Bojan Magajna

THE MINIMAL OPERATOR MODULE OF A BANACH MODULE

It is well known that each Banach space X can be linearly isometrically embedded into a commutative C*-algebra, hence X can be regarded as an operator space. If X has some additional algebraic structure, we may ask whether an embedding can be found which preserves this additional structure. For example, if X is a Banach algebra, the criterion for X to have a bicontinuous isomorphic representation as an algebra of operators on a Hilbert space was obtained in [20] (see also [3] and [5]). Here we shall consider the case when X is a Banach bimodule over a pair of C*-algebras A and B. Using the abstract characterisation of operator bimodules developed by Christensen, Effros and Sinclair in [6] we show first that a normed A, B-bimodule can be represented isometrically as an operator bimodule if and only if

A transitivity theorem for algebras of elementary operators

Let A be a C*-algebra and E the algebra of all elementary operators on A, and let a = (a 1 ,..., a n ), b = (b 1 ,..., b n ) ∈ A n . It is proved that b is contained in the closure of the set {(Ea 1 ,..., Ea n ): E ∈ E} if and only if each complex linear combination Σ j=1 n l j b j is contained in the closed two-sided ideal generated by Σ j=1 n l j a j . In particular, a bounded linear operator on A preserves all closed two-sided ideals if and only if it is in the strong closure of E

Uniform approximation by elementary operators

On a separable C∗-algebra A every (completely) bounded map which preserves closed two sided ideals can be approximated uniformly by elementary operators if and only if A is a finite direct sum of C∗-algebras of continuous sections vanishing at ∞ of locally trivial C∗-bundles of finite type.

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.

The norm of a symmetric elementary operator

The norm of the operator x → a*xb + b*xa on A = B(H) (or on any prime C*-algebra A) is computed for all a, b E A and is shown to be equal to the completely bounded norm.

Dual operator systems

We characterize weak* closed unital vector spaces of operators on a Hilbert space H .M ore 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.

Pointwise approximation by elementary complete contractions

A complete contraction on a C * -algebra A, which preserves all closed two sided ideals J, can be approximated pointwise by elementary complete contractions if and only if the induced map on B ⊗ A/J is contractive for every C * -algebra B, ideal J in A and C * -tensor norm on B ⊗ A/J. A lifting obstruction for such an approximation is also obtained.

Bicommutants and ranges of derivations

In this paper, originally published in Linear and Multilinear Algebra 61 (9) (2013) 1161–1180, there was an error in the last step of the proof of Theorem 2.2. Let R be an algebra over a field F, let U and V be left R-modules, U∗ and V ∗ the dual spaces regarded as right modules over R and LR(U, V ) the space of all module homomorphisms from U to V. The suggestion that the general case can be reduced to finitely generated modules by inverse limits, does not work. The theorem is not true without assuming something about U or V. The following requires only a minor modification of arguments from the correct part of the proof.

Approximation of Maps on C ∗ -algebras by Completely Contractive Elementary Operators

Let \(\overline{{\rm E}_1({\rm A})}^{p.n.}\)be the closure in the point-norm topology of the set of all completely contractive elementary operators on a C* -algebra A. If \(\psi \leq \phi\) are completely positive maps on A and \(\phi \in \overline{{\rm E}_1({\rm A})}^{p.n.}\), then \(\psi \in \overline{{\rm E}_1({\rm A})}^{p.n.}\). A completely positive contraction on o a von Neumann algebra R is \(\overline{{\rm E}_1({\rm R})}^{p.n.}\) in if and only if the normal and the singular part of o are both in \(\overline{{\rm E}_1({\rm R})}^{p.n.}\). Maps on R admitting pointwise approximation by sequences of elementary complete contractions may have additional properties that are not shared by all maps in\(\overline{{\rm E}_1({\rm R})}^{p.n.}\). A specific example on B(H) is also studied.

On C^*-extreme points

Each weak* compact C*-convex set in a hyperfinite factor (in particular in B(H)) is the weak* closure of the C*-convex hull of its C*-extreme points.