Extending Characters of Fixed Point Algebras

A dynamical system is a triple $(A,G,\alpha)$, consisting of a unital locally convex algebra $A$, a topological group $G$ and a group homomorphism $\alpha:G\rightarrow\Aut(A)$, which induces a continuous action of $G$ on $A$. Further, a unital locally convex algebra $A$ is called continuous inverse algebra, or CIA for short, if its group of units $A^{\times}$ is open in $A$ and the inversion $\iota:A^{\times}\rightarrow A^{\times},\,\,\,a\mapsto a^{-1}$ is continuous at $1_A$. For a compact manifold $M$, the Fr\'echet algebra of smooth functions $C^{\infty}(M)$ is the prototype of such a continuous inverse algebra. We show that if $A$ is a complete commutative CIA, $G$ a compact group and $(A,G,\alpha)$ a dynamical system, then each character of $B:=A^G$ can be extended to a character of $A$. In particular, the natural map on the level of the corresponding spectra $\Gamma_A\rightarrow\Gamma_B$, $\chi\mapsto\chi_{\mid B}$ is surjective.