Center of Minimal Tensor Product of $C^{\ast}$-Algebras

Question

Let $A$ and $B$ be $C^{\ast}-$Algebras. Assume that $A \otimes B$ denotes the minimal (spatial) tensor product. Is it true that $Z(A \otimes B)= Z(A) \otimes Z(B)$ where $Z$ is used for center.

Note that $Z(A) \otimes Z(B)\subset Z(A \otimes B)$. Also as $Z(A) \otimes Z(B)$ is $*$-subalgebra Of $A \otimes B$ so using injectivity of min the identity map extends uniquely to an $*$-isometric map say $\theta: Z(A)\otimes Z(B) \to Z(A\otimes B)$. We only need to show $\theta$ is surjective. Any ideas?

Answer 1

This was shown by Haydon and Wassermann:

Haydon, Richard, and Simon Wassermann. "A Commutation Result for Tensor Products of C*‐Algebras." Bulletin of the London Mathematical Society 5.3 (1973): 283-287.

for the minimal case.

The general case was done by Rob Archbold in

Archbold, Robert J. "On the Centre of a Tensor Product of C*‐Algebras." Journal of the London Mathematical Society 2.3 (1975): 257-262.