Hilbert modules and tensor products of operator spaces

Abstract

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.