B is said to be exact if it sends admissible complexes to admissible ones. If F is exact, then, as follows from the definition of the derived category, there exists a unique triangular functor F : D(A) → D(B) such that F ◦ QA = QB ◦ H(F ). It is easy to see that the restriction of F to D−(A)(resp.,D+(A)) is the left (resp., right) derived functor of F ; moreover, (10.2) is the identity morphism. Recall that an object P ∈ Ob (A)issaidtobeprojective if for any admissible epimorphism X → Y in A the induced map Hom A(P, X) → Hom A(P, Y ) is surjective. One says that A has enough projectives if for any X ∈ Ob (A) there are a projective object P ∈ Ob (A) and an admissible epimorphism P → X. Example 10.8. Under the assumptions of Example 10.3, assume in addition that the functor : A → B has a left adjoint F : B → A and that B = Bspl. Then for any admissible epimorphism Y → Z in AB the morphism Y → Z is a retraction in B, and therefore for any E ∈ Ob (B)themapHomB(E,Y ) → Hom B(E,Z) is surjective. Since F and are adjoint, we conclude that F (E) is projective in AB. Furthermore, since the composition → F → is the identity morphism of (see, for example, [39, 4.1.1]), we have that for any X ∈ Ob (A) the morphism F (X) → X is a retraction in B, or, equivalently, that F (X) → X is an admissible epimorphism. Therefore, AB has enough projectives. Moreover, an object X is projective if and only if the canonical morphism F (X) → X is a retraction. Suppose that A is an exact category with enough projectives. Then for any complex X ∈ Ob (C−(A)) there are a complex P ∈ Ob (C−(A)) consisting of projectives and a quasi-isomorphism P → X. Any two complexes P, P with this property are homotopy equivalent (and therefore quasi-isomorphic). Any such complex P together with a quasi- isomorphism P → X is called a projective resolution of X.LetP be the full subcategory of A consisting of projectives. Then, as follows from the above discussion, the canonical functor D−(P) → D−(A) is an equivalence of triangulated categories. On the other hand, any acyclic complex consisting of projectives is homotopy equivalent to a zero complex; − − − hence QP : H (P) → D (P) is an equivalence. The left derived functor LF : D (A) → D−(B) can now be defined as the composition of the functors

− − ∼ − ∼ − − H (F ) − QB − D (A) ←− D (P) ←− H (P) → H (A) −−−−→ H (B) −−→ D (B). Notice that having enough projectives is a rather strict sufficient condition for the existence of a left derived functor. For much more general conditions, see [32] or (in the abelian case) [18, 30]. An object I ∈ Ob (A)issaidtobeinjective if it is projective as an object of A◦.As follows from the above, for the existence of the right derived functor of a covariant functor or of the left derived functor of a contravariant functor it suffices that A has enough injectives, i.e., for any X ∈ Ob (A) there is an admissible monomorphism X → I,whereI is injective. Similarly, for the existence of the right derived functor of a contravariant functor it suffices that A has enough projectives. If the category B is quasi-abelian, then for any n ∈ Z the cohomology functor Hn : C−(B) → B sends quasi-isomorphisms to isomorphisms. This is proved the same way as for abelian categories, using the long cohomology exact sequence (see [34]).7 Therefore, the functor Hn extends to a functor D−(B) → B, also denoted Hn.The

7In fact, a simple argument shows that this is true in any exact category B with kernels and cokernels. We skip the proof, since we do not need such a level of generality.

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composition H−n ◦ LF : D−(A) → B (and its restriction to A) is called the n-th classical left derived functor of F and is denoted LnF . Similarly, the n-th classical right derived functor RnF : D+(A) → B is deﬁned as the composition Hn ◦ RF . Example 10.9. Suppose that A has enough projectives. Fix some Y ∈ Ob (A)and consider the functor Hom A( · ,Y): A → Ab, where Ab is the (abelian) category of abelian groups. Its right derived functor is denoted n RH o m A( · ,Y), and its n-th classical right derived functor is denoted Ext A( · ,Y). The functor RHom A( · ,Y) extends to a bifunctor − + + RH o m A : D (A) × D (A) → D (Ab) (the proof is the same as in the case of abelian categories; see, for example, [23, I.6] or [30, 1.10]). Notice that we have a natural isomorphism n ∼ Ext A(X, Y ) = Hom D(A)(X, Y [n]) (ibid. or [18, III.6.15]). Suppose A, B,andC are exact categories and F : A → B and G: B → C are additive covariant functors. Suppose that F , G,andGF have left derived functors. It follows from the deﬁnition of the derived functor that there is a morphism of functors c:LG ◦ LF → L(GF ) making the following diagram commute:

(LG)sF / − LG ◦ LF ◦ QA LG ◦ QB ◦ H (F )

− cQA sG(H (F )) / − − L(GF ) ◦ QA QC ◦ H (G) ◦ H (F ) sGF If c is an isomorphism, then we say that the theorem on the derived functor of the composition holds for F and G. For some of the suﬃcient conditions guaranteeing this, see [32] or (in the abelian case) [18, 30]. In particular, this theorem holds if A and B have enough projectives, and F sends projectives to projectives. Suppose A and B are exact categories and F : A → B is an additive covariant functor having a derived functor. As before, we identify A and B with the full subcategories of D−(A) and, respectively, D−(B), and consider F and LF as functors from A to D−(B). Then the morphism (10.2) gives rise to a morphism of functors LF → F . An object X ∈ Ob (A)issaidtobe(left)F -acyclic if LF (X) → F (X) is an isomorphism in D−(B). Example 10.10. A projective object is acyclic relative to any functor (see [32]). Example 10.11. If F is exact, then any object of A is F -acyclic. Example 10.12. Suppose A is a ⊗-algebra and Y is a left A-⊗-module. Let Vect denote the abelian category of vector spaces. Then Y is ﬂat (in the sense of [27, 25]) if and only if Y is acyclic relative to the functor X ⊗A ( · ): A-mod → Vect for any right A-⊗-module X. The author is grateful to A. Ya. Helemskii for his attention to this work. References

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Russian Peoples’ Friendship University, 117198 Moscow, Russia E-mail address: [email protected]; [email protected]

Translated by ALEX MARTSINKOVSKY

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