Given a separable, infinite-dimensional Hilbert space , what is the real rank of the minimal tensor product ?
Background/Motivation: The real rank is a noncommutative dimension theory that was introduced by Brown and Pedersen in . It associates to each C*-algebra a number (its real rank) . The lowest and most interesting value is zero. One can think of C*-algebras of real rank zero as zero-dimensional, noncommutative spaces. (There are other noncommutative dimension theories, and they lead to different concepts of zero- or low-dimensional noncommutative spaces.) For topological spaces and , the product theorem for covering dimension shows that under certain weak assumptions (for example, both spaces are metric, or compact) we have . One might therefore expect that , but it was shown in  that this does not hold: there exists C*-algebras and such that
Later, Osaka showed in  that one can even take . Indeed, it is well known that von Neumann algebras have real rank zero, and hence so does . Osaka showed that does not have real rank zero. This raises the question of what the real rank of is, see Question 3.3 in .
- L.G. Brown, G.K. Pedersen, C*-algebras of real rank zero, Journal of Functional Analysis. 99 (1991) 131–149. https://doi.org/10.1016/0022-1236(91)90056-b.
- K. Kodaka, H. Osaka, Real Rank of Tensor Products of C*-Algebras, Proceedings of the American Mathematical Society. 123 (1995) 2213–2215. https://doi.org/10.2307/2160959.
- H. Osaka, Certain C*-algebras with non-zero real rank and extremal richness, MATH. SCAND. 85 (1999) 79. https://doi.org/10.7146/math.scand.a-13886.