Semiprojectivity with and without a group action

Abstract

Abstract. The equivariant version of semiprojectivity was recently introduced by the first author. We study properties of this notion, in particular its relation to ordinary semiprojectivity of the crossed product and of the algebra itself.We show that equivariant semiprojectivity is preserved when the action is restricted to a cocompact subgroup. Thus, if a second countable compact group acts semiprojectively on a C * -algebra A, then A must be semiprojective. This fails for noncompact groups: we construct a semiprojective action of Z on a nonsemiprojective C * -algebra.We also study equivariant projectivity and obtain analogous results, however with fewer restrictions on the subgroup. For example, if a discrete group acts projectively on a C * -algebra A, then A must be projective. This is in contrast to the semiprojective case.We show that the crossed product by a semiprojective action of a finite group on a unital C * -algebra is a semiprojective C * -algebra. We give examples to show that this does not generalize to all compact groups.Equivariant semiprojectivity was introduced in [Phi12], by applying the usual definition of semiprojectivity to the category of unital G-algebras (C * -algebras with actions of the group G) with unital G-equivariant * -homomorphisms. See Definition 1.1 below. The purpose of [Phi12] was to show that certain actions of compact groups on various specific C * -algebras are semiprojective. In particular, it is shown that any action of a second countable compact group on a finite dimensional C * -algebra is semiprojective, and that for n < ∞, quasifree actions of second countable compact groups on the Cuntz algebras O n are semiprojective.In this paper we study equivariant semiprojectivity more abstractly. We also introduce equivariant projectivity and carry out a parallel study of it. We extend the definition to allow actions by general locally compact groups, and we consider the nonunital version of equivariant semiprojectivity.From the work in [Phi12], it is not even clear whether a semiprojective action of a noncompact group can exist. One reason for skepticism was that the trivial

Crossed products by spectrally free actions

“…Let C = C 0 ((0, 1]) ⊗ C(Z n ) be the cone over C(Z n ), with the action of Z n obtained by letting Z n act on Z n by translation and letting Z n act trivially on C 0 ((0, 1]). Then C is equivariantly projective by Proposition 2.10 of [30]. Therefore C is equivariantly semiprojective.…”

“…The analogous statement for other groups is known only for cyclic groups whose order is a power of 2. (See Proposition 2.10 of [30], where equivariant projectivity is proved.) Second, the combinatorics in proofs such as that of Proposition 4.10 would need to be much more complicated, and may not give strong enough inequalities to use as hypotheses in an analog of Lemma 4.8.…”

“…Therefore C is equivariantly semiprojective. It follows from Lemma 1.5 of [30] that the unitization C + of C is equivariantly semiprojective in the unital category.…”

Regularity of simple nuclear realC*-algebras under tracial conditions

“…Thus σ(xz) = σ(x)σ(z) and hσ(z) = σ(z)h for each x, z ∈ M r (C), from which it follows that π(f ⊗ x) = f (h)σ(x) defines the required * -homomorphism. , (Ψ(f )(t) = Ψ(f (t)) for 0 < t ≤ 1 and equivariant projectivity is defined in [16] but is here extended to include anti-automorphic actions of Z 2 as well as automorphic actions. In [13], this was called projectivity in the category C * ,τ .…”

Compact group actions with the Rokhlin property

“…The following is essentially Definition 1.1 in [34]; see also [36] Definition 4.1. Let G be a locally compact group, let A be a C ˚-algebra, and let α : G Ñ AutpAq be a continuous action.…”