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

Abstract:

Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, 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 $\text {PF}^*_p(G)$ of a locally compact group. Taking a cue from the recently introduced symmetrized $p$-pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider ${\otimes _{p,q}}$ for Hölder conjugate $p,q\in [1,\infty ]$ – a Banach $*$-algebra analogue of the tensor product ${\otimes _{p,q}}$. 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 $G_1$ and $G_2$ belonging to a large class of discrete groups, 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 for a large class of non-amenable discrete groups possessing both the rapid decay property and Haagerup’s approximation property.

A Banach $*$-algebra $A$ is symmetric if the spectrum $\mathrm {Sp}_A(a^*a)$ is contained in $[0,\infty )$ for every $a\in A$, and rigidly symmetric if $A\otimes _{\gamma } B$ is symmetric for every C*-algebra $B$. A theorem of Kügler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kügler’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. In particular, a C*-algebra is rigidly symmetric if and only if it is type I. This strongly settles a question of Leptin and Poguntke from 1979 [J. Functional Analysis 33 (1979), pp. 119—134] and corrects an error in the literature.