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

Abstract

For any pair $M,N$ of von Neumann algebras such that the algebraic tensor product $M\otimes N$ admits more than one ${\mathrm{C}}^{\ast }$-norm, the cardinal of the set of ${\mathrm{C}}^{\ast }$-norms is at least $2^{{\aleph }_0}$. Moreover there is a family with cardinality $2^{{\aleph }_0}$ of injective tensor product functors for ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-norms on $\IB \otimes M$ has cardinality equal to $2^{2^{{\aleph }_0}}$.

Cited by in corpus (1)

• Operator-Valued Kirchberg Theory