This is an announcement for the paper "A continuum of $\mathrm{C}^*$-norms on $\IB(H)\otimes \IB(H)$ and related tensor products" by Narutaka Ozawa and Gilles Pisier.
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 $2^{2^{\aleph_0}}$.
Archive classification: math.OA
Submitted from: pisier@math.tamu.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1404.7088
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