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.


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


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