Haagerup Tensor Product Research Articles

A Factorization Problem for Normal Completely Bounded Mappings

Given an operator space X and a von Neumann algebra A, we consider a contractive mapping q:A⊗ehX⊗ehA→NCB(X*, A) formally defined by q(∑aj⊗xjk⊗bk)=∑xjk⊗ajbk, from the extended Haagerup tensor product A⊗ehX⊗ehA into the space of w*-continuous completely bounded maps from X* into A. We characterize elements of the range space Im(q) by a factorization property involving decomposable operators and investigate various properties of that space. In the case when X=B∗ is the predual of a von Neumann algebra B, Im(q) is included in the space DEC(B, A) of decomposable operators from B into A. Regarding q as having values in that space, we show that q is a quotient map onto its range. Then we prove that DEC(B, A) is a normal dual operator A-bimodule and that Im(q)⊂DEC(B, A) is a strong operator A-bimodule.

Hopf C*-Algebras

In this paper we define and study Hopf C*-algebras. Roughly speaking, a Hopf C*-algebra is a C*-algebra A with a comultiplication φ: A → M(A ⊗ A) such that the maps a ⊗ b ↦ φ(a)(1 ⊗ b) and a ↦ (a ⊗ 1)φ(b) have their range in A ⊗ A and are injective after being extended to a larger natural domain, the Haagerup tensor product A ⊗ h A. In a purely algebraic setting, these conditions on φ are closely related to the existence of a counit and antipode. In this topological context, things turn out to be much more subtle, but nevertheless one can show the existence of a suitable counit and antipode under these conditions. The basic example is the C*-algebra C0(G) of continuous complex functions tending to zero at infinity on a locally compact group where the comultiplication is obtained by dualizing the group multiplication. But also the reduced group C*-algebra C r ∗ ( G ) of a locally compact group with the well-known comultiplication falls in this category. In fact all locally compact quantum groups in the sense of Kustermans and the first author (such as the compact and discrete ones) as well as most of the known examples are included. This theory differs from other similar approaches in that there is no Haar measure assumed. 2000 Mathematics Subject Classification: 46L65, 46L07, 46L89.

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 $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ 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.

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.

Compact Schur multipliers

Compact Schur multipliers on the algebra B(t) of all bounded linear operators on an infinite-dimensional separable complex Hilbert space 7will be identified as the elements of the Haagerup tensor product co 9h Co (the completion of co (&co in the Haagerup norm). Other ideals of Schur multipliers related to compact operators will also be characterized.

Hilbert $C^*$-Module Representation on Haagerup Tensor Products and Group Systems

Using the Hilbert C^* -module representation associated with completely multi-positive linear maps [Heo], we give another representation on Haagerup tensor product without the bridging maps. We also construct covariant representations of covariant group systems on Hilbert C^* -modules.

The “maximal” tensor product of operator spaces

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

The n -Fold Central Haagerup Tensor Product

This paper defines the n-fold central Haagerup tensor product ⊗ i = 1 ℒ h n R of a von Neumann algebra R, and shows that the map θ z : ⊗ i = 1 ℒ h n R → CB ( R n − 1 , R ) given by θ([a1⊗a2⊗…⊗an])(x1, x2, …,xn−1)=a1x1a2x2…xn−1an is an isometry.

On some “stability” properties of the full C*-algebra associated to the free group F∞

Let C*(F∞) be the full C*-algebra associated to the free group of countably many generators and SnC*(F∞) be the class of all n-dimensional operator subspaces of C*(F∞). In this paper, we study some stability properties of SnC*(F∞). More precisely, we will prove that for any E0, E1 in SnC*(F∞), the Haagerup tensor product E0⊗hE1 and the operator space obtained by complex interpolation Eθ are (1 + ∈)-contained in C*(F∞) for arbitrary ∈>0. On the other hand, we will show an extension property for WEPC*-algebras.

Matrix Regular Operator Spaces

The concept of the regular (or Riesz) norm on ordered real Banach spaces is generalized to matrix ordered complex operator spaces in a way that respects the matricial structure of the operator space. A norm on an ordered real Banach spaceEis regular if: (1) −x⩽y⩽ximplies that ‖y‖⩽‖x‖; and (2) ‖y‖<1 implies the existence ofx∈Esuch that ‖x‖<1 and −x⩽y⩽x. A matrix ordered operator space is called matrix regular if, at each matrix level, the restriction of the norm to the self-adjoint elements is a regular norm. In such a space, elements at each matrix level can be written as linear combinations of four positive elements. The concept of the matrix ordered operator space is made specific in such a way as to be a natural generalization of ordered real and complex Banach spaces. For the case whereVis a matrix ordered operator space, a natural cone is defined on the operator spaceX*⊗hV⊗hX, with ⊗hindicating the Haagerup tensor product, so as to make it a matrix ordered operator space. Exploiting the advantages gained by takingXto be the column Hilbert spaceHc, an equivalence is established between the matrix regularity of a space and that of its operator dual. This concept of matrix regularity also provides for more accessible proofs of the Christensen–Sinclair Multilinear Representation and Multilinear Decomposition theorems.

A remark on the tensor product of two maximal operator spaces

Given a Banach space E, let us denote by Max(E) the largest operator space structure on E. Recently Paulsen-Pisier and, independently, Junge proved that for any Banach spaces E, F, isomorphically where and respectively denote the Haagerup tensor product and the spatial tensor product of operator spaces. In this paper we show that, in general, this equality does not hold completely isomorphically.

Ideal Spaces of the Haagerup Tensor Product of C*-Algebras

Following the work of Allen, Sinclair and Smith on the primitive ideal space of the Haagerup tensor product A ⊗ hB of C*-algebras A and B, we investigate the hull-kernel topology and use this to determine various other ideal spaces and their topologies in relation to the corresponding ideal spaces of A and B. We study the semi-continuity of norm functions I → ||x + I||(x ∈ A ⊗h B) on these ideal spaces and identify the separated points of Prim (A ⊗h B). Finally, we exhibit several conditions each of which is equivalent to the quasi-standardness of A ⊗h B.

Strong Operator Modules and the Haagerup Tensor Product

Strong operator modules over a von Neumann algebra R are introduced and the so-called extended Haagerup tensor product over R of strong modules is studied. In the case R=C and the spaces are weak* closed this product agrees with the weak* Haagerup tensor product of Blecher and Smith. If C is the center of R, the extended Haagerup tensor product R is a Banach algebra containing the central tensor product R \haagc R of Chatterjee and Smith and has very nice properties concerning slice maps and the relative commutants of subalgebras. It is shown that R is a dual Banach space, and all weak* closed two-sided ideals of the algebra R are determined. 1991 Mathematics Subject Classification: 46L05

Hilbert modules and tensor products of operator spaces

The classical identification of the predual of B(H) (the algebra of all bounded operators on a Hilbert space H) with the projective operator space tensor product H⊗H is extended to the context of Hilbert modules over commutative von Neumann algebras. Each bounded module homomorphism b between Hilbert modules over a general C∗-algebra is shown to be completely bounded with ‖b‖cb = ‖b‖. The so called projective operator tensor product of two operator modules X and Y over an abelian von Neumann algebra C is introduced and if Y is a Hilbert module, this product is shown to coincide with the Haagerup tensor product of X and Y over C. 0. Introduction. Recently the theory of tensor products of operator spaces has evolved considerably (see e.g. [6], [18]). The present paper is an attempt to put a part of this theory in a broader context of operator modules in which the role of the compex field C is played by a von Neumann algebra. It is well known, for example, that B(H) (the space of all bounded linear operators on a Hilbert space H) is isometric to the dual of the projective tensor product H ∧ ⊗H. (In [15] and [2] a more recent improvement of this result can be found and in [12] there is even an extension to general von Neumann algebras instead of B(H).) Here we shall present a generalization of this classical result to Hilbert modules. To achieve this, we have first to extend some parts of the theory of tensor products of operator spaces to operator modules. We have tried to make this paper accessible to everyone familiar with basic notions of functional analysis and operator algebras (and the definition of algebraic tensor product of vector spaces), so all the necessary background concerning operator spaces, completely bounded mappings and Hilbert modules will be explained below. (For a more complete treatment, however, see [28], [32] and [11] for operator spaces and [23], [27], [30] and [20] for Hilbert modules.) 1991 Mathematics Subject Classification: Primary 46L05. The paper is in final form and no version of it will be published elsewhere.

Standard Duals of Operator Bimodules

In this paper, we introduce the notion of standard duals of operator bimodules. They coincide with the standard duals of operator spaces when the operator bimodules are nothing more than operator spaces. Using the Haagerup tensor product of operator sesquimodules and operator bimodules, we give a representation for the standard duals of operator bimodules in terms of the operator space standard duals. We also study the problem whether an operator bimodule is completely isometrically embedded in its second standard dual and its connection with the behavior of the adjoint operation. In addition, we also find out the standard dual of a direct limit of a sequence of operator bimodules.

On the central Haagerup tensor product

For a large class of C*-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

The Central Haagerup Tensor Product and Maps between von Neumann Algebras

We introduce the central Haagerup tensor product R ⊗ F h R for a von Neumann algebra R , and we show that the natural injection into tbe space CB( R , R ) of completely bounded maps on R is isometric. This is used to study mappings between von Neumann algebras and to give a new characterization of injectivity. This tensor product is also studied for C*-algebras A , and we show that the injection into CB( A , A ) need not always be isometric, depending on the topology of the primitive ideal space of A .

The Dual of the Haagerup Tensor Product

The weak*-Haagerup tensor product Jt ®w.hjV of two von Neumann algebras is related to the Haagerup tensor product M ®h Jf in the same way that the von Neumann algebra tensor product is related to the spatial tensor product. Many of the fundamental theorems about completely bounded multilinear maps may be deduced from elementary properties of the weak*-Haagerup tensor product. We show that X* w.h Y* = (X®h Y)* for all operator spaces A'and Y. The weak*-Haagerup tensor product has simple characterizations and behaviour with reference to slice map properties. The tensor product of two (not necessarily self-adjoint) operator algebras is proven to have many strong commutant properties. All operator spaces possess a certain approximation property which is related to this tensor product. The connection between bimodule maps and commutants is explored.

Completely bounded module maps and the Haagerup tensor product

We obtain a representation theorem for module maps defined on the algebra K( H) of compact operators. This is used to show that the Haagerup tensor product of two operator algebras enjoys strong versions of Tomita's commutant theorem and the slice map property. We also give a general theorem concerning the automatic complete boundedness of module maps.

Self-duality for the Haagerup tensor product and Hilbert space factorizations

D. Blecher and V. Paulsen showed that the Haagerup tensor product V ⊗ h W for operator spaces V and W preserves inclusions. It is proved to also preserve complete quotient maps, and to be self-dual in the sense that it induces the Haagerup norm on the algebraic tensor product V ∗ ⊗ W ∗ . The full operator dual space (V ⊗ h W) ∗ is computed. It coincides with the natural operator space \\ ̃ gG 2(V, W ∗) of maps ϑ: V → W ∗ which have completely bounded factorizations through Hilbert spaces (with vectors identified with row matrices). More generally, one has the natural complete isometry \\ ̃ gG 2(V ⊗ h W, X) ≊ \\ ̃ gG 2(V, \\ ̃ gG 2(W, X)) . Given Hilbert spaces H and K with vectors regarded as column matrices, it is shown that one may identify the operator spaces B( H, K) and CB( H, K).