Real rank of B(H) tensor B(H)

Given a separable, infinite-dimensional Hilbert space $\Huge{H}$ , what is the real rank of the minimal tensor product $\mathcal{B}(H)\otimes \mathcal{B}(H)$ ?

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 $A$ a number (its real rank) $\mathrm{rr}(A)\in\{0,1,\ldots,\infty\}$ . 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 $X$ and $Y$ , the product theorem for covering dimension shows that under certain weak assumptions (for example, both spaces are metric, or compact) we have $\dim(X\times Y)\leq\dim(X)+\dim(Y)$ . One might therefore expect that $\mathrm{rr}(A\otimes B)\leq\mathrm{rr}(A)+\mathrm{rr}(B)$ , but it was shown in [2] that this does not hold: there exists C*-algebras $A$ and $B$ such that

$\mathrm{rr}(A)=\mathrm{rr}(B)=0, \text{ yet } \mathrm{rr}(A\otimes B)>0.$

Later, Osaka showed in [3] that one can even take $A=B=\mathcal{B}(H)$

. Indeed, it is well known that von Neumann algebras have real rank zero, and hence so does $\mathcal{B}(H)$

. Osaka showed that $\mathcal{B}(H)\otimes\mathcal{B}(H)$

does not have real rank zero. This raises the question of what the real rank of $\mathcal{B}(H)\otimes\mathcal{B}(H)$

is, see Question 3.3 in [3].

[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.

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