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Zbl 1199.46143
Sakai, Shôichirô
Recent topics on $C^*$-algebras (consistency and independency) and Kadison-Singer problem.
(English)
[A] Dykema, Ken (ed.) et al., Von Neumann algebras in Sibiu. Proceedings of the conference, Sibiu, Romania, June 9--16, 2007. Bucharest: Theta. Theta Series in Advanced Mathematics 10, 103-109 (2008). ISBN 978-973-87899-4-4

Summary: In [in: Advances in quantum dynamics. AMS-IMS-SIAM joint summer research conference, South Hadley, MA, USA, June 16--20, 2002. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 335, 247--251 (2003; Zbl 1049.46046)], the author asked whether the Calkin algebra has an outer $*$-automorphism. Recently, {\it N. C. Phillips} and {\it N. Weaver} [Duke Math. J. 139, No.~1, 185--202 (2007; Zbl 1220.46040)] and {\it I. Farah} [All automorphisms on the Calkin algebra are inner, preprint, arXiv:0705.3085] have proved that the statement ``the Calkin algebra has an outer $*$-automorphism" is independent of ZFC. First, we explain these matters briefly. Next, we discuss the outstanding problem of Kadison-Singer and show some new results on the problem.
MSC 2000:
*46L40 Automorphisms of C*-algebras
46L05 General theory of C*-algebras
03E35 Consistency and independence results (set theory)
03E50 Continuum hypothesis and generalizations (logic)

Keywords: $C^*$-algebra; $W^*$-algebra; $*$-automorphism; Kadison-Singer problem; Calkin algebra; consistency; independence

Citations: Zbl 1220.46040; Zbl 1049.46046

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