New tensor products of C*-algebras and characterization of type I C*-algebras as rigidly symmetric C*-algebras

Abstract

We construct several new classes of bifunctors $(A,B)\mapsto A{\otimes }_{\alpha }B$, where $A{\otimes }_{\alpha }B$ is a cross norm completion of $A\odot B$ for each pair of C*-algebras $A$ and $B$. For the first class of bifunctors considered $(A,B)\mapsto A{\otimes }_{p}B$ ($1\le p\le \infty$), $A{\otimes }_{p}B$ is a Banach algebra cross-norm completion of $A\odot B$ constructed in a fashion similar to $p$-pseudofunctions of a locally compact group. We also consider ${\otimes }_{p,q}$ for Hölder conjugate $p,q\in [1,\infty ]$ -- a Banach $\ast$-algebra analogue of the tensor product ${\otimes }_p$. By taking enveloping C*-algebras of $A{\otimes }_{p,q}B$, we arrive at a third bifunctor $(A,B)\mapsto A{\otimes }_{{\mathrm{C}}_{p,q}^{\ast }}B$ where the resulting algebra $A{\otimes }_{{\mathrm{C}}_{p,q}^{\ast }}B$ is a C*-algebra. For groups belonging to a large class of non-amenable discrete groups possessing both the rapid decay and Haagerup property, we show that the tensor products ${\mathrm{C}}_{\mathrm{r}}^{\ast }(G_1){\otimes }_{{\mathrm{C}}_{p,q}^{\ast }}{\mathrm{C}}_{\mathrm{r}}^{\ast }(G_2)$ coincide with a Brown-Guentner type C*-completion of ${\mathrm{\ell }}^1(G_1\times G_2)$ and conclude that if $2\le p^{\prime }<p\le \infty$, then the canonical quotient map ${\mathrm{C}}_{\mathrm{r}}^{\ast }(G){\otimes }_{{\mathrm{C}}_{p,q}^{\ast }}{\mathrm{C}}_{\mathrm{r}}^{\ast }(G)\to {\mathrm{C}}_{\mathrm{r}}^{\ast }(G){\otimes }_{{\mathrm{C}}_{p^{\prime },q^{\prime }}^{\ast }}{\mathrm{C}}_{\mathrm{r}}^{\ast }(G)$ is not injective. A Banach $\ast$-algebra $A$ is \emph{rigidly symmetric} if $A{\otimes }_{\gamma }B$ is symmetric for every C*-algebra $B$. A theorem of Kugler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kugler's theorem by showing for C*-algebras $A$ and $B$ that $A{\otimes }_{\gamma }B$ is symmetric if and only if $A$ or $B$ is type I.

The misspelling of the name of the author W. Kugler (reference [34]) was corrected