Proper Actions of Groups on C*-Algebras

Group actions on topological graphs

“…Recall from [Rie90] that the action α of a locally compact group G on a C * -algebra A is proper if there is a dense α-invariant * -subalgebra A 0 of A such that for every a, b ∈ A 0 the functions …”

“…[Rie04, Theorem 5.7] implies that the action α is proper and saturated with respect to A 0 , and so by [Rie90] the reduced crossed product A × α,r G is Morita equivalent to a generalized fixed-point algebra A α . It was shown in [KQR08, Proposition 3.1] that Fix(A, α, ϕ) coincides with the algebra A α of [Rie90].…”

“…It was shown in [KQR08, Proposition 3.1] that Fix(A, α, ϕ) coincides with the algebra A α of [Rie90].…”

Naturality of Rieffel’s Morita Equivalence for Proper Actions

“…There has nevertheless been considerable interest in situations where there is a useful analogue of A α in the multiplier algebra M(A) [5,8,23,30,33,36,37]. Here we are particularly interested in the proper actions introduced by Rieffel in [36]; in the motivating example, A = C 0 (T ) is commutative, G acts properly on the right of T , α is the action rt of G by right translation on functions, and the algebra C 0 (T /G), which we can view as a subalgebra of C b (T ) = M(C 0 (T )), is an excellent substitute for the missing fixed-point algebra.…”

PARAMETRIZED STRICT DEFORMATION QUANTIZATION OF C*-BUNDLES AND HILBERT C*-MODULES

“…Since η * (Y ) is the total space of a principal -bundle over the compact space Y , the action of on η * (Y ) is free and proper, so the action of on C 0 (η * (Y )) σ is also proper in the sense of Rieffel [20]. In particular, the fixed-point algebra C 0 (η * (Y )) σ makes sense and is the parametrized strict deformation quantization of C 0 (Y ).…”