Real rank of tensor products of 𝐶*-algebras

Kodaka, Kazunori; Osaka, Hiroyuki

doi:10.1090/S0002-9939-1995-1264820-4

Real rank of tensor products of $C^ \ast$-algebras
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by Kazunori Kodaka and Hiroyuki Osaka

Proc. Amer. Math. Soc. 123 (1995), 2213-2215

DOI: https://doi.org/10.1090/S0002-9939-1995-1264820-4

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Abstract:

We study the real rank of tensor products of ${C^ \ast }$-algebras. From the dimension theory: $\dim (X \times Y) \leq \dim X + \dim Y$, it is naturally hoped that $RR(A \otimes B) \leq RR(A) + RR(B)$. We then prove that it is false generally. Moreover, we point out that (FS)-property for ${C^ \ast }$-algebras is not stable under taking tensor products.

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