Fixed-point algebra of an action

For an automorphism $\alpha$ of a unital operator algebra $A$ , its fixed-point algebra is

A^\alpha=\{a\in A:\alpha(a)=a\}.

For a group action use $A^G=\{a:\alpha_g(a)=a\text{ for every }g\in G\}$ . These are unital $*$ -subalgebras, norm closed in a $C^*$ -algebra and ultraweakly closed for normal actions on a von Neumann algebra.

Ergodicity

A scalar fixed algebra is the defining condition for an ergodic operator-algebra action.