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

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 $2^{{\mathrm{\aleph }}_0}$. Moreover there is a family with cardinality $2^{{\mathrm{\aleph }}_0}$ of injective tensor product functors for ${\mathrm{C}}^{\ast }$-algebras in Kirchberg’s sense. Let $\text{IB}={\prod }_n{M}_n$. We also show that, for any non-nuclear von Neumann algebra $M\subset \text{IB}({\mathrm{\ell }}_2)$, the set of ${\mathrm{C}}^{\ast }$-norms on $\IB \otimes M$hascardinalityequalto$2^{2^{\aleph_0}}$.