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 [1]. 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 [2] that this does not hold: there exists C*-algebras
and
such that
![Rendered by QuickLaTeX.com A=B=\mathcal{B}(H)](http://hannesthiel.org/wp-content/ql-cache/quicklatex.com-638afc2962fad7da1fe036fbcdf66237_l3.png)
![Rendered by QuickLaTeX.com \mathcal{B}(H)](http://hannesthiel.org/wp-content/ql-cache/quicklatex.com-b9c964516885a96fcb2d5e3db50b3a3d_l3.png)
![Rendered by QuickLaTeX.com \mathcal{B}(H)\otimes\mathcal{B}(H)](http://hannesthiel.org/wp-content/ql-cache/quicklatex.com-fc1c441ce1e5e206a20eeecf35b9dfad_l3.png)
![Rendered by QuickLaTeX.com \mathcal{B}(H)\otimes\mathcal{B}(H)](http://hannesthiel.org/wp-content/ql-cache/quicklatex.com-fc1c441ce1e5e206a20eeecf35b9dfad_l3.png)
- [1]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.
- [2]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.
- [3]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.