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.

background or methods

Theory of Operator Algebras II

The “maximal” tensor product of operator spaces

Consider a completely bounded Fourier multiplier phi of a locally compact group G, and take 1 <= p <= infinity. One can associate to phi a Schur multiplier on the Schatten classes S_p(L^2 G), as well as a Fourier multiplier on Lp(LG), the non-commutative Lp-space of the group von Neumann algebra of G. We prove that the completely bounded norm of the Schur multiplier is not greater than the completely bounded norm of the Lp-Fourier multiplier. When G is amenable we show that equality holds, extending a result by Neuwirth and Ricard to non-discrete groups. 
For a discrete group G and in the special case when p > 2 is an even integer, we show the following. If there exists a map between Lp(LG) and an ultraproduct of Lp(M) \otimes S_p(L^2 G) that intertwines the Fourier multiplier with the Schur multiplier, then G must be amenable. This is an obstruction to extend the Neuwirth-Ricard result to non-amenable groups.

Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces

Around 1967, Arveson invented a striking noncommutative generalization of classical $H^\infty$, known as {\em subdiagonal algebras}, which include a wide array of examples of interest to operator theorists. Their theory extends that of the generalized $H^p$ spaces for function algebras from the 1960s, in an extremely remarkable, complete, and literal fashion, but for reasons that are `von Neumann algebraic'. Most of the present paper consists of a survey of our work on Arveson's algebras, and the attendant $H^p$ theory, explaining some of the main ideas in their proofs, and including some improvements and short-cuts. The newest results utilize new variants of the noncommutative Szeg\"{o} theorem for $L^p(M)$, to generalize many of the classical results concerning outer functions, to the noncommutative $H^p$ context. In doing so we solve several of the old open problems in the subject. 
We include full proofs, for the most part, of the simpler `antisymmetric algebra' special case of our results on outers.

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Von Neumann algebraic H^p theory

We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations.

A spectral sequence to compute L2-Betti numbers of groups and groupoids

Given a type I von Neumann algebra M with a faithful normal semi-finite trace �, let S0(M,�) be the algebra of all �-compact operators affiliated with M. We give a complete description of all derivations on the algebra S0(M,�). In particular, we prove that if M is of type I∞ then every derivation on S0(M,�) is spatial.

Theory of Operator Algebras II

Citations

The “maximal” tensor product of operator spaces

Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces

Von Neumann algebraic H^p theory

A spectral sequence to compute L2-Betti numbers of groups and groupoids

Complete description of derivations on -compact operators for type I von Neumann algebras

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Citations

The “maximal” tensor product of operator spaces

Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces

Von Neumann algebraic H^p theory

A spectral sequence to compute L2-Betti numbers of groups and groupoids

Complete description of derivations on -compact operators for type I von Neumann algebras

Related Papers (5)

Fundamentals of the Theory of Operator Algebras

Operator algebras and quantum statistical mechanics

C-Algebras and Their Automorphism Groups

C[*]-algebras and W[*]-algebras

Completely Bounded Maps and Operator Algebras

C[]-algebras and W[]-algebras