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 $(A,B)\mapsto A\otimes_p B$ ($1\leq p\leq\infty$), $A\otimes_p B$ 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 H\"older conjugate $p,q\in [1,\infty]$ -- a Banach $*$-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 $(A,B)\mapsto A\otimes_{\mathrm C^*_{p,q}} B$ where the resulting algebra $A\otimes_{\mathrm C^*_{p,q}} B$ 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 $\mathrm C^*_{\mathrm r}(G_1)\otimes_{\mathrm C^*_{p,q}}\mathrm C^*_{\mathrm r}(G_2)$ coincide with a Brown-Guentner type C*-completion of $\mathrm \ell^1(G_1\times G_2)$ and conclude that if $2\leq p'<p\leq\infty$, then the canonical quotient map $\mathrm C^*_{\mathrm r}(G)\otimes_{\mathrm C^*_{p,q}}\mathrm C^*_{\mathrm r}(G)\to \mathrm C^*_{\mathrm r}(G)\otimes_{\mathrm C^*_{p',q'}}\mathrm C^*_{\mathrm r}(G)$ is not injective. A Banach $*$-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.