Question:Is the minimal tensor product of two C*-algebras pure whenever one of them is?

**Background**: Following Winter [1], a C*-algebra 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 -stability, where denotes the Jiang-Su algebra, and a C*-algebra is said to be -stable if . Rørdam [2] showed that every -stable C*-algebra is pure, and the Toms-Winter conjecture predicts that a separable, unital, simple, nuclear C*-algebra is -stable if (and only if) it is pure. Moreover, it was shown in [4] that a C*-algebra is pure if and only if , that is, its Cuntz semigroup tensorially absorbs the Cuntz semigroup of . One may therefore think of pureness as -stability at the level of the Cuntz semigroup.

Using that is tensorially self-absorbing (even strongly self-absorbing in the sense of [3]), one sees that the minimal tensor product is -stable whenever is -stable. Indeed, one has

This raises the question above, which is largely unexplored. For the special case that and is pure, this was asked by Chris Phillips after my talk on pure C*-algebras in Shanghai 2024. For the case and a simple, pure C*-algebra with stable rank one and vanishing -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.