| 概要 | Let  be a separable infinite dimensional Hilbert space, for example, the space  of all square summable sequences. Let ) be the algebra of all continuous linear maps from to~ and let ) be the closure in of the set of continuous linear maps which have finite rank. Then is an ideal in ,) and we can form the quotient algebra  / K (H).) It is called the Calkin algebra, and is an example of a C*-algebra. Question: Does the Calkin algebra have outer automorphisms, that is, automorphisms not of the form  for suitable  ?
It turns out that this question is undecidable in ZFC. If one assumes the Continuum Hypothesis, then outer automorphisms exist. In fact, there are more automorphisms than there are possible choices for~ (This is joint work with Nik Weaver.) However, Ilijas Farah proved that it is consistent with ZFC that  ~has no outer automorphisms.
This talk is intended for a general audience. I will describe some of the background, and give several of the ideas of the proof that the Continuum Hypothesis implies the existence of outer automorphisms of the Calkin algebra.
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