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Multiplier operator algebras and applications

David P. Blecher† and Vrej Zarikian

Department of , University of Houston, Houston, TX 77204; and Department of Mathematics, University of Texas, Austin, TX 78712

Edited by Richard V. Kadison, University of Pennsylvania, Philadelphia, PA, and approved November 14, 2003 (received for review August 18, 2003)

The one-sided multipliers of an X are a key to Mᒉ(X) may be described as consisting of the linear maps T : X 3 ‘‘latent operator algebraic structure’’ in X. We begin with a survey X satisfying of these multipliers, together with several of the applications that T͑x͒ x they have had to operator algebras. We then describe several new Ͱͫ ͬͰ Յ Ͱͫ ͬͰ [1] results on one-sided multipliers, and new applications, mostly to y y one-sided M-ideals. (the precise statement may be found in Theorem 3.2). Because 1. Introduction the criterion in Eq. 1 is so simple, it is easy to see that left multipliers remain left multipliers under most of the basic or every closed linear subspace X of B(H), for a functional analytic operations: subspaces, tensor products, du- H, D.P.B. has introduced two subalgebras F ality, interpolation, and so on. This is extremely useful in our

Aᒉ͑X͒ ʚ Mᒉ͑X͒ ʚ B͑X͒, theory and applications. This note is addressed to a general reader with a knowledge where B(X) is the space of bounded linear maps on X. The first, of ; the background facts needed may be found Mᒉ(X), is an (we will define this term formally in the first six chapters of ref. 4. in the next section). The second, Aᒉ(X), is a C*-algebra and, indeed, is a W*-algebra (that is, it is an ‘‘abstract von Neumann 2. Operator Spaces and Operator Algebras algebra’’) if X is weak* closed in B(H). The elements of these We begin by recalling a few basic definitions from operator space algebras are maps on X, called left multipliers of X. Of course theory. there is a matching theory of right multipliers, but we shall avoid A concrete operator space is a linear subspace X of B(H), for mentioning this because it is essentially identical (with appro- a Hilbert space H. For such a subspace X we may view the space ϫ priate modifications such as replacing ‘‘left’’ by ‘‘right’’ and Mm,n(X)ofm n matrices with entries in X as the corresponding (n) (m) ‘‘column’’ by ‘‘row,’’ and so on). By a one-sided multiplier we subspace of B(H , H ). Thus, this matrix space has a natural ʦ ގ mean of course a left or a right multiplier. , for each m, n .Anabstract operator space is a vector ʈ ⅐ ʈ ʦ ގ The following list gives the main areas where the theory of space X with a norm n on Mn(X), for all n , which is multiplier algebras is being applied currently. We will discuss completely isometric to a concrete operator space. The latter 3 each point in greater detail below. term, complete , refers to a T : X Y such that (i) Most importantly perhaps, the algebras Aᒉ(X) and Mᒉ(X) are key to ‘‘latent operator algebraic structure’’ in the operator ʈ͓T͑x ͔͒ʈ ϭ ʈ͓x ͔ʈ space X. We discuss this in further detail in sections 3 and 4. ij n ij n

(ii) These algebras allow one to define and develop a non- for all n ʦ ގ,[xij] ʦ Mn(X). A complete contraction is defined commutative M-ideal theory; this is treated in sections 5 and 6. similarly, simply replacing ‘‘ϭ’’ by ‘‘Յ’’ in the last centered We recall for now that ‘‘classical M-ideals’’ are a basic tool for equation. There is also a celebrated theorem of Ruan, charac- Banach spaces, and they have an extensive and useful theory (1). terizing operator spaces in terms of two simple conditions on the

We develop a ‘‘one-sided,’’ or noncommutative, generalization matrix norms. The interested reader should consult ref. 5 for this MATHEMATICS of M-ideals, which we call left and right M-ideals, and we show and for other facts listed in this section, or the texts (6, 7). D.P.B. that this class includes examples that are of interest to operator is also completing a text on related matters with Le Merdy. algebraists. We then are able to extend much of the classical We saw above that if X is an operator space, then so is Mm,n(X). M-ideal theory to our setting. In particular, we will reserve the symbols Cn(X) and Rn(X) for (iii) For many key examples, the spaces Aᒉ(X) and Mᒉ(X)do Mn,1(X) and M1,n(X), respectively. consist precisely of the maps on X of most interest. For example, Every operator space X has a dual object that is also an if X is a C*-module (see, for example, refs. 2 and 3 for the operator space. Namely, we write X* for the dual definition and basic theory of these objects), then these spaces of X, with the norm of a matrix [fij] ʦ Mn(X*) given by consist of the maps that C*-algebraists are interested in [see ʈ͓ ͔ʈ ϭ ͕ʈ͓ ͑ ͔͒ʈ ͓ ͔ ʦ ͑ ͑ ͖͒͒ Example 4.1 iii]. Many of our results on left multipliers and right fij n sup fij xkl : xkl Ball Mn X . M-ideals may then be viewed as generalizations of important We call X*, equipped with this ‘‘operator space structure,’’ a dual facts about C*-modules and their ‘‘dual variant’’ W*-modules. operator space. An early result in operator space theory, due to Thus, the multiplier algebras facilitate a generalization of some D.P.B. and V. I. Paulsen, and independently to E. G. Effros and C*- and W*-module techniques to operator spaces. We will not Z. J. Ruan, states that the canonical map from X into X** is a however emphasize this aspect in this note. complete isometry. (iv) Left multipliers work admirably in settings involving dual Every Banach space X may be made into an operator space in spaces, unlike some earlier techniques. We will amplify this point canonical ways. The simplest of these is called MIN(X); this is at the end of sections 4 and 6. obtained by assigning the smallest matrix norms {ʈ ⅐ ʈn}nՆ2 with The notion of left multipliers above, and the corresponding algebras Aᒉ(X) and Mᒉ(X), are best viewed as a sequence of equivalent definitions, some of which we will meet in sections 3 This paper was submitted directly (Track II) to the PNAS office. and 4. No one is best for all purposes; each seems good for some †To whom correspondence should be addressed at: Department of Mathematics, University purposes and unsuitable for others. However, one of these of Houston, Houston, TX 77204-3008. E-mail: [email protected]. definitions appears to be ‘‘most useful.’’ Namely the unit ball of © 2004 by The National Academy of Sciences of the USA

www.pnas.org͞cgi͞doi͞10.1073͞pnas.0305296101 PNAS ͉ January 20, 2004 ͉ vol. 101 ͉ no. 3 ͉ 727–731 Downloaded by guest on September 26, 2021 respect to which X becomes an operator space (note that we do Mᒉ(X): namely, as the linear maps T : X 3 X such that there exists not mention n ϭ 1 here; this is just the usual norm on X). some completely isometric embedding X B(K), for a Hilbert Another way of describing MIN(X) is to embed X isometrically space K, and an operator a ʦ B(K), with into a commutative C*-algebra C(K) (this may be done with K ϭ Ball(X*) for example, using the Hahn-Banach theorem). Then T͑x͒ ϭ ax, x ʦ X. MIN(X) is simply X endowed with the canonical matrix norms ʈ ⅐ ʈ inherited from the C*-algebra M (C(K)) Х C(K, M ). A very We define the ‘‘multiplier norm’’ of T to be the infimum of the n n n ʈ ʈ nice situation occurs when a given ‘‘noncommutative͞operator quantities a for all K,a as above. Finally, the matrix norms on ϭ space theory,’’ applied to MIN(X) spaces, is just the associated Mᒉ(X) are specified by viewing a matrix t [Tij]inMn(Mᒉ(X)) ϭ classical theory. We will see an illustration of this in section 5; as a map Lt from Cn(X) Mn,1(X) to itself, via the formula ϭ ¥ right M-ideals of MIN(X) turn out to be precisely the classical Lt([xi]) [ jTijxj]. It turns out that Lt is also a left multiplier, and ʈ ʈ M-ideals of X. we define [Tij] Mn(Mᒉ(X)) to be the ‘‘multiplier norm’’ of Lt. A concrete operator algebra is a subalgebra A of B(H). For simplicity in this exposition, we shall always assume that A also Proposition 3.1. For any operator space X, Mᒉ(X) is an operator has an identity of norm 1. By an (abstract) operator algebra we will algebra. mean a unital algebra that is also an operator space, which is The last result may be proved in several ways. In fact, it is more completely isometrically homomorphic to a concrete operator or less obvious from the equivalent definition of left multipliers algebra. There is an operator algebraic version of Ruan’s theo- in terms of the noncommutative or injective rem, due to D.P.B., Z. J. Ruan, and A. M. Sinclair (8): envelope (13, 15). Not surprisingly, one may also prove Propo- sition 3.1 by using Theorem 2.1. However, one of the attractive Theorem 2.1. Up to completely isometric homomorphism, the features of the left multiplier approach is that it automatically abstract operator algebras are precisely the unital algebras A,which yields theorems such as 2.1 as corollaries and, moreover, allows ʈ ʈ ϭ ʈ ʈ Յ ʈ ʈ ʈ ʈ are also operator spaces, satisfying 1 1 and ab n a n b n, for a relaxation of some of the hypotheses of such characterization ʦ ގ ʦ all n and a, b Mn(A). results. This was the main motivation of D.P.B. for the intro- As noted later, this theorem may be proved easily from the duction of operator space multipliers (13). Thus, for example, theory of left multipliers. Operator algebras may be regarded as Theorem 2.1 follows in just a few lines by combining Proposition the noncommutative function algebras. Indeed, note that every 3.1 with the following result (see the proof of theorem 1.11 in ref. function algebra or is an operator algebra (being 17 for details; this deduction was independently noticed by V. I. a subalgebra of a commutative C*-algebra). A main tool for Paulsen). studying a general operator algebra A is Arveson’s noncommu- tative Shilov boundary (9–12). This is, loosely speaking, the Theorem 3.2. [Blecher, Effros, and Zarikian (16)] A linear map smallest C*-algebra containing A: it is a quotient of any other T : X 3 X on an operator space X is in Ball(Mᒉ(X)) if and only if C*-algebra containing A. Thus, for the A ϭ A(ބ), for example, the smallest C*-algebra containing A is C(ޔ), the Txij xij continuous functions on the circle. More generally, for an Ͱͫ ͬͰ Յ Ͱͫ ͬͰ, yij yij operator space X, the noncommutative Shilov boundary may be viewed as a smallest C*-module (or ‘‘ternary of operators’’) all n ʦ ގ and [xij], [yij] ʦ Mn(X). containing X completely isometrically. Because we are aiming at The norms in the last centered equation are just the norms in an elementary presentation of our subject, we will not give a M2n,n(X). We remark that later simplifications of our original more precise definition of this ‘‘boundary,’’ referring the inter- proof of this result were given by V. I. Paulsen (7), W. Werner, ested reader to refs. 12 and 13 for more details. However, it is and D.P.B. (In fact, Theorem 3.2 was inspired by a similar in the background and will be mentioned, in passing, from time theorem of W. Werner in an ordered context. Later, W. Werner to time. (14) showed how one may view operator space multipliers within 3. Multipliers and Algebraic Structure his ordered context and also how Theorem 3.2 may be deduced from his earlier result.) With the extra structure consisting of the additional matrix The condition in Theorem 3.2 is purely in terms of the vector norms on an operator algebra, one might expect to not have to space structure and matrix norms on X; and yet it often encodes rely as heavily on other structure, such as the product. Indeed ‘‘operator algebraic structure.’’ For example, if A is an operator one may ask questions like the following: Can one recover the algebra (with an identity of norm 1), then Mᒉ(A) Х A completely product on an operator algebra A from its norm (or matrix isometrically homomorphically. In fact, the ‘‘regular represen- norms)? Or could one describe the right ideals of A without tation’’ ␭ : A 3 B(A) defined by ␭(a)(b) ϭ ab is a completely referring to the product? The answers to these kinds of questions are essentially in the affirmative, as we shall see in the sequel, the isometric homomorphism onto Mᒉ(A). key being the operator space multipliers mentioned section 1. From this last fact, we can provide a recipe to recover the These objects were introduced by D.P.B. in 1999 (see ref. 13) and product in an operator algebra. Let us suppose that A is an independently by W. Werner but in an ‘‘ordered’’ context (see operator algebra with a forgotten product. For simplicity, let us ref. 14, for example). Soon after the main results in ref. 13 were first suppose that we do remember the identity element e. Form ␪ 3 ␪ ϭ found, Blecher and Paulsen (15) established an alternative Mᒉ(A) using Theorem 3.2, and define : Mᒉ(A) A by (T) T(e). Then it follows from the fact in the last paragraph that the approach to one-sided multipliers. A year later, Blecher, Effros, Ϫ Ϫ and Zarikian (16) found the very practical criterion shown in Eq. forgotten product on A is ab ϭ ␪(␪ 1(a)␪ 1(b)), for a, b ʦ A.If 1. Each of these contributed to the growing list of equivalent we have forgotten the identity element e too, then one may only definitions of one-sided multipliers. hope to recover the product modulo the ‘‘unitary subgroup’’ of As we said in section 1, for any operator space X we have two A, but this may be done in a similar manner to the previous case subalgebras Aᒉ(X) ʚ Mᒉ(X) ʚ B(X). These can be defined in (see the remarks on p. 316 of ref. 16 for details). terms of the ‘‘noncommutative Shilov boundary’’ (13) or in Of course, once one has such a recipe for recovering the terms of the ‘‘injective envelope’’ (15). However, for the sake of product, one would expect Banach–Stone-type theorems for a simpler exposition we will not emphasize these approaches but, complete between operator algebras to be more or instead, will begin with the following equivalent definition of less immediate. D.P.B. has discussed this elsewhere, and of

728 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0305296101 Blecher and Zarikian Downloaded by guest on September 26, 2021 course such results all have their origin in Kadison’s well known decomposition, or Murray–von Neumann equivalencies, in Banach–Stone theorem for isometries between C*-algebras (18). Aᒉ(X) will have consequences for the structure of X. At the very In summary, we are clearly beginning to substantiate our least, the theorem implies that for a X, the algebra statement that left multipliers are a key to ‘‘operator algebraic Aᒉ(X) will have lots of orthogonal projections (if it is nontrivial) structure.’’ Another fact along these lines is that the operator and that these projections may be used in the ways that von module actions (in the sense of ref. 19) of an operator algebra A Neumann algebraists and operator theorists are familiar with. on an operator space X are in a bijective correspondence with These projections are called left M-projections on X, and they completely contractive unital homomorphisms ␪ : A 3 Mᒉ(X). form a basic ingredient in the ‘‘noncommutative M-ideal’’ theory Indeed, if A is a C*-algebra, then the correspondence is also with presented in the last two sections. unital *-homomorphisms from A into Aᒉ(X). This result, origi- In the remainder of this section we continue to discuss duality. nally due to Blecher (see ref. 13, and see ref. 15 for an alternative We recall that operator algebra questions often come in two approach), also follows in just a few lines from Theorem 3.2. varieties, a norm͞C*-algebraic type or a weak*͞von Neumann algebraic type. One often proves a theorem of the first type by first 4. The C*-Algebra Aᒉ(X) proving an analogous result of the second type, and then deducing As was the case for left multipliers, there are several equivalent the desired result by going to the second dual. For function algebras definitions of Aᒉ(X). For example, it may be defined to be the and algebras of operators there is often a technical obstacle to this ‘‘diagonal C*-algebra’’ of Mᒉ(X); that is, the span of the Her- approach; namely, that the Shilov boundary (or injective envelope) mitian elements in that operator algebra. Alternatively, although rarely works well in tandem with duality. Although operator space we will not use this formulation, Aᒉ(X) may be defined to be the multipliers may be defined in terms of the noncommutative Shilov set of maps T : X 3 X for which there is a map S : X 3 X with boundary (or injective envelope), they behave extremely well with ͗ ͘ ϭ ͗ ͘ ʦ respect to duality. This is mostly because of the existence of the Tx, y x, Sy , x, y X, criterion in Theorem 3.2. For example, using this criterion it is easy where the ‘‘inner product’’ in the last formula is the operator- to see that valued one inherited from the inner product on a noncommu- T ʦ Ball͑Mᒉ͑X͒͒ if and only if T** ʦ Ball͑Mᒉ͑X**͒͒. [2] tative Shilov boundary of X.IfH is a Hilbert space for example, and if Hc is H thought of as a fixed column in B(H), then the inner Using duality properties of multipliers, one may prove versions product above on X ϭ Hc may be taken to be the usual inner of many of the basic theorems concerning operator algebras and product. A similar remark holds for C*-modules, which yields the their modules that are appropriate to dual spaces and the weak* facts stated next. . For example, the left multiplier approach and Theo- rem 3.2 were key to the first proof (see ref. 17) of the following Example 4.1. (i) For a Hilbert space H, Mᒉ(Hc) ϭ Aᒉ(Hc) ϭ B(H) ‘‘nonselfadjoint version’’ of Sakai’s characterization of von Neu- (see the notation above). mann algebras: (ii) ForaC*-algebra A, Aᒉ(A) is the usual C*-algebra multiplier algebra M(A) of A, whereas Mᒉ(A) is the usual left multiplier Theorem 4.3. (Le Merdy–Blecher) An operator algebra A is com- algebra LM(A)(see 3.12 in ref. 20). pletely isometrically isomorphic (via a weak* homeomorphism) to (iii) Generalizing examples i and ii, ifYisaC*-module, then a weak* closed unital subalgebra of B(H), if and only if A is a dual Mᒉ(Y) is the space of bounded module maps on Y, whereas Aᒉ(Y) operator space. is the C*-algebra of adjointable maps in the usual C*-module sense (see ref. 3). 5. Right Ideals in Operator Spaces These examples show that in certain cases the spaces Aᒉ(X) and The one-sided multipliers above are intimately related to the Mᒉ(X) do consist precisely of the maps on X of most interest. generalization of the classical M-ideal notion to ‘‘right M-ideals’’ However, we must issue a warning: for a ‘‘general space,’’ both in operator spaces made by Blecher, Effros, and Zarikian (16). MATHEMATICS Mᒉ(X) and Aᒉ(X) may be one-dimensional! In fact, the preva- First, we recall the classical definitions, due to Alfsen and Effros lence of spaces with only ‘‘trivial’’ one-sided multipliers is (1, 21). These are based on the notion of the ϱ direct sum of two disappointing, until one remembers that this is simply reflecting Banach spaces; namely, the algebraic sum of the spaces, with the the lack of any trace of operator algebra or operator module norm ʈ(x, y)ʈ ϭ max{ʈxʈ, ʈyʈ}. An M-projection on a Banach space structure in such spaces. Thus, one should not expect the classical X is an idempotent linear map P : X 3 X such that the map from 1 or noncommutative L spaces to have any nontrivial one-sided X 3 X ᮍϱ X given by multipliers. This is also related to a lack of ‘‘noncommutative [P͑x͒, ͑I Ϫ P͒͑x͒͒ [3͑ۋM-structure,’’ as we shall see later. On the other hand, one would x expect L1 to have ‘‘noncommutative L-structure.’’ Interestingly, classical and noncommutative Lp spaces for p Ն 2 may have is an isometry. This is simply saying that X ϭ P(X) ᮍϱ (I Ϫ P)(X). nontrivial left multipliers and nontrivial ‘‘noncommutative M- The range of an M-projection is a called an M-summand.A structure.’’ This may be seen by first noting that by Example 4.1 closed subspace J of a Banach space X is called an M-ideal if JЌЌ i above, L2 has plenty of left multipliers. So does Lϱ by the fourth is the range of an M-projection on X**. These objects have an last paragraph of section 3. Using Theorem 3.2, it is not hard to extensive and useful theory. Thus, in the text (1) the reader will see that the complex interpolation method, applied to two left find an extensive ‘‘calculus’’ for M-ideals, by which we mean a multipliers, produces a left multiplier on the interpolated space, collection of hundreds of theorems concerning the properties of which in this case is Lp (with a slightly nonstandard ‘‘operator M-ideals, together with a long list of examples. For example, the space structure’’). M-ideals in a C*-algebra are just the closed two-sided ideals (see A result that is extremely important in applications is the refs. 21 and 22). Analysts, realizing that they have encountered following fact from ref. 16: an M-ideal in the course of a calculation, often need only to consult such a text to obtain a desired conclusion. Theorem 4.2. If X is a dual operator space, then Aᒉ(X) is a It is therefore natural to ask whether there is a noncommu- W*-algebra. tative, one-sided notion of M-ideals that, at the very least, has the This result opens another door to the use of von Neumann property that the right M-ideals in MIN(X) are the classical algebra methods in operator space theory. For example, the type M-ideals in X and the right M-ideals in C*-algebras are the closed

Blecher and Zarikian PNAS ͉ January 20, 2004 ͉ vol. 101 ͉ no. 3 ͉ 729 Downloaded by guest on September 26, 2021 right ideals. One could then hope to connect such objects to the examples. In fact, we claim that the result is essentially a ‘‘one-sided rich existing one-sided ideal theory of C*-algebras, which is M-ideal phenomenon.’’ Indeed, Theorem 6.1 is immediate from intimately related to ‘‘’’ (see, for ex- Example 5.1 iv, and the following general result: ample, refs. 23 and 24). Our goal in the rest of this article is to describe such a notion, to relate our definition to one- Theorem 6.2. The closed span of a family {J␭ : ␭ ʦ ⌳} of right sided multipliers and then begin to assemble a ‘‘calculus’’ for M-ideals in an operator space X is a right M-ideal in X. one-sided M-ideals. Such a program was proposed in 2000 by We sketch the proof of this theorem because it is quite instructive E. G. Effros, and this was initiated in ref. 16 and in the Ph.D. and shows the noncommutative nature of our arguments. By the thesis of Zarikian (25). definition of right M-ideals, and basic Banach space duality theory It turns out that there are several equivalent definitions of left (passage to the second dual X**), one sees that the theorem will M-projections on an operator space X. For example, they may be follow if it is the case that the weak* closed span of a family of right defined to be the idempotent maps P : X 3 X such that the map M-summands of a dual space Y is again a right M-summand of Y. in Eq. 3 is a complete isometry but now viewed as a map from X to By Theorem 4.2, Aᒉ(Y)isaW*-algebra. By basic properties of ϭ M2,1(X) C2(X). Alternatively, the set of left M-projections on X families of projections in von Neumann algebras, it is easy to reduce is exactly the set of orthogonal projections P in the C*-algebra the theorem to the assertion that the weak* closed span of two right Aᒉ(X). The range of a left M-projection is a right M-summand.As M-summands is again a right M-summand. In fact, a much more ۋ before, we call a closed subspace J of an operator space X a right ЌЌ striking result is true: the map P P(Y) is a lattice isomorphism M-ideal if J isarightM-summand in X**. between the lattice of projections in the W*-algebra Aᒉ(Y) and the The last few definitions are stated only in terms of the vector lattice of right M-summands of Y. The ‘‘meet’’ in the latter lattice space structure and matrix norms. The following examples show is ‘‘intersection,’’ while the ‘‘join’’ is given by the weak* closed span. that they often encode important algebraic information. In particular, we claim that if P, Q are two left M-projections, then the weak* closure of P(Y) ϩ Q(Y)is(P ∧ Q)(Y). To see this, we use Example 5.1. (i) The M-ideals in a Banach space X are exactly the the fact that P ∧ Q is the weak* limit in Aᒉ(Y)of(P ϩ Q)1/n, which right M-ideals in MIN(X). in fact is valid for any two projections P, Q in a von Neumann The right M ideals in a C algebra are exactly the closed right (ii) - *- algebra. If x is in (P ∧ Q)(Y), one can show that x ϭ (P ∧ Q)(x)is ideals. the weak* limit of (P ϩ Q)1/n(x). By a argument, one (iii) The right M-ideals in a C*-module are exactly the closed can see that (P ϩ Q)1/n(x)isinthenorm closure of P(Y) ϩ Q(Y). submodules. Thus x, and hence (P ∧ Q)(Y), is contained in the weak* closure of (iv) The right M-ideals in an operator algebra are exactly the P(Y) ϩ Q(Y). The reverse containment is quite obvious. closed right ideals with a left contractive . There are two main techniques for proving results about one- As we said earlier, one would not expect spaces like L1 to have sided M-ideals. We just saw an illustration of the first technique: interesting one-sided M-structure. In the sequel (26) to the paper ⅐ (16), we show for example that preduals of von Neumann heavy use of the fact that Aᒉ( )isaW*-algebra for dual spaces, and algebras have no nontrivial one-sided M-ideals. Rather, such the consequent permission to use facts and spaces will be interesting from the standpoint of ‘‘one-sided spectral theory. The second main technique is to first establish a L-ideal’’ theory, where the latter is the ‘‘dual theory’’ to that of result for Aᒉ(X)orMᒉ(X), often using Theorem 3.2 for this, and then one-sided M-ideals. to use the fact that the projections in these algebras are just the left M-projections. We give just one illustration of this latter technique, 6. The Calculus of One-Sided M-Ideals an application to dual spaces. It follows from Eq. 2 that for any In this section, we briefly present some new results and also operator space X, we may regard Mᒉ(X) as a subalgebra of Mᒉ(X**). describe a lengthy study that we have undertaken of one-sided It is easy to see that this inclusion embeds Aᒉ(X)asaC*-subalgebra M-ideals and multipliers. As mentioned in the last section, we of the W*-algebra Aᒉ(X**). More is true if X is a dual operator wish to generalize the basic theorems from classical M-ideal space; in this case, we claim that there is a canonical conditional theory, to develop a theory that will be useful for some problems expectation E from the W*-algebra Aᒉ(X**) onto its subalgebra in ‘‘noncommutative functional analysis.’’ As we shall see here, Aᒉ(X) and, moreover, that E is weak* continuous with respect to the this often reduces to proving that left multipliers have certain weak* for which these algebras are W*-algebras (see properties, or to von Neumann algebra techniques and 4.2). We sketch the key idea in the proof of this claim. We theory [recall that Aᒉ(X) is a von Neumann algebra if X is a dual define E by space]. Indeed, most of the basic theory of classical M-ideals does ϭ ␫* ؠ ؠ ␫ ͒ ͑ generalize but usually requires quite different, ‘‘noncommuta- E T Y T X , tive’’ arguments! And perhaps surprisingly, the results do not where Y is a predual of X and ␫ is the canonical complete become untidy in the noncommutative framework, although Y isometry from Y into Y**. If T ʦ Ball(Mᒉ(X**)) and x,y ʦ X, then sometimes we have to add reasonable hypotheses. As one would expect, one also finds truly new phenomena in the new frame- ͑ ͒͑ ͒ ␫*͑ ͑␫ ͑ ͒͒͒ ͰͫE T x ͬͰ ϭ Ͱͫ Y T X x ͬͰ work (one example of this is the fact that we mentioned above y ␫*͑␫ ͑y͒͒ in Theorem 4.2 of the existence of nontrivial one-sided M-ideals Y X in Lp spaces). ͑␫ ͑ ͒͒ Յ ͰͫT X x ͬͰ To illustrate some of the points above, consider the following ␫ ͑y͒ result: X ␫ ͑ ͒ Յ X x Theorem 6.1. The closed span in an operator algebra of a family of Ͱͫ ͬͰ ␫X͑y͒ closed right ideals, each possessing a left contractive approximate identity for itself, is a right ideal also possessing a left contractive ϭ ͰͫxͬͰ approximate identity. y , A result like this is useful in practice, because the presence of an approximate identity is often key to calculations involving approx- where the second inequality follows from Theorem 3.2. A similar imations. It is also interesting that this result is not true for Banach argument with matrices shows that E(T) satisfies the criterion in algebras, as one may easily see by testing three- or four-dimensional Theorem 3.2. Thus, E is a contraction from Mᒉ(X**) into Mᒉ(X).

730 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0305296101 Blecher and Zarikian Downloaded by guest on September 26, 2021 It is then easy to show that E maps Aᒉ(X**) into Aᒉ(X). Checking To see this, suppose that J,P,Q are as discussed in the the remaining assertions of the claim is then routine. paragraph above. If x ʦ X is in the unit ball of the weak* closure The claim established in the last paragraph has for example the of J, by Goldstine’s theorem we may choose a net (xt) in Ball(J) following corollaries: converging in the weak* topology of X** to P(x). Thus, xt ϭ ϭ Corollary 6.3. The weak* closure of a one-sided M-ideal in a dual Q(xt) E(P)(xt) converges in the weak* topology of X to operator space X is a one-sided M-summand of X. E(P)(x) ϭ Q(x) ϭ x. To see this, suppose that J is the right M-ideal, and consider These corollaries are also true for general dual Banach spaces, the canonical left M-projection P of X** onto JЌЌ. Let Q ϭ E(P), with ‘‘one-sided’’ removed throughout. This may be seen by where E is the conditional expectation above. By a weak* applying these results to MIN(X) for a Banach space X. We were continuity argument one shows that Q is a left M-projection onto surprised to find no trace of such results in the M-ideal literature. the weak* closure of J.

Corollary 6.4. (Kaplansky density for M-ideals) The unit ball of a D.P.B. was supported in part by a grant from the National Science one-sided M-ideal in a dual operator space is weak* dense in the Foundation. V.Z. was supported by a National Science Foundation Vertical unit ball of its weak* closure. Integration of Research and Education Postdoctoral Fellowship.

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