*-automorphism - Knowlpedia

Let $A$ be a $C^*$ -algebra. A $*$ -automorphism of $A$ is a $*$ -isomorphism $\alpha:A\to A$ ; equivalently, it is a bijective complex-linear map that preserves multiplication and involution. Its inverse is again a $*$ -automorphism. Under composition, all such maps form the automorphism group $\operatorname{Aut}(A)$ . If $A$ is unital, every $*$ -automorphism preserves $1_A$ automatically. No topology on $\operatorname{Aut}(A)$ , continuity in an external parameter, or choice of an implementing operator is included in the definition.

Inner and outer automorphisms

For a unitary $u$ in a unital $C^*$ -algebra, conjugation

\operatorname{Ad}_u(a)=uau^*

is a $*$ -automorphism. Such automorphisms are inner; automorphisms not of this form are outer. For a nonunital algebra, inner automorphisms are naturally implemented by unitaries in the multiplier algebra. The quotient by inner automorphisms records genuinely external symmetries.

Preserved structure

Every $*$ -automorphism is isometric and preserves spectra, positivity, ideals, approximate identities, and functional calculus. It therefore acts on the state space by pullback and transports representations by composition. For commutative $A=C_0(X)$ , $*$ -automorphisms correspond contravariantly to homeomorphisms of $X$ , giving the basic geometric model.

Dynamical conventions and near-misses

A $C^*$ -dynamical system requires an action $g\mapsto\alpha_g$ by $*$ -automorphisms together with a stated continuity condition; individual automorphisms are automatically norm-continuous as maps $A\to A$ , but this does not make the parameter map continuous. A conjugate-linear multiplicative involution, an anti-automorphism that reverses products, or a merely isometric linear bijection is not a $*$ -automorphism unless it also satisfies every algebraic requirement in the core.

References

Gert K. Pedersen, C-Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.2 and Chapter 8 on automorphisms, inner implementation, and automorphism groups.
Gerard J. Murphy, C-Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §§2.1–2.2 on $*$ -isomorphisms, commutative $C^*$ -algebras, and preserved structure.