Real rank of   B(H)\otimes B(H)

Von | Juni 20, 2020

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 andY, 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. [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. [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. [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.

Automorphisms of the Calkin algebra

Von | Juni 20, 2020

Does the Calkin algebra \mathcal{Q} admit an automorphism that induces the flip on  K_1(\mathcal{Q})?

Background/Motivation: Let H be a separable, infinite-dimensional Hilbert space. The Calkin algebra \mathcal{Q}:=\mathcal{B}(H)/\mathcal{K}(H) is the quotient of the bounded, linear operators on H by the closed, two-sided ideal of compact operators. The important problem of whether the Calkin algebra has outer automorphisms was eventually shown to be independent of the usual set theoretic axioms ZFC: Phillips and Weaver, ​[1]​, proved that the Continuum Hypothesis implies that \mathcal{Q} has many outer automorphism. This was complemented by a result of Farah, ​[2]​, who showed that the Open Coloring Axiom implies that all automorphisms of \mathcal{Q} are inner.

Given a C*-algebra A, every automorphism \alpha\colon A\to A induces a group automorphism K_1(A)\to K_1(A). It is known that K_1(\mathcal{Q})\cong\mathbb{Z}. Hence, an automorphism of \mathcal{Q} either induces the identity or the flip on K_1(\mathcal{Q}). Every inner automorphism of a C*-algebra acts trivially on K-theory. Therefore, the result of Farah shows that assuming the Open Coloring Axiom, every automorphism of \mathcal{Q} induces the identity on K_1(\mathcal{Q}). As it turns out, the outer automorphisms \alpha constructed by Phillips and Weaver are locally inner, that is, for every element a\in\mathcal{Q} there exists a unitary u\in\mathcal{Q} such that \alpha(a)=uau^*. (The unitary depends on the element.) Since the generator of K_1(\mathcal{Q}) is represented by a unitary in \mathcal{Q}, it follows that every locally inner automorphism of \mathcal{Q} induces the identity on K_1(\mathcal{Q}) as well. One may consider automorphisms of \mathcal{Q} that induce the flip on K_1(\mathcal{Q}) as ‚very outer‘, and the question is if such automorphisms exists (assuming some set-theoretic axioms).

  1. [1]
    N.C. Phillips, N. Weaver, The Calkin algebra has outer automorphisms, Duke Math. J. 139 (2007) 185–202. https://doi.org/10.1215/s0012-7094-07-13915-2.
  2. [2]
    I. Farah, All automorphisms of the Calkin algebra are inner, Ann. Math. 173 (2011) 619–661. https://doi.org/10.4007/annals.2011.173.2.1.