Toke Meier Carlsen

The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources ☆

Abstract We catalogue the primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources. Each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph; the primitive ideals in the Cuntz–Krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group. The Cuntz–Krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic.

CO-UNIVERSAL ALGEBRAS ASSOCIATED TO PRODUCT SYSTEMS, AND GAUGE-INVARIANT UNIQUENESS THEOREMS

Let X be a product system over a quasi-lattice ordered group. 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-universalC � -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 realise 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.

CUNTZ–PIMSNER C*-ALGEBRAS ASSOCIATED WITH SUBSHIFTS

By using C*-correspondences and Cuntz–Pimsner algebras, we associate to every subshift (also called a shift space) 𝖷 a C*-algebra , which is a generalization of the Cuntz–Krieger algebras. We show that is the universal C*-algebra generated by partial isometries satisfying relations given by 𝖷. We also show that is a one-sided conjugacy invariant of 𝖷.

C*-crossed products and shift spaces

We use Exels C*-crossed products associated to non-invertible dynamical systems to associate a C*-algebra to arbitrary shift space. We show that this C*-algebra is canonically isomorphic to the C*-algebra associated to a shift space given by Carlsen [Cuntz–Pimsner C*-algebras associated with subshifts, Internat. J. Math. (2004) 28, to appear, available at arXiv:math.OA/0505503], has the C*-algebra defined by Carlsen and Matsumoto [Some remarks on the C*-algebras associated with subshifts, Math. Scand. 95 (1) (2004) 145–160] as a quotient, and possesses properties indicating that it can be thought of as the universal C*-algebra associated to a shift space. We also consider its representations and its relationship to other C*-algebras associated to shift spaces. We show that it can be viewed as a generalization of the universal Cuntz–Krieger algebra, discuss uniqueness and present a faithful representation, show that it is nuclear and satisfies the Universal Coefficient Theorem, provide conditions for it being simple and purely infinite, show that the constructed C*-algebras and thus their K-theory, K0 and K1, are conjugacy invariants of one-sided shift spaces, present formulas for those invariants, and present a description of the structure of gauge invariant ideals. (Less)

Partial actions and KMS states on relative graph C⁎-algebras ☆

Abstract The relative graph C ⁎ -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 ⁎ -algebra C ⁎ ( 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 ⁎ ( E , R ) induced by a function on the edge set, we characterize the KMS β 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 graphs with finitely many vertices.

Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras

We prove that ample groupoids with sigma-compact unit spaces are equivalent if and only if they are stably isomorphic in an appropriate sense, and relate this to Matuis notion of Kakutani equivalence. We use this result to show that diagonal-preserving stable isomorphisms of graph C*-algebras or Leavitt path algebras give rise to isomorphisms of the groupoids of the associated stabilised graphs. We deduce that the Leavitt path algebras

On the Exel Crossed Product of Topological Covering Maps

L_Z(E_2)

Graph algebras and orbit equivalence

and

Augmenting dimension group invariants for substitution dynamics

L_Z(E_{2-})

INDEX MAPS IN THE K-THEORY OF GRAPH ALGEBRAS

are not stably *-isomorphic.