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Outer automorphisms of injective \(C^*\)-algebras. (English) Zbl 0559.46026
Let A be a \(C^*\)-subalgebra of a \(C^*\)-algebra B and let \(\beta\) be an inner automorphism of B which leaves A invariant. When is the restriction of \(\beta\) to A an inner automorphism of A? That is, when is \(\beta\) implemented by a unitary in A, if A is unital, or by a unitary in M(A), the multiplier algebra of A, if A is not unital? A deep theorem of Kishimoto, which builds on the important earlier work of Elliott and Lance, shows that when A is separable and simple and when B is the second dual of A then the answer is ”always”. We proved in the previous work that when A is simple and B is the regular completion of A then the answer is also ”always”. We shall prove a much stronger result than we did. Let \(\alpha\) be an outer *-automorphism of A, where A is \(\alpha\)- simple. Let B be the injective envelope of A, which is characterized by the following two properties: First, B is injective. Secondly, if \(\phi\) is a completely positive map from B to B such that \(\phi(a+\lambda l)=a+\lambda l\) for all \(a\in A\) and all \(\lambda\in C\) then \(\phi\) is the identity map on B. Then Theorem 3.6 implies that \(\alpha\) has a unique extension to an outer *-automorphism of B.
The following elementary example illustrates what can go wrong. Let H be an infinite dimensional Hilbert space, let B be the algebra of all bounded operators on H and let A be the subalgebra of B generated by the identity of B and the algebra of compact operators on H. Then each unitary in B induces an automorphism of A which, in general, will not be inner.

46L40 Automorphisms of selfadjoint operator algebras
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