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 $\alpha :G\to Aut\left(A\right)$ that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units $A^{\times }$ is open in A and the inversion map $\iota :A^{\times }\to A^{\times }$, $a\mapsto a^{-1}$ is continuous at $1_A$. Given a dynamical system $(A,G,\alpha )$ with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A.

1. Introduction

Let $\sigma :P\times G\to P$ be a smooth action of a Lie group G on a manifold P. It is well-known (see e.g., [1], Proposition 2.1) that $\sigma$ induces a smooth action of G on the unital Fréchet algebra $C^{\infty }\left(P\right)$ of smooth functions on P defined by ${\alpha }_{\sigma }:G\times C^{\infty }\left(P\right)\to C^{\infty }\left(P\right)$, $(g,f)\mapsto f\circ {\sigma }_g$. The corresponding fixed point algebra is given by

$$\begin{array}{c}\hfill C^{\infty }{\left(P\right)}^G:=\{f\in C^{\infty }\left(P\right):(\forall g\in G){\alpha }_{\sigma }(g,f)=f\}.\end{array}$$

The origin of this short article is, in a manner of speaking, “commutative geometry”, namely the question whether each character $\chi :C^{\infty }{\left(P\right)}^G\to \mathbb{C}$ extends to a character $\tilde{\chi }:C^{\infty }\left(P\right)\to \mathbb{C}$ (cf. [2,3]).

One possible way to approach this problem is to classify the characters under consideration. Indeed, it follows from ([1], Lemma A.1) that each character $\chi :C^{\infty }\left(P\right)\to \mathbb{C}$ is an evaluation in some point $p\in P$, that is, of the form ${\delta }_p:C^{\infty }\left(P\right)\to \mathbb{C}$, $f\mapsto f\left(p\right)$. If the action $\sigma$ is additionally free and proper, then the orbit space $P/G$ has a unique manifold structure such that the canonical quotient map $q:P\to P/G$, $p\mapsto \left[p\right]$ is a submersion. Moreover, in this situation, the map

$$\begin{array}{c}\hfill \mathsf{\Phi }:C^{\infty }{\left(P\right)}^G\to C^{\infty }(P/G),f\mapsto \left(\left[p\right]\mapsto f\left(p\right)\right)\end{array}$$

is an isomorphism of unital Fréchet algebras showing that each character $C^{\infty }{\left(P\right)}^G\to \mathbb{C}$ is of the form ${\delta }_{\left[p\right]}\circ \mathsf{\Phi }$ for some $p\in P$ which may simply be extended by ${\delta }_p$.

In this note, however, we approach the above problem in a more systematic way. In fact, given a dynamical system $(A,G,\alpha )$ with a complete commutative continuous inverse algebra (CIA) A and a compact group G, we show that each character of the corresponding fixed point algebra

$$\begin{array}{c}\hfill A^G:=\{a\in A:(\forall g\in G)\alpha \left(g\right)\left(a\right)=a\}\end{array}$$

extends to a character of A (Theorem 2). Our approach is motivated by the following three facts:

(i)

Our initial question is, after all, of purely topological nature.

(ii)

If P is compact, then $C^{\infty }\left(P\right)$ is the prototype of a complete commutative CIA.

(iii)

CIA’s provide a class of algebras for which characters are automatically continuous (cf. [4], Lemma 2.3).

We would also like to mention that CIAs are naturally encountered in K-theory and noncommutative geometry, usually as dense unital subalgebras of C${}^{*}$-algebras. Finally, we point out that a classical result for actions of finite groups can be found in ([5], Chapter 5, §2.1, Corollary 4).

2. Preliminaries and Notations

All algebras are assumed to be complex. The spectrum of an algebra A is the set ${\Gamma }_A:={Hom}_{\mathrm{alg}}(A,\mathbb{C})\backslash \left\{0\right\}$ (endowed with the topology of pointwise convergence on A) and its elements are called characters. Moreover, given a compact group G, we denote by $\widehat{G}$ the (countable) set of equivalence classes of finite-dimensional irreducible representations of G. For $\pi \in \widehat{G}$ we write ${\chi }_{\pi }$ for the function defined by $G\mapsto \mathbb{C}$, $g\mapsto \mathrm{tr}\left(\pi \right(g\left)\right)$ and we put $d_{\pi }:={\chi }_{\pi }\left(1_G\right)$ for the corresponding dimension. We also need the following well-known structure theorem for dynamical systems:

Lemma 1.

([6], [Lemma 3.2 and Theorem 4.22]). Let $(A,G,\alpha )$ be a dynamical system with a complete unital locally convex algebra A and a compact group G. Furthermore, given $\pi \in \widehat{G}$ and $a\in A$, let

$$\begin{array}{c}\hfill P_{\pi }\left(a\right):=d_{\pi }{\int }_G\overline{{\chi }_{\pi }}\left(g\right)\left(\alpha \left(g\right)\left(a\right)\right)dg,\end{array}$$

where $dg$ denotes the normalized Haar measure on G. Then the following assertions hold:

(a)

For each$\pi \in \widehat{G}$the map$P_{\pi }:A\to A$is a continuous G-equivariant projection onto the G-invariant subspace$A_{\pi }:=P_{\pi }\left(A\right)$. In particular,$A_{\pi }$is algebraically and topologically a direct summand of A.

(b)

The module direct sum$A_{\mathrm{fin}}:={⨁}_{\pi \in \widehat{G}}{A}_{\pi }$is a dense subalgebra of A.

3. Extension Results

In this section our main results are stated and proved. We begin with some general statements on the extendability of ideals.

Lemma 2.

Let$(A,G,\alpha )$be a dynamical system with a complete unital locally convex algebra A and a compact group G. Then the following assertions hold:

(a)

If I is a proper left ideal in$A^G$, then$A_{\mathrm{fin}}·I={⨁}_{\pi \in \widehat{G}}{A}_{\pi }·I$defines a proper left ideal in$A_{\mathrm{fin}}$that contains I.

(b)

If I is a proper closed left ideal in$A^G$and J is the closure of$A_{\mathrm{fin}}·I$in$A_{\mathrm{fin}}$, then J is a proper closed left ideal in$A_{\mathrm{fin}}$that contains I.

Proof. 

(a) We first observe that $A^G$ coincides with $A_1$ (where 1 stands for the equivalence class of the trivial representation). Hence $I\subseteq A^G$ is contained in $A_{\mathrm{fin}}$ and thus $A_{\mathrm{fin}}·I$ is the left ideal of $A_{\mathrm{fin}}$ generated by I. Using the integral formula for $P_{\pi }$ from Lemma 1, we see that $A_{\pi }·I\subseteq A_{\pi }$, entailing that the sum in part (a) is direct. To see that $A_{\mathrm{fin}}·I$ is proper, we assume the contrary, that is,

$$\begin{array}{c}\hfill 1_A\in A_{\mathrm{fin}}·I=\underset{\pi \in \widehat{G}}{⨁}{A}_{\pi }·I.\end{array}$$

Then $1_A\in A^G$ implies that $1_A\in A^G·I=I$, which contradicts the fact that I is a proper left ideal of $A^G$. We conclude that $A_{\mathrm{fin}}·I$ is a proper left ideal in $A_{\mathrm{fin}}$ that contains I.

(b) Part (a) and the definition of J imply that J is a closed left ideal in $A_{\mathrm{fin}}$ that contains I. To see that J is proper, we again assume the contrary, that is, $1_A\in J$. Then there exists a net ${\left(a_{\gamma }\right)}_{\gamma \in \Gamma }$ in $A_{\mathrm{fin}}·I$ such that ${lim}_{\gamma }{a}_{\gamma }=1_A$ and the continuity of the projection map $P_1:A\to A$ onto the fixed point algebra $A^G$ implies that

$$\begin{array}{c}\hfill 1_A=P_1\left(1_A\right)=P_1\left(\underset{\gamma }{lim}{a}_{\gamma }\right)=\underset{\gamma }{lim}{P}_1\left(a_{\gamma }\right).\end{array}$$

Since I is closed in $A^G$ and $P_1\left(a_{\gamma }\right)\in A^G·I=I$ for all $\gamma \in \Gamma$, we conclude that $1_A\in I$. This contradicts the fact that I is a proper ideal of $A^G$ and therefore J is a proper closed left ideal in $A_{\mathrm{fin}}$ that contains I. ☐

Lemma 3.

Let A be a topological algebra and B a dense subalgebra of A. If I is a proper closed left ideal in B, then$\overline{I}$is a proper closed left ideal in$\overline{B}=A$.

Proof. 

It is easily seen that $\overline{I}$ is a closed left ideal in $\overline{B}=A$. Moreover, we have $I=\overline{I}\cap B$. Indeed, the inclusion $“\subseteq ”$ is obvious and for the other inclusion we use the fact that I is closed in B. Consequently, if $\overline{I}$ is not proper, that is, $\overline{I}=A$, then $I=B$, which yields a contradiction. Hence, $\overline{I}$ is a proper closed left ideal in A. ☐

We are now ready to state and prove our main extension results.

Theorem 1.

(Extending ideals). Let $(A,G,\alpha )$ be a dynamical system with a complete unital locally convex algebra A and a compact group G. Then each proper closed left ideal in $A^G$ is contained in a proper closed left ideal in A.

Proof. 

Let I be a proper closed left ideal in $A^G$. Then Lemma 2 (b) implies that I is contained in a proper closed left ideal in $A_{\mathrm{fin}}$. Since $A_{\mathrm{fin}}$ is a dense subalgebra of A by Lemma 1 (b), the claim is a consequence of Lemma 3. ☐

Theorem 2.

(Extending characters). Let $(A,G,\alpha )$ be a dynamical system with a complete commutative CIA A and a compact group G. Then each character $\chi :A^G\to \mathbb{C}$ is continuous and extends to a continuous character $\tilde{\chi }:A\to \mathbb{C}$.

Proof. 

Let $\chi :A^G\to \mathbb{C}$ be a character. Since $A^G$ carries the structure of a CIA in its own right, it follows from ([4], Lemma 2.3) that $\chi$ is continuous which shows that $I:=ker\chi$ is a proper closed ideal in $A^G$. Hence, Theorem 1 implies that I is contained in a proper closed ideal in A. In particular, it is contained in a proper maximal ideal J of A which, according to ([7], Lemma 2.2.2) and ([4], Lemma 2.3), is the kernel of some continuous character $\tilde{\chi }:A\to \mathbb{C}$. Since I is a maximal ideal in the unital algebra $A^G$ and

$$\begin{array}{c}\hfill I=I\cap A^G\subseteq J\cap A^G\subseteq A^G,\end{array}$$

we conclude that $I=J\cap A^G$. Therefore, the decomposition $A^G=I\oplus \mathbb{C}=(J\cap A^G)\oplus \mathbb{C}$ finally proves that $\tilde{\chi }$ extends $\chi$. ☐

Remark 1.

It is not clear how to extend Theorem 2 beyond the class of CIAs. For instance, given a non-compact manifold P, the set $C_c^{\infty }\left(P\right)$ of compactly supported smooth functions on P is a proper ideal in $C^{\infty }\left(P\right)$. As such it is contained in a proper maximal ideal in $C^{\infty }\left(P\right)$ that cannot be closed since $C_c^{\infty }\left(P\right)$ is dense in $C^{\infty }\left(p\right)$. However, in the more general situation of a complete commutative unital locally convex algebra A, a similar argument as in the proof of Theorem 2 shows that each continuous character $\chi :A^G\to \mathbb{C}$ can be extended to a character $\tilde{\chi }:A\to \mathbb{C}$.

We conclude with the following two immediate corollaries.

Corollary 1.

Suppose we are in the situation of Theorem 2. Then the natural map on the level of spectra ${\Gamma }_A\to {\Gamma }_{A^G}$, $\chi \mapsto {\chi }_{\mid A^G}$ is surjective.

Corollary 2.

Let$(C^{\infty }\left(P\right),G,\alpha )$be a dynamical system with a compact manifold P and a compact group G. Then each character$\chi :C^{\infty }{\left(P\right)}^G\to \mathbb{C}$extends to a character$\tilde{\chi }:C^{\infty }\left(P\right)\to \mathbb{C}$.

Remark 2.

Given a dynamical system$(C^{\infty }\left(P\right),G,\alpha )$with a compact manifold P and a compact group G, we would like to describe${\Gamma }_{C^{\infty }{\left(P\right)}^G}$as a set of points associated to P and G. As already explained in the introduction, it is not hard to see that${\Gamma }_{C^{\infty }{\left(P\right)}^G}$is homeomorphic to$P/G$if G is a Lie group and α is induced by a free and smooth action of G on P. However, even if we do not have any additional information, it is still possible to show that the map

$$\begin{array}{c}\hfill P/G\to {\Gamma }_{C^{\infty }{\left(P\right)}^G},q\left(p\right)\mapsto {\delta }_p\end{array}$$

is a homeomorphism (see e.g., [2], Proposition 8.7) and Corollary 2 may be used to verify its surjectivity.