Exel's crossed product for non-unital C*-algebras

Abstract

Abstract. We consider a family of dynamical systems (A, α, L) in which α is an endomorphism of a C * -algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C * -algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

Stacey Crossed Products Associated to Exel Systems

“…5, we consider a directed graph E and the Exel system (C * (E ∞ ), α, L) introduced in [9], and obtain a new description of the graph algebra C * (E) as a Stacey crossed product of the core (which generalises results of Rørdam and Kwaśniewski for finite graphs [20,36]). This led us to revisit Exel systems involving other endomorphisms of the UHF core in Cuntz algebras, which we do in Sect.…”

“…As in [9], a Toeplitz-covariant representation of (A, α, L) in a C * -algebra B consists of a nondegenerate homomorphism π : A → B and an element S ∈ M (B) such that Sπ(a) = π(α(a))S and S * π(a)S = π(L(a)).…”

“…In [9,Sect. 4], the Exel crossed product A α,L N of a possibly non-unital C * -algebra A by N is the quotient of T (A, α, L) by the ideal generated by…”

“…There is a dual actionα of T on A α,L N such thatα z (j(a)) = j(a) andα z (t) = zt. Theorem 4.1 of [9] says that there is an isomorphism…”

“…Since E has no sinks, σ is surjective. Since E is column-finite, σ is a local homeomorphism which is proper in the sense that inverse images of compact sets are compact (see [9,Sect. 2.2]).…”