Dual Operator Spaces Research Articles

ANY DUAL OPERATOR SPACE IS WEAKLY LOCALLY REFLEXIVE

Abstract We introduce the notion of weakly local reflexivity in operator space theory and prove that any dual operator space is weakly locally reflexive.

Crossed products of dual operator spaces by locally compact groups

If $ \alpha $ is an action of a locally compact group $ G $ on a dual operator space $ X $, then two generally different notions of crossed products are defined, namely the Fubini crossed product $ X\mathbin {\rtimes^{\mathcal {F}} _{\alpha }} G $ an

Kirchberg’s conjecture in the system of Hilbert space factorable mapping spaces

This study aims to introduce the notions of injectivity, local reflexivity, exactness, and nuclearity in the system ( $$\left( {{\rm{\Gamma }}_2^c\left( { \cdot , \cdot } \right),{\rm{\gamma }}_2^c\left( \cdot \right)} \right)$$ ). We find that every dual operator space is injective in the system ( $$\left( {{\rm{\Gamma }}_2^c\left( { \cdot , \cdot } \right),{\rm{\gamma }}_2^c\left( \cdot \right)} \right)$$ ) and nuclearity is equivalent to exactness in this system. As a corollary, we prove that Kirchberg’s conjecture on the equivalence of exactness and local reflexivity for C* -algebras is false in this system, i.e., there exists a C*-algebra $${\cal A}$$ that is locally reflexive in this system but is not exact in this system.

Morita embeddings for dual operator algebras and dual operator spaces

We define a relation < for dual operator algebras. We say that B < A if there exists a projection p in A such that B and pAp are Morita equivalent in our sense. We show that < is transitive, and we investigate the following question: If A < B and B < A, then is it true that A and B are stably isomorphic? We propose an analogous relation < for dual operator spaces, and we present some properties of < in this case.

STABLE PROPERTIES OF HYPERRELEXIVITY

AbstractRecently, a new equivalence relation between weak* closed operator spaces acting on Hilbert spaces has appeared. Two weak* closed operator spaces ${\mathcal{U}}$, ${\mathcal{V}}$ are called weak TRO equivalent if there exist ternary rings of operators ${\mathcal{M}}$i, i = 1, 2 such that ${\mathcal{U}}=[{\mathcal{M}}_2{\mathcal{V}}{\mathcal{M}}_1^*]^{-w^*}, {\mathcal{V}}=[{\mathcal{M}}_2^*{\mathcal{U}}{\mathcal{M}}_1]^{-w^*}.$ Weak TRO equivalent spaces are stably isomorphic, and conversely, stably isomorphic dual operator spaces have normal completely isometric representations with weak TRO equivalent images. In this paper, we prove that if ${\mathcal{U}}$ and ${\mathcal{V}}$ are weak TRO equivalent operator spaces and the space of I × I matrices with entries in ${\mathcal{U}}$, MIw(${\mathcal{U}}$), is hyperreflexive for suitable infinite I, then so is MIw(${\mathcal{V}}$). We describe situations where if ${\mathcal{L}}$1, ${\mathcal{L}}$2 are isomorphic lattices, then the corresponding algebras Alg($\mathcal{L}$1), Alg($\mathcal{L}$2) have the same complete hyperreflexivity constant.

Dual Operator Algebras with Normal Virtual h-Diagonal

We study the class of dual operator algebras admitting a normal virtual h-diagonal (i.e. a diagonal in the normal Haagerup tensor product), this property can be seen as a dual operator space version of amenability. After giving several characterizations of these algebras, we show that this class is stable under algebraic perturbations and cb-isomorphisms with small bound. We also prove some perturbation results for the Kadison-Kastler metric.

Dual operator systems

We characterize weak* closed unital vector spaces of operators on a Hilbert space $H$. More precisely, we first show that an operator system, which is the dual of an operator space, can be represented completely isometrically and weak* homeomorphically as a weak* closed operator subsystem of $B(H)$. An analogous result is proved for unital operator spaces. Finally, we give some somewhat surprising examples of dual unital operator spaces.

Nuclearity for dual operator spaces

In this short paper, we study the nuclearity for the dual operator space V* of an operator space V. We show that V* is nuclear if and only if V*** is injective, where V*** is the third dual of V. This is in striking contrast to the situation for general operator spaces. This result is used to prove that V** is nuclear if and only if V is nuclear and V** is exact.

Stable isomorphism of dual operator spaces

We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations ϕ and ψ of X and Y, respectively, and ternary rings of operators M 1 , M 2 such that ϕ ( X ) = [ M 2 ∗ ψ ( Y ) M 1 ] − w ∗ and ψ ( Y ) = [ M 2 ϕ ( X ) M 1 ∗ ] − w ∗ . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras.

Weak* exactness for dual operator spaces

We introduce a new tensor product ⊗:σ,∨ and study the weak* condition C′, which is also called weak* exactness, for dual operator spaces. Our definition of weak* condition C′ is equivalent to Kirchberg's notion of weak exactness in the case of von Neumann algebras. We also study the connection of weak* exact W*-TROs with their linking von Neumann algebras and study the structure of exact (respectively, nuclear) W*-TROs.

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.

Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the ‘nonselfadjoint analogue’ of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak * -continuous on dual spaces. In particular, if X is a subspace of a C * -algebra A, and if a ∈ A satisfies aX ⊂ X , then we show that the function x ↦ ax on X is automatically weak * continuous if either (a) X is a dual operator space, or (b) a * X ⊂ X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C * -subalgebra. Applications include a new characterization of the σ -weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W * -modules to the framework of modules over such algebras. We also give a Banach module characterization of σ -weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.

Type Decomposition and the Rectangular AFD Property for W*-TRO’s

AbstractWe study the type decomposition and the rectangular AFD property for W*-TRO’s. Like von Neumann algebras, every W*-TRO can be uniquely decomposed into the direct sum of W*- TRO's of type I, type II, and type III. We may further considerW*-TRO's of type Im,n with cardinal numbers m and n, and considerW*-TRO's of type IIλ,μ with λ, μ = 1 or ∞. It is shown that every separable stable W*-TRO (which includes type I∞, ∞, type II∞, ∞ and type III) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for W*-TRO’s. One of our major results is to show that a separable W*-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular space (equivalently, a rectangular space).

Completely bounded maps into certain Hilbertian operator spaces

We prove a factorization of completely bounded maps from a C*-algebra A (or an exact operator space E ⊂ A) to ℓ2 equipped with the operator space structure of (C,R)θ (0 < θ < 1) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if F denotes ℓ2 equipped with the operator space structure of (C,R)θ, then u: A → F is completely bounded if and only if there are states f, g on A and C > 0 such that for all a ∈ A, ‖ua‖2≤Cf(a*a)1−θg(aa*)θ⁠. This extends the case θ = 1/2 treated in a recent paper with Shlyakhtenko (2002). The constants we obtain tend to 1 when θ → 0 or θ→ 1, so that we recover, when θ = 0 (or θ = 1), the case of mappings into C (or into R), due to Effros and Ruan. We use analogs of “free Gaussian” families in nonsemifinite von Neumann algebras. As an application, we obtain that, if 0 < θ < 1, (C,R)θ does not embed completely isomorphically into the predual of a semifinite von Neumann algebra. Moreover, we characterize the subspaces S ⊂ R ⊕ C such that the dual operator space S* embeds (completely isomorphically) into M* for some semifinite von Neumann algebra M: the only possibilities are S = R, S = C, S = R ∩ C and direct sums built out of these three spaces. We also discuss when S ⊂ R ⊕ C is injective, and give a simpler proof of a result due to Oikhberg on this question. In the appendix, we present a proof of Junge's theorem that OH embeds completely isomorphically into a noncommutative L1-space. The main idea is similar to Junge's, but we base the argument on complex interpolation and Shlyakhtenko's generalized circular systems (or “generalized free Gaussian”), which somewhat unifies Junge's ideas with those of our work with Shlyakhtenko (2002).

Amplification of completely bounded operators and Tomiyama's slice maps

Let ( M, N) be a pair of von Neumann algebras, or of dual operator spaces with at least one of them having property S σ , and let Φ be an arbitrary completely bounded mapping on M . We present an explicit construction of an amplification of Φ to a completely bounded mapping on M ⊗ N . Our approach is based on the concept of slice maps as introduced by Tomiyama, and makes use of the description of the predual of M ⊗ ̄ N given by Effros and Ruan in terms of the operator space projective tensor product (cf. Effros and Ruan (Internat. J. Math. 1(2) (1990) 163; J. Operator Theory 27 (1992) 179)). We further discuss several properties of an amplification in connection with the investigations made in May et al. (Arch. Math. (Basel) 53(3) (1989) 283), where the special case M= B( H) and N= B( K) has been considered (for Hilbert spaces H and K ). We will then mainly focus on various applications, such as a remarkable purely algebraic characterization of w ∗ -continuity using amplifications, as well as a generalization of the so-called Ge–Kadison Lemma (in connection with the uniqueness problem of amplifications). Finally, our study will enable us to show that the essential assertion of the main result in May et al. (1989) concerning completely bounded bimodule homomorphisms actually relies on a basic property of Tomiyama's slice maps.

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.

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.

The Haagerup invariant for tensor products of operator spaces

In [7, 8] Haagerup introduced two isomorphism invariants and for C*-algebras and von Neumann algebras , based on appropriate forms of the completely bounded approximation property defined below. These definitions have obvious extensions to operator spaces and dual operator spaces respectively, and in [16] we established the multiplicativity of A on the ultraweakly closed spatial tensor product of two dual operator spaces and :