Extending Characters of Fixed Point Algebras

Abstract

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 $α:G\rightarrow\Aut(A)$, 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 $\iota :A^{\times }\to A^{\times },a\mapsto a^{-1}$ is continuous at $1_A$. For a compact manifold $M$, the Fréchet 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 ${\Gamma }_A\to {\Gamma }_B$, $\chi \mapsto {\chi }_{\mid B}$ is surjective.

All comments are welcome

Cited by in corpus (1)

• A Geometric Approach to Noncommutative Principal Torus Bundles