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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.

##### MSC:
 46L40 Automorphisms of selfadjoint operator algebras
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