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Continuous and Smooth Envelopes of Topological Algebras

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Continuous and Smooth Envelopes of Topological Algebras CONTINUOUS AND SMOOTH ENVELOPES OF TOPOLOGICAL ALGEBRAS S.S.Akbarov December 8, 2020 arXiv:1303.2424v11 [math.FA] 28 Feb 2017 2 § 0 Geometries as categorical constructions Observation tools and visible image. The part of mathematics that studies the constructions on the objects called manifolds (or varieties) can be divided into four domains: – algebraic geometry (which can be perceived as a science studying the structure generated on algebraic varieties M by the algebra (M) of polynomials), P – complex geometry (where the algebras (M) of holomorphic functions on complex manifolds M play the same role), O – differential geometry (with the algebras (M) of smooth functions on smooth manifolds M), E – topology (where the algebras (M) of continuous functions on topological spaces M can be considered as structure algebras). C The obvious parallels between these disciplines inspire an idea that there must exist a universal scheme inside mathematics that explains these similarities and allows to discuss the differences in formal terms. Such a scheme indeed exists, and the idea leading to it is borrowed from physics and can be expressed in the formula: the visible picture depends on the observation tools. As an example, in astronomy the visible image of an object under study that appears in the observer’s mind when he uses optical telescope differs from what he sees with his own eyes, or when he uses radio telescope, or X-ray telescope, etc. It turns out that with certain understanding of the terms “observation tools” and “visible image” in math- ematics one can form a general view at least on the last three disciplines in this list, – complex geometry, differential geometry and topology, – and they will be reflections of one common reality, the pictures that ap- pear as results of the choice of a concrete set of tools. This leads to an intriguing picture, where it becomes possible to compare these “geometries as disciplines”, to find common features, differences, generalizations, new examples, and so on. It is convenient to assume that the common reality which these geometries reflect is some, enough wide, category of topological associative algebras, for instance, the category SteAlg of stereotype algebras1 (possibly, with some supplementary structures, like involution). Then for the formalization of the scheme of observation, which we discuss here, the following two agreement are sufficient: 1) by the observation tools one means morphisms of a given class Φ in this category, 2) the object in study (an algebra) A and its visible image (another algebra) E are connected through a natural morphism A E (like an original and a photo), and the class Ω of such morphisms, called “class of representations” is→ given from the very beginning. 2 Ω Under these assumptions the “visible image” E of an object A can be interpreted as its envelope EnvΦ A, generated by the class of tools (morphisms) Φ in the given class of representations (morphisms) Ω. Every concrete choice of classes Φ and Ω give birth some “projection” of functional analysis (which is understood here as a theory of topological algebras) into geometry (understood as a theory of functional algebras in “generalized sense”). Historically one of the first examples of an envelope of a topological algebra was the Arens-Michael envelope, introduced by J. L. Taylor in [63]. This construction was studied in detail by A. Yu. Pirkovaskii in his researches on “noncommutative complex geometry” [52]. In the work [3] it was applied by the author to a generalization of Pontryagin duality to a class of (not necessarily commutative) complex Lie groups. Up to the further terminological corrections (see [4]), these investigations can be considered as an application of this scheme of observation with a result as a categorical construction of complex geometry. Later Yu. N. Kuznetsova obtained analogous results in [41], where she constructed a variant of duality theory in topology with homomorphisms into C∗-algebras as observation tools. Again, up to later correction in [4] and in this paper, the results of [41] can be interpreted as a way for categorical construction of topology. In the present paper we suggest an analogous way for categorical construction of differential geometry. We describe here a construction of smooth envelope of stereotype algebra, we study its properties and build a generalization of Pontryagin duality for some class of real Lie groups. Pictorially the results of this paper and the papers we mentioned here can be presented in the following table: 1See definition of stereotype algebra below on page 52. 2See definition of envelope on page 15. 0. GEOMETRIES AS CATEGORICAL CONSTRUCTIONS 3 § Discipline Complex Differential Topology geometry geometry Key Algebra O(M) Algebra E(M) Algebra C(M) example of holomorphic functions of smooth functions of continuous functions of visible on a complex on a smooth on a topological image manifold M manifold M space M Observation Homomorphisms Differential Involutive tools into Banach involutive homomorphisms Φ algebras homomorphisms into C∗-algebras into C∗-algebras with joined self-adjoint nilpotent elements Class of Dense Dense Dense representations epimorphisms epimorphisms epimorphisms Ω Visible Holomorphic envelope Smooth envelope Continuous envelope image of an object A EnvO A EnvE A EnvC A Key EnvO A = O(M) EnvE A = E(M) EnvC A = C(M) representation for a subalgebra for a subalgebra for a subalgebra A ⊆O(M) A ⊆E(M) A ⊆C(M) ⋆ EnvO ⋆ ⋆ EnvE ⋆ ⋆ EnvC ⋆ O (G) ✤ / Oexp(G) E (G) ✤ / EnvE E (G) C (G) ✤ / EnvC C (G) Reflexivity O ❴ O ❴ O ❴ diagram ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ❴ ❴ ❴ O(G) o ✤ Oexp(G) E(G) o ✤ K∞(G) C(G) o ✤ K(G) EnvO EnvE EnvC ⋆ F b ⋆ F b ⋆ F b Reflexivity O (G) ✤ / O(G) E (G) ✤ / E(G) C (G) ✤ / C(G) diagram ⋆ O ❴⋆ ⋆ O ❴⋆ ⋆ O ❴⋆ for ❴ ❴ ❴ commutative ⋆ b ⋆ b ⋆ b O(G) o ✤ O (G) E(G) o ✤ E (G) C(G) o ✤ C (G) groups F F F Complex geometry. The second column in this table was chronologically first, so it is logical to start the explanation with it. Complex geometry studies complex manifolds with supplementary structures like Hermitian metrics, or connextions, or curvature, etc. [28]. Usually a complex manifold is defined by its sheaf of holomorphic functions, and for a mathematician with functional-analytic mentality this construction poses a psychological problem. However among complex manifolds there is a subclass, which does not require the notion of sheaf for its description, its objects are called Stein manifold [27],[62]. For our purposes they are good, since, first, they visually illustrate our idea, and, second, the passage from them to the general case (of a manifold defined by sheaf) in the category theory seems to be easy, since a sheaf itself is a simple categorical construction (however, the formal generalization was not constructed yet). A Stein manifold M is defined by its algebra (M) of holomorphic functions (and respectively, all the supplementary structures, like metrics on M, also canO be defined as constructions on (M)). As a corollary, it O 4 is logical to consider the algebra (M) as a key example of visible image in complex geometry. In this science it has been noticed long ago (seeO [52], [3], [4]) that (M) can be restored from a (sufficiently wide) subalgebra A (M) with the help of the system of Banach quotientO algebras A/p for various continuous submultiplicative seminorms⊆ O p on A – then (M) is just a projective limit of A/p: O (M) = lim A/p. (0.1) O ←−p Of course, this holds not for any A (M): if we want (0.1) to be true, A must be sufficiently close to (M). An important example was found by⊆ O A. Yu. Pirkovskii [52]: if M is an affine algebraic manifold, thenO (M) can be restored in this way by the subalgebra A = (M) of polynomials on M. O Up to the details insufficient in the first approximation,P the equality (0.1) can be understood as the propo- sition that (M) is an envelope of A in the class Ω = DEpi of dense epimorphisms3 of the category SteAlg of stereotpye algebrasO with respect to the clas Φ = Mor(SteAlg, BanAlg) of homomorphisms into Banach algebras: (M)= EnvDEpi A. (0.2) O Mor(SteAlg,BanAlg) The projective limit on the right side of (0.1) is called the Arens-Michael envelope, and for the arbitrary stereotype algebras A it doesn’t coincide with the obect on the right side of (0.2), that is why in [4] the author DEpi introduced a new term for EnvMor(SteAlg,BanAlg) A, the holomorphic envelope. In the table at the page 3 this envelope is denoted by EnvO A. Up to this notation, the equality (0.2) is the proposition in the sixth cell of the second column of the table. We call it key representation having in mind that it describes the mechanism of discerning objects as a key example in this science, the algebra (M). The main result in [3] is the diagram in the next to last cellO of the column: Env ⋆(G) ✤ O / ⋆ (G) (0.3) O Oexp O ❴ ⋆ ⋆ ❴ (G) o ✤ exp(G) O EnvO O Here (G) is the algebra of holomorphic functions on a complex Lie group G, exp(G) the algebra of holomorphic O ⋆ ⋆ O functions of exponential type on G, (G) and exp(G) the dual convolution algebras of analytic functionals, and ⋆ the operation of passing to theO dual stereotypeO space4. Diagram (0.3) shows that the objects in its corners satisfy some reflexivity conditions. For example, if we ⋆ denote the composition ⋆ Env by then (G) and exp(G) become reflexive with respect to : ◦ O O O b A ∼= A.
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