Multipliers between two operator spaces
In a recent survey paper we introduced one-sided multipliers between two different operator spaces. Here we give some basic theory for these maps.
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arXiv subjects
David P. Blecher
Publications and source records attributed to David P. Blecher.
The Calculus of One-Sided $M$-Ideals and Multipliers in Operator Spaces
The theory of one-sided $M$-ideals and multipliers of operator spaces is simultaneously a generalization of classical $M$-ideals, ideals in operator algebras, and aspects of the theory of Hilbert $C^*$-modules and their maps. Here we give a systematic exposition of this theory; a reference tool for `noncommutative functional analysts' who may encounter a one-sided $M$-ideal or multiplier in their work.
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A double commutant theorem for operator algebras
Every unital nonselfadjoint operator algebra possesses canonical and functorial classes of faithful (even completely isometric) Hilbert space representations satisfying a double commutant theorem generalizing von Neumann's classical result. Examples and complementary results are given.
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One-sided M-Ideals and Multipliers in Operator Spaces, I
The theory of M-ideals and multiplier mappings of Banach spaces naturally generalizes to left (or right) M-ideals and multiplier mappings of operator spaces. These subspaces and mappings are intrinsically characterized in terms of the matrix norms. In turn this is used to prove that the algebra of left adjointable mappings of a dual operator space X is a von Neumann algebra. If in addition X is an operator A--B-bimodule for $C^{*}$-algebras A and B, then the module operations on X are automatically weak$^{*}$ continuous. One sided L-projections are introduced, and analogues of various results from the classical theory are proved. An assortment of examples is considered.
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Complete isometries - an illustration of noncommutative functional analysis
This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from `noncommutative functional analysis', more specifically the new field of {\em operator spaces.} In our illustration we show how the classical characterization of (possibly non-surjective) isometries between function algebras generalizes to operator algebras. We give some variants of this characterization, and a new proof which has some advantages.
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Logmodularity and isometries of operator algebras
We generalize some facts about function algebras to operator algebras, using the `noncommutative Shilov boundary' or $C^*$-envelope first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are given.
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Complete isometries into C*-algebras
We give various characterizations of into (not necessarily onto) complete isometries between $C^*$-algebras, generalizing a classical result of Holsztynski. Our results are related to a natural embedding of the noncommutative Shilov boundary in a second dual.
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One-Sided Projections on C*-algebras
In [BEZ] the notion of a complete one-sided M-ideal for an operator space X was introduced as a generalization of Alfsen and Effros' notion of an M-ideal for a Banach space [AE72]. In particular, various equivalent formulations of complete one-sided M-projections were given. In this paper, some sharper equivalent formulations are given in the special situation that $X = \mathcal{A}$, a $C^*$-algebra (in which case the complete left M-projections are simply left multiplication on $\mathcal{A}$ by a fixed orthogonal projection in $\mathcal{A}$ or its multiplier algebra). The proof of the first equivalence makes use of a technique which is of interest in its own right--a way of ``solving'' multi-linear equations in von Neumann algebras. This technique is also applied to show that preduals of von Neumann algebras have no nontrivial complete one-sided M-ideals. In addition, we show that in a $C^*$-algebra, the intersection of finitely many complete one-sided M-summands need not be a complete one-sided M-summand, unlike the classical situation.
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One-sided ideals and approximate identities in operator algebras
A left ideal of any $C^*$-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Indeed left ideals in $C^*$-algebras may be characterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a $C^*$-algebra. We study left ideals from the perspective of `Hamana theory' and using the multiplier algebras introduced by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approximate identity, including a Banach-Stone theorem for these algebras, and an analysis of the `multiplier operator algebra'.
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The ideal envelope of an operator algebra
A left ideal of any C*-algebra is an example of an operator algebra with a right contractive approximate identity (r.c.a.i.). Conversely, we show here and in a `pre-quel' to this paper [B], that operator algebras with r.c.a.i. should be studied in terms of a certain left ideal of a C*-algebra. We study operator algebras and their multiplier algebras from the perspective of `Hamana theory' and using the multiplier algebras introduced by the first author.
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Multipliers and dual operator algebras
In a previous paper we showed how the main theorems characterizing operator algebras and operator modules, fit neatly into the framework of the `noncommutative Shilov boundary', and more particularly via the left multiplier operator algebra of an operator space. As well as giving new characterization theorems, the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated. Recent progress of the author with Effros and Zarikian now enables weak*-versions of these characterization theorems. For example, we prove a result analogous to Sakai's famous characterization of von Neumann algebras as the C*-algebras with predual, namely, that the $σ$-weakly closed unital (not-necessarily-selfadjoint) subalgebras of $B(H)$ for a Hilbert space $H$, are exactly the unital operator algebras which possess an operator space predual. This removes one of the hypotheses from an earlier characterization due to Le Merdy. We also show that the multiplier operator algebras of dual operator spaces are dual operator algebras. Using this we refine several known results characterizing dual operator modules.
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The Shilov boundary of an operator space - and the characterization theorems
We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C$^*-$modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C$^*-$algebras of an operator space, which generalize the algebras of adjointable operators on a C$^*-$module, and the `imprimitivity C$^*-$algebra'. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify, and strengthen several theorems characterizing operator algebras and modules, in a way that seems to give more information than other current proofs. We also include some general notes on the `commutative case' of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about `function modules'.
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Multipliers of operator spaces, and the injective envelope
We study the injective envelope I(X) of an operator space X, showing amongst other things that it is a self-dual C$^*-$module. We describe the diagonal corners of the injective envelope of the canonical operator system associated with X. We prove that if X is an operator $A-B$-bimodule, then A and B can be represented completely contractively as subalgebras of these corners. Thus, the operator algebras that can act on X are determined by these corners of I(X) and consequently bimodules actions on X extend naturally to actions on I(X). These results give another characterization of the multiplier algebra of an operator space, which was introduced by the first author, and a short proof of a recent characterization of operator modules, and a related result. As another application, we extend Wittstock's module map extension theorem, by showing that an operator $A-B$-bimodule is injective as an operator $A-B$-bimodule if and only if it is injective as an operator space.
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Isomorphisms of function modules, and generalized approximation in modulus
For a function algebra A we investigate relations between the following three topics: isomorphisms of singly generated A-modules, Morita equivalence bimodules, and `real harmonic functions' with respect to A. We also consider certain groups which are naturally associated with A. We illustrate the notions considered with several examples.
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On function and operator modules
Let $A$ be a unital Banach algebra. We give a characterization of the left Banach $A$-modules $X$ for which there exists a commutative unital $C^*$-algebra $C(K)$, a linear isometry $i\colon X\to C(K)$, and a contractive unital homomorphism $θ\colon A\to C(K)$ such that $i(a\cdotp x) =θ(a)i(x)$ for any $a\in A, x\in X$. We then deduce a "commutative" version of the Christensen-Effros-Sinclair characterization of operator bimodules. In the last section of the paper, we prove a $w^*$-version of the latter characterization, which generalizes some previous work of Effros and Ruan.
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A Morita theorem for algebras of operators on Hilbert space
We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule.
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Modules over operator algebras, and the maximal C^*-dilation
We continue our study of the general theory of possibly nonselfadjoint algebras of operators on a Hilbert space, and modules over such algebras, developing a little more technology to connect `nonselfadjoint operator algebra' with the C$^*-$algebraic framework. More particularly, we make use of the universal, or maximal, C$^*-$algebra generated by an operator algebra, and C$^*-$dilations. This technology is quite general, however it was developed to solve some problems arising in the theory of Morita equivalence of operator algebras, and as a result most of the applications given here (and in a companion paper) are to that subject. Other applications given here are to extension problems for module maps, and characterizations of C$^*-$algebras.
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On Morita's Fundamental Theorem for C*-algebras
We give a solution, via operator spaces, of an old problem in the Morita equivalence of C*-algebras. Namely, we show that C*-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule. An operator module over a C$^*-$algebra $\A$ is a closed subspace of some B(H) which is left invariant under multiplication by $π(\A)$, where $π$ is a *-representation of $\A$ on $H$. The category $_{\A}HMOD$ of *-representations of $\A$ on Hilbert space is a full subcategory of the category $_{\A}OMOD$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the C$^*-$algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a C$^*-$algebra argument using compactness of the quasistate space of a C$^*-$algebra, and lowersemicontinuity in the enveloping von Neumann algebra.
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