Extending Characters of Fixed Point Algebras

TL;DR

This paper proves that for certain algebraic dynamical systems involving continuous inverse algebras and compact groups, characters of fixed point algebras can be extended to the larger algebra, ensuring a surjective spectral map.

Contribution

It establishes the extension of characters from fixed point algebras to the original algebra in the setting of complete commutative CIA and compact group actions.

Findings

Characters of fixed point algebras extend to the original algebra.

The spectral map from the algebra to the fixed point algebra is surjective.

Results apply to algebras like smooth functions on compact manifolds.

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 $\alpha:G\rightarrow\Aut(A)$, 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.