Representations of dense subalgebras of C*-algebras with applications to spectral invariance

We show that, for certain groups and actions, every differentiable irreducible Frechet space representation of a dense subalgebra {dollar}G times {lcub}cal S{rcub}(M){dollar} of the transformation group C*-algebra {dollar}G times Csb0(M){dollar} is contained in a *-representation of {dollar}G times Csb0(M){dollar} on a Hilbert space if and only if {dollar}G times Csb0(M){dollar} is CCR. Here G is a closed subgroup of a simply connected nilpotent Lie group, M is a locally compact space on which G acts with closed orbits, and the isotropy subgroups are semidirect products of free finitely generated abelian groups with the abelian group {dollar}IRsp{lcub}n{rcub}{dollar}.; We define a general dense Frechet subalgebra {dollar}G timessp{lcub}L{rcub} A{dollar} of the crossed product {dollar}Lsp1(G,B){dollar}. A is a dense Frechet subalgebra of a Banach algebra B, and L is a submultiplicative length function on {dollar}G{dollar}. The crossed product {dollar}G timessp{lcub}L{rcub} A{dollar} consists of differentiable A-valued functions on G, rapidly vanishing in L In the case of a transformation group C*-algebra {dollar}G times Csb0(M){dollar}, we define a dense subalgebra {dollar}G timessp{lcub}L{rcub} {lcub}cal S{rcub}(M){dollar}, where {dollar}{lcub}cal S{rcub}(M){dollar} consists of G-differentiable functions, rapidly vanishing with respect to some scale on M.; Every irreducible representation of the dense subalgebra {dollar}G times {lcub}cal S{rcub}(M){dollar} is supported on the closure of an orbit. If {dollar}H{dollar} is a closed subgroup of G, we show that {dollar}G times {lcub}cal S{rcub}(G/H){dollar} is Morita equivalent to the group Schwartz algebra {dollar}{lcub}cal S{rcub}(H){dollar} in an appropriate sense, so that the representations of {dollar}G times {lcub}cal S{rcub}(G/H){dollar} correspond to ones of {dollar}{lcub}cal S{rcub}(H){dollar}.; The problem of determining when the irreducible representations of a dense subalgebra extend to representations of a C*-algebra is related to the problem of spectral invariance for the dense subalgebra. If all the differentiable irreducible representations of a dense differentiable m-convex Frechet subalgebra A of a Banach algebra B extend to representations of B, then A is spectral invariant in B.; We also show that A is spectral invariant in B if and only if every simple A-module is contained in a B-module. We use this to give a simple proof that the {dollar}n times n{dollar} matrices {dollar}Msb{lcub}n{rcub}(A){dollar} are spectral invariant in {dollar}Msb{lcub}n{rcub}(B){dollar} if A is spectral invariant in B. (Abstract shortened with permission of author.)...