Let (G, P) be a quasi‐lattice ordered group, and let X be a product system over P of Hilbert bimodules. Under mild hypotheses, we associate to X a C*‐algebra which is co‐universal for injective Nica covariant Toeplitz representations of X which preserve the gauge coaction. Under appropriate amenability criteria, this co‐universal C*‐algebra coincides with the Cuntz‐Nica‐Pimsner algebra introduced by Sims and Yeend. We prove two key uniqueness theorems, and indicate how to use our theorems to realize a number of reduced crossed products as instances of our co‐universal algebras. In each case, it is an easy corollary that the Cuntz‐Nica‐Pimsner algebra is isomorphic to the corresponding full crossed product.
We introduce the notion of orbit equivalence of directed graphs, following
Matsumoto's 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 $C^*$-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 $E$ we
construct a groupoid $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ from the graph
algebra $C^*(E)$ and its diagonal subalgebra $\mathcal{D}(E)$ which generalises
Renault's Weyl groupoid construction applied to $(C^*(E),\mathcal{D}(E))$. We
show that $\mathcal{G}_{(C^*(E),\mathcal{D}(E))}$ recovers the graph groupoid
$\mathcal{G}_E$ without the assumption that every cycle in $E$ has an exit,
which is required to apply Renault's results to $(C^*(E),\mathcal{D}(E))$. We
finish with applications of our results to out-splittings of graphs and to
amplified graphs.Comment: 27 page
We point out incorrect lemmas in some papers regarding the C * -algebras associated with subshifts written by the second named author. To recover the incorrect lemmas and the affected main results, we will describe an alternative construction of C * -algebras associated with subshifts. The resulting C * -algebras are generally different from the originally constructed C * -algebras associated with subshifts and they fit the mentioned papers including the incorrect results. The simplicity conditions and the K-theory formulae for the originally constructed C * -algebras are described. We also introduce a condition called ( * ) for subshifts such that under this condition the new C * -algebras and the original C * -algebras are canonically isomorphic to each other. We finally present a subshift for which the two kinds of algebras have different K-theory groups.
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