Tensor products with pure C*-algebras

Question: Is the minimal tensor product $A \otimes B$ of two C*-algebras pure whenever one of them is?

Background: Following Winter [1], a C*-algebra $A$ is said to be pure if it has very good comparison and divisibility properties, specifically its Cuntz semigroup is almost unperforated and almost divisible. This notion is closely related to that of $\mathcal{Z}$ -stability, where $\mathcal{Z}$ denotes the Jiang-Su algebra, and a C*-algebra $A$ is said to be $\mathcal{Z}$ -stable if $A \cong A \otimes \mathcal{Z}$ . Rørdam [2] showed that every $\mathcal{Z}$ -stable C*-algebra is pure, and the Toms-Winter conjecture predicts that a separable, unital, simple, nuclear C*-algebra is $\mathcal{Z}$ -stable if (and only if) it is pure. Moreover, it was shown in [4] that a C*-algebra $A$ is pure if and only if $\mathrm{Cu}(A) \cong \mathrm{Cu}(A) \otimes \mathrm{Cu}(\mathcal{Z})$ , that is, its Cuntz semigroup tensorially absorbs the Cuntz semigroup of $\mathcal{Z}$ . One may therefore think of pureness as $\mathcal{Z}$ -stability at the level of the Cuntz semigroup.

Using that $\mathcal{Z}$ is tensorially self-absorbing (even strongly self-absorbing in the sense of [3]), one sees that the minimal tensor product $A \otimes B$ is $\mathcal{Z}$ -stable whenever $B$ is $\mathcal{Z}$ -stable. Indeed, one has

$(A \otimes B) \otimes \mathcal{Z} \cong A \otimes (B \otimes \mathcal{Z}) \cong A \otimes B.$

This raises the question above, which is largely unexplored. For the special case that $A=C(X)$ and $B$ is pure, this was asked by Chris Phillips after my talk on pure C*-algebras in Shanghai 2024. For the case $A=C([0,1])$ and $B$ a simple, pure C*-algebra with stable rank one and vanishing $K_1$ -group, a positive answer to the question can likely be deduced from Corollary 2.7 in [5].

[1] W. Winter. Nuclear dimension and Z-stability of pure C*-algebras. Invent. Math. 187 (2012), 259-342.
[2] M. Rørdam. The stable and the real rank of Z-absorbing C*-algebras. Internat. J. Math. 15 (2004), 1065-1084.
[3] A. Toms, W. Winter. Strongly self-absorbing C*-algebras. Trans. Amer. Math. Soc. 359 (2007), 3999-4029.
[4] R. Antoine, F. Perera, H. Thiel. Tensor products and regularity properties of Cuntz semigroups. Mem. Amer. Math. Soc. 251 (2018).
[5] R. Antoine, F. Perera, L. Santiago. Pullbacks, C(X)-algebras, and their Cuntz semigroup. J. Funct. Anal. 260 (2011), 2844-2880.

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