Nathan Brownlowe

Exel's crossed product and relative Cuntz–Pimsner algebras

We consider Exels new construction of a crossed product of a C ∗ -algebra A by an endomorphism α. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz-Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from A into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product. In this paper, we re-examine Exels crossed product, denoted A� α,LN ,a nd identify a family of representations for which A� α,LN is universal. We then show that A� α,LN can be realised as a relative Cuntz-Pimsner algebra as in (6, 11), and use known results for relative Cuntz-Pimsner algebras to study A� α,LN. In particular, we identify conditions which ensure that the canonical map A → A� α,LN is injective, thus answering a question raised by Exel in (3), and partially answered by him in

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

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.

Graph algebras and orbit equivalence

We introduce the notion of orbit equivalence of directed graphs, following Matsumotos notion of continuous orbit equivalence for topological Markov shifts. We show that two graphs in which every cycle has an exit are orbit equivalent if and only if there is a diagonal-preserving isomorphism between their

BOUNDARY QUOTIENTS OF THE TOEPLITZ ALGEBRA OF THE AFFINE SEMIGROUP OVER THE NATURAL NUMBERS

C^*

On C*-algebras associated to right LCM semigroups

-algebras. We show that it is necessary to assume that every cycle has an exit for the forward implication, but that the reverse implication holds for arbitrary graphs. As part of our analysis of arbitrary graphs

The boundary quotient for algebraic dynamical systems

E

Equilibrium states on right LCM semigroup C*-algebras

we construct a groupoid

Leavitt R-algebras over countable graphs embed into L2,R

\mathcal{G}_{(C^*(E),\mathcal{D}(E))}

Subquotients of Hecke C*-algebras

from the graph algebra

Zappa-Szép product groupoids and C∗-blends

C^*(E)