A CHARACTERIZATION OF SATURATED C^*-ALGEBRAIC BUNDLES OVER FINITE GROUPS

Abstract

Let A be a unital C * -algebra. Let (B, E) be a pair consisting of a unital C * -algebra B containing A as a C * -subalgebra with a unit that is also the unit of B, and a conditional expectation E from B onto A that is of index-finite type and of depth 2. Let B 1 be the C * -basic construction induced by (B, E). In this paper, we shall show that any such pair (B, E) satisfying the conditions that A ∩ B = C1 and that A ∩ B 1 is commutative is constructed by a saturated C * -algebraic bundle over a finite group. Furthermore, we shall give a necessary and sufficient condition for B to be described as a twisted crossed product of A by its twisted action of a finite group under the condition that A ∩ B 1 is commutative.2000 Mathematics subject classification: primary 46L08; secondary 46L40.

Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^*$-algebras

“…since t∈G α A st −1 (e A ) = 1 for any s ∈ G by [10,Remark 3.4]. Therefore, we obtain the conclusion.…”

“…Let A = {A t } t∈G be a C * -algebraic bundle over a finite group G. We say that A is saturated if A t A * t = A for all t ∈ G. Since A is unital, in our case we do not need to take the closure in Definition 1.1. If A is saturated, by [10,Corollary 3.2], E A is of index-finite type and its Watatani index, Ind W (E A ) = |G|, where |G| is the order of G.…”

“…Let C 1 be the C * -basic construction of C and e A the Jones' projection for E A . By [10,Lemma 3.7], there is an action α A of G on C 1 induced by A defined as follows: Since A is saturated and A is unital, there is a finite set…”

“…Let C 1 and D 1 be the C * -basic constructions of C and D and e A and e B the Jones' projections for E A and E B , respectively. Then by [10,Lemma 3.7], there are actions α A and α B of G on C 1 and D 1 induced by A and B, respectively. Furthermore, let C 2 and D 2 be the C * -basic constructions of C 1 amd D 1 for the dual conditional expectations E C of E A and E D of E B , which are isomorphic to C 1 ⋊ α A G and D 1 ⋊ α B G, respectively.…”