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Recent topics on $C^*$-algebras (consistency and independency) and Kadison-Singer problem. (English) Zbl 1199.46143
Dykema, Ken (ed.) et al., Von Neumann algebras in Sibiu. Proceedings of the conference, Sibiu, Romania, June 9--16, 2007. Bucharest: Theta (ISBN 978-973-87899-4-4). Theta Series in Advanced Mathematics 10, 103-109 (2008).
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, \url{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. For the entire collection see [Zbl 1162.46004].
MSC:
46L40Automorphisms of $C^*$-algebras
46L05General theory of $C^*$-algebras
03E35Consistency; independence results (set theory)
03E50Continuum hypothesis; Martin’s axiom (logic)