Crossed products by compact group actions with the Rokhlin property

Abstract

Abstract. We present a systematic study of the structure of crossed products and fixed point algebras by compact group actions with the Rokhlin property on not necessarily unital C * -algebras. Our main technical result is the existence of an approximate homomorphism from the algebra to its subalgebra of fixed points, which is a left inverse for the canonical inclusion. Upon combining this with results regarding local approximations, we show that a number of classes characterized by inductive limit decompositions with weakly semiprojective building blocks, are closed under formation of crossed products by such actions. Similarly, in the presence of the Rokhlin property, if the algebra has any of the following properties, then so do the crossed product and the fixed point algebra: being a Kirchberg algebra, being simple and having tracial rank zero or one, having real rank zero, having stable rank one, absorbing a strongly self-absorbing C * -algebra, satisfying the Universal Coefficient Theorem (in the simple, nuclear case), and being weakly semiprojective. The ideal structure of crossed products and fixed point algebras by Rokhlin actions is also studied.The methods of this paper unify, under a single conceptual approach, the work of a number of authors, who used rather different techniques. Our methods yield new results even in the well-studied case of finite group actions with the Rokhlin property.

Rokhlin dimension: duality, tracial properties, and crossed products

“…, θ d : Proof. That (1) implies (2) is easily seen by tensoring each ϕ j with the identity on D and using Lemma 2.3 in [21]. We show that (2) implies (1).…”

“…A) be a unital equivariant homomorphism as in the statement. Upon tensoring with id D and using Lemma 2.3 in [21], we obtain an equivariant embedding…”

“…The concept of Rokhlin dimension for actions of finite groups and actions of the integers was introduced by the second author, Winter and Zacharias in [37] as a generalization of the well-studied Rokhlin property for group actions. (See [31], [38], and [50] for finite group actions with the Rokhlin property, and [36], [21], and [23] for compact group actions.) The Rokhlin property can be viewed as a regularity condition for the group action, which can be used to show that various structural properties pass from a C * -algebra to its crossed product; see for example [36], [50], [58], [21].…”

“…(See [31], [38], and [50] for finite group actions with the Rokhlin property, and [36], [21], and [23] for compact group actions.) The Rokhlin property can be viewed as a regularity condition for the group action, which can be used to show that various structural properties pass from a C * -algebra to its crossed product; see for example [36], [50], [58], [21]. However, particularly in the case of finite group actions, the Rokhlin property is a very restrictive hypothesis to place on the action, as it as it implies that the unit of the algebra can be written nontrivially as a sum of projections indexed by the group.…”

Actions of rigid groups on UHF-algebras

“…Also, (3) holds by construction, while (4) follows from (2) and (3). Finally, the Rokhlin property for θ G ensures that the properties for A G listed in (1) are inherited by A θ G G , by the theorem in the introduction of [16]. This gives (5), and finishes the proof.…”