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

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 ) ↦ A ⊗ α B (A,B)\mapsto A\otimes _{\alpha } B , where A ⊗ α B A\otimes _\alpha B is a cross norm completion of A ⊙ B A\odot B for each pair of C*-algebras A A and B B . For the first class of bifunctors considered ( A , B ) ↦ A ⊗ p B (A,B)\mapsto A{\otimes _p} B ( 1 ≤ p ≤ ∞ 1\leq p\leq \infty ), A ⊗ p B A{\otimes _p} B is a Banach algebra cross-norm completion of A ⊙ B A\odot B constructed in a fashion similar to p p -pseudofunctions PF p ∗ ( G ) \text {PF}^*_p(G) of a locally compact group. Taking a cue from the recently introduced symmetrized p p -pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider ⊗ p , q {\otimes _{p,q}} for Hölder conjugate p , q ∈ [ 1 , ∞ ] p,q\in [1,\infty ] – a Banach ∗ * -algebra analogue of the tensor product ⊗ p , q {\otimes _{p,q}} . By taking enveloping C*-algebras of A ⊗ p , q B A{\otimes _{p,q}} B , we arrive at a third bifunctor ( A , B ) ↦ A ⊗ C p , q ∗ B (A,B)\mapsto A{\otimes _{\mathrm C^*_{p,q}}} B where the resulting algebra A ⊗ C p , q ∗ B A{\otimes _{\mathrm C^*_{p,q}}} B is a C*-algebra.

For G 1 G_1 and G 2 G_2 belonging to a large class of discrete groups, we show that the tensor products C r ∗ ( G 1 ) ⊗ C p , q ∗ C r ∗ ( G 2 ) \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 ℓ 1 ( G 1 × G 2 ) \mathrm \ell ^1(G_1\times G_2) and conclude that if 2 ≤ p ′ > p ≤ ∞ 2\leq p’>p\leq \infty , then the canonical quotient map C r ∗ ( G ) ⊗ C p , q ∗ C r ∗ ( G ) → C r ∗ ( G ) ⊗ C p , q ∗ C r ∗ ( G ) \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 A is symmetric if the spectrum S p A ( a ∗ a ) \mathrm {Sp}_A(a^*a) is contained in [ 0 , ∞ ) [0,\infty ) for every a ∈ A a\in A , and rigidly symmetric if A ⊗ γ B A\otimes _{\gamma } B is symmetric for every C*-algebra B 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 A and B B that A ⊗ γ B A\otimes _{\gamma }B is symmetric if and only if A A or B 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.