A continuum of $\mathrm{C}^*$-norms on $\IB(H)\otimes \IB(H)$ and related tensor products

TL;DR

This paper investigates the vast diversity of C*-norms on tensor products of von Neumann algebras, revealing uncountably many such norms and related tensor product functors, especially in non-nuclear cases.

Contribution

It establishes the existence of a continuum of C*-norms on tensor products of von Neumann algebras and constructs a large family of injective tensor product functors.

Findings

The set of C*-norms on certain tensor products has cardinality at least 2^{\aleph_0}.

For non-nuclear von Neumann algebras, the set of C*-norms has cardinality 2^{2^{\\aleph_0}}.

There are at least 2^{\\aleph_0} injective tensor product functors in Kirchberg's sense.

Abstract

For any pair $M,N$ of von Neumann algebras such that the algebraic tensor product $M\otimes N$ admits more than one $\mathrm{C}^*$-norm, the cardinal of the set of $\mathrm{C}^*$-norms is at least $ {2^{\aleph_0}}$. Moreover there is a family with cardinality $ {2^{\aleph_0}}$ of injective tensor product functors for $\mathrm{C}^*$-algebras in Kirchberg's sense. Let $\IB=\prod_n M_{n}$. We also show that, for any non-nuclear von Neumann algebra $M\subset \IB(\ell_2)$, the set of $\mathrm{C}^*$-norms on $\IB \otimes M$ has cardinality equal to $2^{2^{\aleph_0}}$.