Partial actions and KMS states on relative graph $C^{\ast }$-algebras

Abstract

The relative graph $C^{\ast }$-algebras introduced by Muhly and Tomforde are generalizations of both graph algebras and their Toeplitz extensions. For an arbitrary graph $E$ and a subset $R$ of the set of regular vertices of $E$ we show that the relative graph $C^{\ast }$-algebra $C^{\ast }(E,R)$ is isomorphic to a partial crossed product for an action of the free group generated by the edge set on the relative boundary path space. Given a time evolution on $C^{\ast }(E,R)$ induced by a function on the edge set, we characterize the KMS${}_{\beta }$ states and ground states using an abstract result of Exel and Laca. Guided by their work on KMS states for Toeplitz-Cuntz-Krieger type algebras associated to infinite matrices, we obtain complete descriptions of the convex sets of KMS states of finite type and of KMS states of infinite type whose associated measures are supported on recurrent infinite paths. This allows us to give a complete concrete description of the convex set of all KMS states for a big class of graphs which includes all finite graphs.

33 pages. A few typos corrected. Bibliography updated. This is the version to be published

References in corpus (7)

• Graph inverse semigroups, groupoids and their C*-algebras
• Adding tails to C*-correspondences
• A class of C^*-algebras generalizing both graph algebras and homeomorphism C^*-algebras I, fundamental results
• Modular Index Invariants of Mumford Curves
• KMS weights on groupoid and graph C*-algebras
• Dissipative conformal measures on locally compact spaces
• KMS states for the generalized gauge action on graph algebras

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• Preferred traces on C*-algebras of self-similar groupoids arising as fixed points
• KMS weights on graph C*-algebras II Factor types and ground states
• KMS and ground states on ultragraph C*-algebras
• Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics
• KMS states on the C*-algebras of Fell bundles over groupoids
• Spatial realisations of KMS states on the C*-algebras of higher-rank graphs
• KMS states on the C*-algebra of a higher-rank graph and periodicity in the path space
• Bratteli-Vershik models for partial actions of $\mathbb{Z}$
• C*-dynamical invariants and Toeplitz algebras of graphs
• Reconstructing directed graphs from generalised gauge actions on their Toeplitz algebras
• Exchange rings and real rank zero C*-algebras associated with finitely separated graphs
• Topological entropy for partial actions of the group $\mathbb{Z}$
• Limits of KMS states on Toeplitz algebras of finite graphs
• Finite digraphs and KMS states