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The Calkin algebra has outer automorphisms. (English) Zbl 1220.46040

Let \(H\) be a separable infinite-dimensional Hilbert space, \(L(H)\) the \(C^*\)-algebra of all bounded linear operators on \(H\), \(K(H)\) the \(C^*\)-algebra of all compact operators on \(H\) and \(\mathcal Q = L(H)/K(H)\) the Calkin algebra. An inner automorphism of \(\mathcal Q\) is an automorphism of the form \(x\mapsto u^*xu\), where \(u\in \mathcal Q\) is unitary. A long-standing open problem in operator algebra theory, first formulated by L. G. Brown, R. G. Douglas and P. A. Fillmore [Ann. Math. (2) 105, 265–324 (1977; Zbl 0376.46036)], asks whether every automorphism of \(\mathcal Q\) is inner. In the paper under review, the authors show that the Continuum Hypothesis implies that \(\mathcal Q\) has \(2^{\aleph_1}\) outer automorphisms; in particular, assuming the Continuum Hypothesis, \(\mathcal Q\) has outer automorphisms. To achieve this they construct, for each function \(\epsilon: [0,\aleph_1)\rightarrow\{0,1\}\) (where \([0,\aleph_1)\) is the interval of countable ordinals), an upward directed family \((A_{\beta}^{\epsilon})_{\beta < \aleph_1}\) of separable \(C^*\)-subalgebras of \(\mathcal Q\), indexed by the countable ordinals, whose union equals \(\mathcal Q\), and an automorphism \(\varphi^{\epsilon}\) of \(\mathcal Q\) whose restriction \(\varphi_{\beta}^{\epsilon}\) to \(A_{\beta}^{\epsilon}\) is inner and which is the pointwise limit of the net \(\{\varphi_{\beta}^{\epsilon} \colon \beta < \aleph_1\}\). The statement follows from the fact that different \(\epsilon\)’s give rise to different \(\varphi^{\epsilon}\)’s. The construction of the algebras \(A_{\beta}^{\epsilon}\) and the automorphisms \(\varphi_{\beta}^{\epsilon}\) uses the following theorem, whose proof relies on results of W. Arveson [Duke Math. J. 44, 329–355 (1977; Zbl 0368.46052)] and V. Manuilov and K. E. Thomsen [Proc. Lond. Math. Soc., III. Ser. 88, No. 2, 455–478 (2004; Zbl 1045.19001)]: Let \(A\) be a separable \(C^*\)-algebra, \(B\subseteq C_b([1,\infty),A)/C_0([1,\infty),A)\) be a separable \(C^*\)-subalgebra containing the canonical image of \(A\) in \(C_b([1,\infty),A)/C_0([1,\infty),A)\) and \(E\) be a non-unital \(C^*\)-algebra with countable approximate identity. (Here \(C_b([1,\infty),A)\) (resp. \(C_0([1,\infty),A)\)) is the \(C^*\)-algebra of all bounded (resp. vanishing at infinity) \(A\)-valued functions on \([1,\infty)\).) Then every \(^*\)-homomorphism from \(A\) into \(M(E)/E\) can be extended to a \(^*\)-homomorphism from \(B\) into \(M(E)/E\).
Reviewer’s remark: In a preprint [arxiv:0705.3085], I. Farah has shown that it is consistent with the Zermelo-Fraenkel Set Theory that all automorphisms of the Calkin algebra be inner, and has given an alternative proof of the existence of outer automorphisms assuming the Continuum Hypothesis.

MSC:

46L40 Automorphisms of selfadjoint operator algebras
03E50 Continuum hypothesis and Martin’s axiom
46L05 General theory of \(C^*\)-algebras
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References:

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