Tensor Products in Quantum Functional Analysis: the Non-Matricial Approach

Abstract

As is known, there exists an alternative, "non-matricial" way to present basic notions and results of quantum functional analysis (= operator space theory). This approach is based on considering, instead of matrix spaces, a single space, consisting, roughly speaking, of vectors from the initial linear space equipped with coefficients taken from some good operator algebra. It seems that so far there was no systematical exposition of the theory in the framework of the non-matricial approach. We believe, however, that in a number of topics the non-matricial approach gives a more elegant and transparent theory. In this paper we introduce, using only the non-matricial language, both quantum versions of the classical (Grothendieck) projective tensor product of normed spaces. These versions correspond to the "matricial" Haagerup and operator-projective tensor products. We define them in terms of the universal property with respect to some classes of bilinear operators, corresponding to "matricial" multiplicatively bounded and completely bounded, and then produce their explicit constructions. Among the relevant results, we shall show that both tensor products are actually quotient spaces of some "genuine" projective tensor products. Moreover, the Haagerup tensor product is itself a "genuine" projective tensor product, however not of just normed spaces but of some normed modules.