Cuntz Pimsner Algebra Research Articles

KMS states on C*-algebras associated to local homeomorphisms

For every Hilbert bimodule over a C*-algebra, there are natural gauge actions of the circle on the associated Toeplitz algebra and Cuntz–Pimsner algebra, and hence natural dynamics obtained by lifting these gauge actions to actions of the real line. We study the KMS states of these dynamics for a family of bimodules associated to local homeomorphisms on compact spaces. For inverse temperatures larger than a certain critical value, we find a large simplex of KMS states on the Toeplitz algebra, and we show that all KMS states on the Cuntz–Pimsner algebra have inverse temperature at most this critical value. We illustrate our results by considering the backward shift on the one-sided path space of a finite graph, where we can use recent results about KMS states on graph algebras to see what happens below the critical value. Our results about KMS states on the Cuntz–Pimsner algebra of the shift show that recent constraints on the range of inverse temperatures obtained by Thomsen are sharp.

Operator algebras and subproduct systems arising from stochastic matrices

We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselter's generalization of the Cuntz–Pimsner C*-algebra to the context of subproduct systems. Suppose that X and Y are Arveson–Stinespring subproduct systems associated to two stochastic matrices over a countable set Ω, and let T+(X) and T+(Y) be their tensor algebras. We show that every algebraic isomorphism from T+(X) onto T+(Y) is automatically bounded. Furthermore, T+(X) and T+(Y) are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of ℓ∞(Ω). When Ω is finite, we prove that T+(X) and T+(Y) are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of ℓ∞(Ω). Moreover, we provide an explicit description of the Cuntz–Pimsner algebra O(X) in the case where Ω is finite and the stochastic matrix is essential.

Equilibrium states on the Cuntz–Pimsner algebras of self-similar actions

We consider a family of Cuntz–Pimsner algebras associated to self-similar group actions, and their Toeplitz analogues. Both families carry natural dynamics implemented by automorphic actions of the real line, and we investigate the equilibrium states (the KMS states) for these dynamical systems. We find that for all inverse temperatures above a critical value, the KMS states on the Toeplitz algebra are given, in a very concrete way, by traces on the full group algebra of the group. At the critical inverse temperature, the KMS states factor through states of the Cuntz–Pimsner algebra; if the self-similar group is contracting, then the Cuntz–Pimsner algebra has only one KMS state. We apply these results to a number of examples, including the self-similar group actions associated to integer dilation matrices, and the canonical self-similar actions of the basilica group and the Grigorchuk group.

Zappa–Szép products of semigroups and their [formula omitted]-algebras

Zappa–Szép products of semigroups provide a rich class of examples of semigroups that include the self-similar group actions of Nekrashevych. We use Li's construction of semigroup C⁎-algebras to associate a C⁎-algebra to Zappa–Szép products and give an explicit presentation of the algebra. We then define a quotient C⁎-algebra that generalises the Cuntz–Pimsner algebras for self-similar actions. We indicate how known examples, previously viewed as distinct classes, fit into our unifying framework. We specifically discuss the Baumslag–Solitar groups, the binary adding machine, the semigroup N⋊N×, and the ax+b-semigroup Z⋊Z×.

Formula omitted]-algebras and equivalences for [formula omitted]-correspondences

We study several notions of shift equivalence for C⁎-correspondences and the effect that these equivalences have on the corresponding Pimsner dilations. Among others, we prove that non-degenerate, regular, full C⁎-correspondences which are shift equivalent have strong Morita equivalent Pimsner dilations. We also establish that the converse may not be true. These results settle open problems in the literature. In the context of C⁎-algebras, we prove that if two non-degenerate, regular, full C⁎-correspondences are shift equivalent, then their corresponding Cuntz–Pimsner algebras are strong Morita equivalent. This generalizes results of Cuntz and Krieger and Muhly, Tomforde and Pask. As a consequence, if two subshifts of finite type are eventually conjugate, then their Cuntz–Krieger algebras are strong Morita equivalent. Our results suggest a natural analogue of the Shift Equivalence Problem in the context of C⁎-correspondences. Even though we do not resolve the general Shift Equivalence Problem, we obtain a positive answer for the class of imprimitivity bimodules.

Crossed products by endomorphisms and reduction of relations in relative Cuntz–Pimsner algebras

Starting from an arbitrary endomorphism α of a unital C⁎-algebra A we construct a crossed product. It is shown that the natural construction depends not only on the C⁎-dynamical system (A,α) but also on the choice of an ideal J orthogonal to kerα. The article gives an explicit description of the internal structure of this crossed product and, in particular, discusses the interrelation between relative Cuntz–Pimsner algebras and partial isometric crossed products. We present a canonical procedure that reduces any given C⁎-correspondence to the ‘smallest’ C⁎-correspondence yielding the same relative Cuntz–Pimsner algebra as the initial one. In the context of crossed products this reduction procedure corresponds to the reduction of C⁎-dynamical systems and allows us to establish a coincidence between relative Cuntz–Pimsner algebras and crossed products introduced.

KMS STATES ON FINITE-GRAPH C∗-ALGEBRAS

We study Kubo-Martin-Schwinger(KMS) states on finite-graph C∗-algebras with sinks and sources. We compare finite-graph C∗-algebras with C∗-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature β is large, then the set of extreme β-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational function from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C∗-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshvyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

$C^*$-algebras generalizing both relative Cuntz-Pimsner and Doplicher-Roberts algebras

We introduce and analyse the structure of C*-algebras arising from ideals in right tensor C*-precategories, which naturally generalize both relative Cuntz-Pimsner and Doplicher-Roberts algebras. We establish an explicit intrinsic construction of the considered algebras and a number of key results such as structure theorem, gauge-invariant uniqueness theorem or description of the gauge-invariant ideal structure. In particular, these statements give a new insight into the corresponding results for relative Cuntz-Pimsner algebras and are applied to Doplicher-Roberts associated to C*-correspondences.

Contributions to the theory of $\mathrm{C}^{*}$-correspondences with applications to multivariable dynamics

Motivated by the theory of tensor algebras and multivariable $\mathrm {C}^*$-dynamics, we revisit two fundamental techniques in the theory of $\mathrm {C}^*$-corres- pondences, the “addition of a tail” to a non-injective $\mathrm {C}^*$-correspondence and the dilation of an injective $\mathrm {C}^*$-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for “adding a tail” to a non-injective $\mathrm {C}^*$-correspondence; our scheme includes the “tail” of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable $\mathrm {C}^*$-dynamics. We also exhibit a transparent picture for the dilation of an injective $\mathrm {C}^*$-correspondence to an essential Hilbert bimodule. As an application of our constructs, we prove two results in the theory of multivariable dynamics that extend earlier results. We also discuss the impact of our results on the description of the $\mathrm {C}^*$-envelope of a tensor algebra as the Cuntz-Pimsner algebra of the associated $\mathrm {C}^*$-correspondence.

CUNTZ–PIMSNER ALGEBRAS FOR SUBPRODUCT SYSTEMS

In this paper we generalize the notion of Cuntz–Pimsner algebras of C*-correspondences to the setting of subproduct systems. The construction is justified in several ways, including the Morita equivalence of the operator algebras under suitable conditions and examples are provided to illustrate its naturality. We also demonstrate why some features of the Cuntz–Pimsner algebras of C*-correspondences fail to generalize to our setting, and discuss what we have instead.

$C^{*}$-algebras associated with real multiplication

Noncommutative tori with real multiplication are the irrational rotation algebras that have special equivalence bimodules. Y. Manin proposed the use of noncommutative tori with real multiplication as a geometric framework for the study of abelian class field theory of real quadratic fields. In this paper, we consider the Cuntz-Pimsner algebras constructed by special equivalence bimodules of irrational rotation algebras. We shall show that associated $C^*$-algebras are simple and purely infinite. We compute the K-groups of associated $C^*$-algebras and show that these algebras are related to the solutions of Pell's equation and the unit groups of real quadratic fields. We consider the Morita equivalent classes of associated $C^*$-algebras.

Semicrossed products of operator algebras and their [formula omitted]-envelopes

Let A be a unital operator algebra and let α be an automorphism of A that extends to a ⁎-automorphism of its C⁎-envelope Cenv⁎(A). We introduce the isometric semicrossed product A×αisZ+ and we show that Cenv⁎(A×αisZ+)≃Cenv⁎(A)×αZ. In contrast, the C⁎-envelope of the familiar contractive semicrossed product A×αZ+ may not equal Cenv⁎(A)×αZ. Our main tool for calculating C⁎-envelopes for semicrossed products is a new concept, the relative semicrossed product of an operator algebra by an endomorphism. As an application of our theory, we show that if TX+ is the tensor algebra of a C⁎-correspondence (X,A) and α is a ⁎-extendible automorphism of TX+ that fixes the diagonal elementwise, then the contractive semicrossed product satisfies Cenv⁎(TX+×αZ+)≃OX×αZ, where OX denotes the Cuntz–Pimsner algebra of (X,A). This extends the main result of Davidson and Katsoulis (2010) [6].

The C*-algebra of a vector bundle

We prove that the Cuntz–Pimsner algebra OE of a vector bundle E of rank ≧ 2 over a compact metrizable space X is determined up to an isomorphism of C(X)-algebras by the ideal (1 − [E])K0(X) of the K-theory ring K0(X). Moreover, if E and F are vector bundles of rank ≧ 2, then a unital embedding of C(X)-algebras OE ⊂ OF exists if and only if 1 − [E] is divisible by 1 − [F] in the ring K0(X). We introduce related but more computable K-theory and cohomology invariants for OE and study their completeness. As an application we classify the unital separable continuous fields with fibers isomorphic to the Cuntz algebra Om + 1 over a finite connected CW complex X of dimension d ≦ 2m + 3 provided that the cohomology of X has no m-torsion.

C∗-algebras associated to C∗-correspondences and applications to mirror quantum spheres

The structure of the $C^*$-algebras corresponding to even-dimensional mirror quantum spheres is investigated. It is shown that they are isomorphic to both Cuntz-Pimsner algebras of certain $C^*$-correspondences and $C^*$-algebras of certain labelled graphs. In order to achieve this, categories of labelled graphs and $C^*$-correspondences are studied. A functor from labelled graphs to $C^*$-correspondences is constructed, such that the corresponding associated $C^*$-algebras are isomorphic. Furthermore, it is shown that $C^*$-correspondences for the mirror quantum spheres arise via a general construction of restricted direct sum.

On the nuclearity of certain Cuntz-Pimsner algebras

In the present paper, we give a short proof of the nuclearity property of a class of Cuntz-Pimsner algebras associated with a Hilbert 𝒜-bimodule ℳ︁, where 𝒜 is a separable and nuclear C *-algebra. We assume that the left 𝒜-action on the bimodule ℳ︁ is given in terms of compact module operators and that ℳ︁ is direct summand of the standard Hilbert module over 𝒜 (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Graphs of $C^*$-correspondences and Fell bundles

We define the notion of a �-system of C � -correspondences associated to a higher-rank graph �. Roughly speaking, such a system assigns to each vertex ofa C � - algebra, and to each path ina C � -correspondence in a way which carries compositions of paths to balanced tensor products of C � -correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of C � -correspondences to associate a C � -algebra to each �-system. We then construct a Fell bundle over the path groupoid Gand show that the C � -algebra of the �-system coincides with the reduced cross-sectional algebra of the Fell bundle. We conclude by discussing several examples of our construction arising in the literature.

GAUGE-EQUIVARIANT HILBERT BIMODULES AND CROSSED PRODUCTS BY ENDOMORPHISMS

C*-endomorphisms arising from superselection structures with nontrivial center define a 'rank' and a 'first Chern class'. Crossed products by such endomorphisms involve the Cuntz–Pimsner algebra of a vector bundle having the above-mentioned rank, first Chern class and can be used to construct a duality for abstract (nonsymmetric) tensor categories versus group bundles acting on (nonsymmetric) Hilbert bimodules. Existence and unicity of the dual object (i.e. the 'gauge' group bundle) are not ensured: we give a description of this phenomenon in terms of a certain moduli space associated with the given endomorphism. The above-mentioned Hilbert bimodules are noncommutative analogs of gauge-equivariant vector bundles in the sense of Nistor–Troitsky.

Crossed products by abelian semigroups via transfer operators

AbstractWe propose a generalization of Exel’s crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler’s theory of Toeplitz and Cuntz–Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz–Pimsner algebras of product systems.

REALIZATION OF MINIMAL C*-DYNAMICAL SYSTEMS IN TERMS OF CUNTZ–PIMSNER ALGEBRAS

In the present article, we provide several constructions of C*-dynamical systems [Formula: see text] with a compact group [Formula: see text] in terms of Cuntz–Pimsner algebras. These systems have a minimal relative commutant of the fixed-point algebra [Formula: see text] in [Formula: see text], i.e. [Formula: see text], where [Formula: see text] is the center of [Formula: see text], which is assumed to be non-trivial. In addition, we show in our models that the group action [Formula: see text] has full spectrum, i.e. any unitary irreducible representation of [Formula: see text] is carried by a [Formula: see text]-invariant Hilbert space within [Formula: see text]. First, we give several constructions of minimal C*-dynamical systems in terms of a single Cuntz–Pimsner algebra [Formula: see text] associated to a suitable [Formula: see text]-bimodule ℌ. These examples are labelled by the action of a discrete Abelian group ℭ (which we call the chain group) on [Formula: see text] and by the choice of a suitable class of finite dimensional representations of [Formula: see text]. Second, we present a more elaborate contruction, where now the C*-algebra [Formula: see text] is generated by a family of Cuntz–Pimsner algebras. Here, the product of the elements in different algebras is twisted by the chain group action. We specify the various constructions of C*-dynamical systems for the group [Formula: see text], N ≥ 2.

Bundles of [formula omitted]-categories, II: [formula omitted]-dynamical systems and Dixmier–Douady invariants

We introduce a cohomological invariant arising from a class in nonabelian cohomology. This invariant generalises the Dixmier–Douady class and encodes the obstruction to a C ∗ -algebra bundle being the fixed-point algebra of a gauge action. As an application, the duality breaking for group bundles vs. tensor C ∗ -categories with nonsimple unit is discussed in the setting of Nistor–Troitsky gauge-equivariant K-theory: there is a map assigning a nonabelian gerbe to a tensor category, and “triviality” of the gerbe is equivalent to the existence of a dual group bundle. At the C ∗ -algebraic level, this corresponds to studying C ∗ -algebra bundles with fibre a fixed-point algebra of the Cuntz algebra and in this case our invariant describes the obstruction to finding an embedding into the Cuntz–Pimsner algebra of a vector bundle.