Beatriz Abadie, Mauricio Achigar
Given a correspondence X over a C*-algebra A, we construct a C*-algebra and a Hilbert C*-bimodule over it whose crossed product is isomorphic to the augmented Cuntz-Pimsner C*-algebra of X. This construction enables us to establish a condition for two augmented Cuntz-Pimsner C*-algebras to be Morita equivalent.
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We investigate how a correspondence coaction gives rise to a coaction on the associated Cuntz-Pimsner algebra. We apply this to recover a recent result of Hao and Ng concerning Cuntz-Pimsner algebras of crossed products of correspondences by actions of amenable groups.
We give a simple and complete picture on the classification of relative Cuntz--Pimsner algebras (and so also of gauge-equivariant representations) using their intuitive parametrisation by kernel--covariance pairs.
Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspo…
We interpret the construction of relative Cuntz-Pimsner algebras of correspondences in terms of the correspondence bicategory, as a reflector into a certain sub-bicategory. This generalises a previous characterisation of absolute Cuntz-Pimsner algebras of proper correspondences as colimits in the correspondence bicategory.
We extend the definition of an extension of a right Hilbert module to the setting of Hilbert bimodules and show that an extension of Hilbert bimodules induces an extension of Cuntz-Pimsner algebras. We also study the Cuntz-Pimsner algebra associated to the multiplier bimodule and show that an extension can be realised as a restricted direct-sum bimodule.
In this article we study how decompositions of a quasi-lattice ordered group $(G,P)$ relate to decompositions of the Nica-Toeplitz algebra $\mathcal{NT}_\mathbf{X}$ and Cuntz-Nica-Pimsner algebra $\mathcal{NO}_\mathbf{X}$ of a compactly aligned product system $\mathbf{X}$ over $P$. In particular, we are interested in the situation where $(G,P)$ may be realised as the semidirect product of quasi-lattice ordered groups. Our results generalise Deaconu's work on iterated Toeplitz and Cuntz-Pimsner algebras - we show that the Nica-Toeplitz algebra and Cuntz-Nica-Pimsner algebra of a compactly align…
We construct a functor that maps $C^*$-correspondences to their Cuntz-Pimsner algebras. The objects in our domain category are $C^*$-correspondences, and the morphisms are the isomorphism classes of $C^*$-correspondences satisfying certain conditions. As an application, we recover a well-known result of Muhly and Solel. In fact, we show that functoriality leads us to a more generalized result: strongly Morita equivalent $C^*$-correspondences have Morita equivalent Cuntz-Pimsner algebras.
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.
We show that the passage from a $C^\ast$-correspondence to its Cuntz-Pimsner $C^\ast$-algebra gives a functor on a category of $C^\ast$-correspondences with appropriately defined morphisms. Applications involving topological graph $C^\ast$-algebras are discussed, and an application to crossed-product correspondences is presented in detail.
Given two correspondences $X$ and $Y$ and a discrete group $G$ which acts on $X$ and coacts on $Y$, one can define a twisted tensor product $X\boxtimes Y$ which simultaneously generalizes ordinary tensor products and crossed products by group actions and coactions. We show that, under suitable conditions, the Cuntz-Pimsner algebra of this product, $\mathcal O_{X\boxtimes Y}$, is isomorphic to a "balanced" twisted tensor product $\mathcal O_X\boxtimes_\mathbb T\mathcal O_Y$ of the Cuntz-Pimsner algebras of the original correspondences. We interpret this result in several contexts and connect it…
Let H be a full Hilbert bimodule over a C*-algebra A. We show that the Cuntz-Pimsner C*-algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact C*-algebras. In the case that A is a finite dimensional C*-algebra, we also show that the Brown-Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz-Pimsner algebra associated to an A,A Hilbert bimodule is zero.
The extended Cuntz-Pimsner algebra E(H), introduced by Pimsner, is constructed from a Hilbert B,B-bimodule H over a C*-algebra B. In this paper we investigate the Haagerup invariant Λ(.) for these algebras, the main result being that Λ(E(H))=Λ(B) when H is full over B. In particular, E(H) has the completely bounded approximation property if and only if the same is true for B.
Exactness of the Cuntz-Pimsner Construction
本文将 Cuntz-Pimsner 构造视为函子，证明了其代数理论的基本结果是该构造函子性的直接推论，并证明了该函子在目标与定义域范畴中是正合的。
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*-correspondenc…
In this thesis we generalise the six-term exact sequence in graded $KK$-theory obtained in a paper of Kumjian, Pask and Sims (2017) to allow correspondences with non-compact left action. In particular, this allows us to compute the graded $KK$-theory of row-infinite graphs. We develop the theory necessary for following the arguments of Kumjian, Pask and Sims and of Pimsner (1997), with detailed sections on Hilbert modules, $C^*$-correspondences, Crossed products, Toeplitz algebras, Cuntz-Pimsner algebras and $KK$-theory.
Given a finitely aligned $k$-graph $Λ$, we let $Λ^i$ denote the $(k-1)$-graph formed by removing all edges of degree $e_i$ from $Λ$. We show that the Toeplitz-Cuntz-Krieger algebra of $Λ$, denoted by $\mathcal{T}C^*(Λ)$, may be realised as the Toeplitz algebra of a Hilbert $\mathcal{T}C^*(Λ^i)$-bimodule. When $Λ$ is locally-convex, we show that the Cuntz-Krieger algebra of $Λ$, which we denote by $C^*(Λ)$, may be realised as the Cuntz-Pimsner algebra of a Hilbert $C^*(Λ^i)$-bimodule. Consequently, $\mathcal{T}C^*(Λ)$ and $C^*(Λ)$ may be viewed as iterated Toeplitz and iterated Cuntz-Pimsner al…
Passing $C^*$-correspondence Relations to the Cuntz-Pimsner algebras
本文构造了一个将 $C^*$-对应映射到其 Cuntz-Pimsner 代数的函子，并由此推广了 Muhly-Solel 及 Muhly-Pask-Tomforde 关于 Morita 等价保持性的经典结果。
We consider Exel's 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.
Necessary and sufficient conditions for when every non-zero ideal in a relative Cuntz-Pimsner ring contains a non-zero graded ideal, when a relative Cuntz-Pimsner ring is simple, and when every ideal in a relative Cuntz-Pimsner ring is graded, are given. A "Cuntz-Krieger uniqueness theorem" for relative Cuntz-Pimsner rings is also given and condition (L) and condition (K) for relative Cuntz-Pimsner rings are introduced.
We compute the sheaf cohomology of a groupoid built from a local homeomorphism of a locally compact space $X$. In particular, we identify the twists over this groupoid, and its Brauer group. Our calculations refine those made by Kumjian, Muhly, Renault and Williams in the case $X$ is the path space of a graph, and the local homeomorphism is the shift. We also show how the C*-algebra of a twist may be identified with the Cuntz-Pimsner algebra constructed from a certain C*-correspondence.
$k$-graph algebras are iterated Cuntz-Pimsner algebras -- from the bottom up
本文提出一种将k-图C*代数表示为Cuntz-Pimsner代数的新方法，通过从底层开始迭代，利用产品系统和最近的定理，展示Cuntz-Pimsner构造在enchilada范畴中的函子性。
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 se…
Starting from an arbitrary endomorphism αof 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 orthogonal to kernel of α. 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 …
A Cuntz-Krieger Uniqueness Theorem for Cuntz-Pimsner Algebras
本文引入C*-对应的条件(S)作为图的条件(L)的类比，并据此证明Cuntz-Pimsner代数的Cuntz-Krieger唯一性定理及单纯性充分条件，同时建立拓扑箭图条件(L)与对应条件(S)的等价关系。
Let $A$ be a separable unital C*-algebra and let $π: A \ra \Lc(\Hf)$ be a faithful representation of $A$ on a separable Hilbert space $\Hf$ such that $π(A) \cap \Kc(\Hf) = \{0 \}$. We show that $\Oc_E$, the Cuntz-Pimsner algebra associated to the Hilbert $A$-bimodule $E = \Hf \ot_{\C} A$, is simple and purely infinite. If $A$ is nuclear and belongs to the bootstrap class to which the UCT applies, then the same applies to $\Oc_E$. Hence by the Kirchberg-Phillips Theorem the isomorphism class of $\Oc_E$ only depends on the $K$-theory of $A$ and the class of the unit.
We give a construction of a nuclear $C^\ast$-algebra associated with an amalgamated free product of groups, generalizing Spielberg's construction of a certain Cuntz-Krieger algebra associated with a finitely generated free product of cyclic groups. Our nuclear $C^\ast$-algebras can be identified with certain Cuntz-Krieger-Pimsner algebras. We will also show that our algebras can be obtained by the crossed product construction of the canonical actions on the hyperbolic boundaries, which proves a special case of Adams' result about amenability of the boundary action for hyperbolic groups. We wil…
Discrete inclusions from Cuntz-Pimsner algebras
本文研究了从Cuntz-Pimsner代数中获得的离散包含性质，证明其为C*-离散，并探讨了相关双模的结构和特性。
Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences
本文研究了群oid对应关系诱导的C*-对应关系的相对Cuntz-Pimsner代数，并展示了其作为特定群oid的C*-代数的结构，统一了拓扑图和自相似群的C*-代数的构造方法。
Morphisms of Cuntz-Pimsner algebras from completely positive maps
本文引入由完全正映射给出左作用的正对应半范畴，并利用KSGNS构造建立到C*-对应范畴的半函子，从而研究Cuntz-Pimsner代数间的态射及其典型例子。
The algebraic Cuntz-Pimsner rings are naturally $\mathbb{Z}$-graded rings that generalize both Leavitt path algebras and unperforated $\mathbb{Z}$-graded Steinberg algebras. We classify strongly, epsilon-strongly and nearly epsilon-strongly graded algebraic Cuntz-Pimsner rings up to graded isomorphism. As an application, we characterize noetherian and artinian fractional skew monoid rings by a single corner automorphism.
KMS states of quasi-free dynamics on Pimsner algebras
本研究刻画了右希尔伯特C*-双模上连续单参数酉等距群诱导的拟自由动力学的KMS态，推广了此前Cuntz代数等的相关结果，还给出了非酉等距情形下基于KMS权的一般刻画。
Given a locally compact group $G$ and a unitary representation $ρ:G\to U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a $C^*$-correspondence ${\mathcal E}(ρ)={\mathcal H}\otimes_{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(ρ)}$. We prove that for $G$ compact, ${\mathcal O}_{{\mathcal E}(ρ)}$ is strong Morita equivalent to a graph $C^*$-algebra. If $λ$ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(λ)}$ is simple and purely infinite, with the same …
There are many different crossed products by an endomorphism of a C*-algebra, and constructions by Exel and Stacey have proved particularly useful. Here we show that every Exel crossed product is isomorphic to a Stacey crossed product (though by a different endomorphism of a different C*-algebra), that every Stacey crossed product is an Exel crossed product, and answer some questions raised by Ionescu and Muhly. While this manuscript is not yet in its final form(s) several people have already used our results, so we are posting it now.
Simplicity of Cuntz-Pimsner algebras of quantum graphs
本文研究了无量子源的量子图的Cuntz-Pimsner代数单纯性，给出了充分条件并定义了到相对代数的满射，首次构造了具有不同量子Cuntz-Krieger与局部代数的量子图实例。
The article discusses the interrelation between relative Cuntz-Pimsner algebras and partial isometric crossed products, and presents a procedure that reduces any given Hilbert bimodule to the "smallest" Hilbert bimodule yielding the same relative Cuntz-Pimsner algebra as the initial one. In the context of crossed products this reduction procedure corresponds to reduction of C*-dynamical systems.
Cuntz-Nica-Pimsner algebras associated to product systems over quasi-lattice ordered groupoids
本文刻画拟格序群胚上紧对齐积系统的Cuntz-Nica-Pimsner代数，并证明特定条件下Morita等价双模的Fell丛完全截面$C^*$-代数与其同构。
We describe the flows of weights for von Neumann algebras arising from KMS states for the gauge actions of the Cuntz-Pimsner algebras in terms of Poisson (tail) boundaries.
We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of directed graphs for gradings induced by $\{0,1\}$-valued labellings of their edge sets.
Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles II: Nuclear dimension
本文证明最小作用且底空间紧、无限、有限覆盖维数时，向量丛扭曲的部分自同构的 Cuntz--Pimsner 代数在 Elliott 程序下可分类，并建立迹与共形测度的双射，推广了部分交叉积及破轨子代数的相关结果。
Cuntz-Nica-Pimsner algebras of product systems over groupoids
本文在拟格序广群上引入了保持规范余作用的内射 Nica 协变 Toeplitz 表示的余泛 $C^*$-代数，推广了群情形的 Cuntz-Nica-Pimsner 代数，并证明了相关的规范不变唯一性定理。
To every $C^*$ correspondence over a $C^*$-algebra one can associate a Cuntz-Pimsner algebra generalizing crossed product constructions, graph $C^*$-algebras, and a host of other classes of operator algebras. Cuntz-Pimsner algebras come equipped with a `gauge action' by the circle group and its finite subgroups. For unital Cuntz-Pimsner algebras, we derive necessary and sufficient conditions for the gauge actions (by either the circle or its closed subgroups) to be free.
Local factors and Cuntz-Pimsner algebras
利用Cuntz-Pimsner代数重新表述Hasse-Weil zeta函数在无穷远处的局部因子，解决Deninger和Serre提出的开放问题。
For Cuntz-Pimsner algebras of bi-Hilbertian bimodules of finite Jones-Watatani index satisfying some side conditions, we give an explicit isomorphism between the $K$-theory exact sequences of the mapping cone of the inclusion of the coefficient algebra into a Cuntz-Pimsner algebra, and the Cuntz-Pimsner exact sequence. In the process we extend some results by the second author and collaborators from finite projective bimodules to finite index bimodules, and also clarify some aspects of Pimsner's `extension of scalars' construction.
Unifying two notions of an action and coaction of a locally compact group on a $C^*$-cor\-re\-spond\-ence we introduce a coaction $(σ,δ)$ of a Hopf $C^*$-algebra $S$ on a $C^*$-cor\-re\-spond\-ence $(X,A)$. We show that this coaction naturally induces a coaction $ζ$ of $S$ on the associated Cuntz-Pimsner algebra $\mathcal{O}_X$ under the weak $δ$-invariancy for the ideal $J_X$. When the Hopf $C^*$-algebra $S$ is defined by a well-behaved multiplicative unitary, we construct a $C^*$-cor\-re\-spond\-ence $(X\rtimes_σ\widehat{S},A\rtimes_δ\widehat{S})$ from $(σ,δ)$ and show that it has a represen…
We show that if $G$ is a second countable locally compact Hausdorff étale groupoid carrying a suitable cocycle $c:G\to\mathbb{Z}$, then the reduced $C^*$-algebra of $G$ can be realised naturally as the Cuntz-Pimsner algebra of a correspondence over the reduced $C^*$-algebra of the kernel $G_0$ of $c$. If the full and reduced $C^*$-algebras of $G_0$ coincide, we deduce that the full and reduced $C^*$-algebras of $G$ coincide. We obtain a six-term exact sequence describing the $K$-theory of $C^*_r(G)$ in terms of that of $C^*_r(G_0)$.
We consider C*-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invariant under the dual action of the circle group. We then prove the analogous statement for the case of product systems, generalizing a theorem of Arveson from the case of product systems of Hilbert spaces.
Suppose $A$ is a $C^*$-algebra and $H$ is a $C^*$-correspondence over $A$. If $H$ is regular in the sense that the left action of $A$ is faithful and is given by compact operators, then we compute the $K$-theory of $\mathcal{O}_A(H) \rtimes \mathbb{T}$ where the action is the usual gauge action. The case where $A$ is an AF-algebra is carefully analyzed. In particular, if $A$ is AF, we show $\mathcal{O}_A(H) \rtimes \mathbb{T}$ is AF. Combining this with Takai duality and an AF-embedding theorem of N. Brown, we show the conditions AF-embeddability, quasidiagonality, and stable finiteness are eq…
Let $P$ be a completely positive map on $M_n(\mathbb{C})$ and let $E_P$ be the associated \emph{GNS}-$C^*$-correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of $E_P$, $\mathcal{O}(E_P)$, is strongly Morita equivalent to the Cuntz algebra $\mathcal{O}_{d(P)}$, where $d(P)$ is the index of $P$.
For bi-Hilbertian $A$-bimodules, in the sense of Kajiwara--Pinzari--Watatani, we construct a Kasparov module representing the extension class defining the Cuntz--Pimsner algebra. The construction utilises a singular expectation which is defined using the $C^*$-module version of the Jones index for bi-Hilbertian bimodules. The Jones index data also determines a novel quasi-free dynamics and KMS states on these Cuntz--Pimsner algebras.
Stabilization theorem and symmetric structure of Cuntz--Pimsner algebras
本文通过建立稳定Cuntz-Pimsner代数的交叉积分解定理，揭示了其中隐含的对称结构，并利用该结构研究了这些代数的简单性，分类了理想、迹权重和KMS权重，同时修正和扩展了Kitamura、Schweizer以及Laca-Neshveyev等人的经典结果。
Given a $\mathbb{Z}$-graded ring $A$ and a subring $R\subseteq A$, it is natural to ask whether $A$ can be realised as the Cuntz-Pimsner ring of some $R$-system. In this paper, we derive sufficient conditions on $A$ and $R$ for this to be the case. As an application, we give conditions under which the Steinberg algebra $A_K(\mathcal{G})$ associated to a $\mathbb{Z}$-graded groupoid $\mathcal{G}=\sqcup_{n\in \mathbb{Z}} \mathcal{G}_n$ can be realised as the Cuntz-Pimsner ring of an $A_K(\mathcal{G}_0)$-system.
Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras
该研究引入乘积系统同伦概念，证明同伦乘积系统的Cuntz-Nica-Pimsner代数K-理论同构，应用于k-图C*-代数并将相关K-理论独立性问题转化为Yang-Baxter型方程解空间的道路连通性问题。
We provide a Cuntz-Pimsner model for graph of groups $C^*$-algebras. This allows us to compute the $K$-theory of a range of examples and show that graph of groups $C^*$-algebras can be realised as Exel-Pardo algebras. We also make a preliminary investigation of whether the crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain trees satisfies Poincaré duality in $KK$-theory. By constructing a $K$-theory duality class we compute the $K$-homology of these crossed products.
We show how the fine structure in shift-tail equivalence, appearing in the noncommutative geometry of Cuntz-Krieger algebras developed by the first two authors, has an analogue in a wide range of other Cuntz-Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz-Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz- and Cuntz-Krieger alge…
We interpret the Cuntz-Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz-Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz-Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.
Let (G,P) be a quasi-lattice ordered group and let X be a compactly aligned product system over P of Hilbert bimodules. Under mild hypotheses we associate to X a C*-algebra which we call the Cuntz-Nica-Pimsner algebra of X. Our construction generalises a number of others: a sub-class of Fowler's Cuntz-Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz-Pimsner algebras of Hilbert bimodules; the C*-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems …
By using C*-correspondences and Cuntz-Pimsner algebras, we associate to every subshift (also called a shift space) $X$ a C*-algebra $O_X$, which is a generalization of the Cuntz-Krieger algebras. We show that $O_X$ is the universal C*-algebra generated by partial isometries satisfying relations given by $X$. We also show that $O_X$ is a one-sided conjugacy invariant of $X$.
When a locally compact group acts on a C*-correspondence, it also acts on the associated Cuntz-Pimsner algebra in a natural way. Hao and Ng have shown that when the group is amenable the Cuntz-Pimsner algebra of the crossed product correspondence is isomorphic to the crossed product of the Cuntz-Pimsner algebra. In this paper, we have a closer look at this isomorphism in the case where the group is not necessarily amenable. We also consider what happens at the level of Toeplitz algebras.
For every one-sided shift space $X$ over a finite alphabet, left special elements are those points in $X$ having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner $C^*$-algebra $\mathcal{O}_X$ has nuclear dimension 1 when $X$ is minimal and the number of left special elements in $X$ is finite. This is done by describing thoroughly the cover of $X$ which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.
We give a short proof of the nuclearity property of a class of Cuntz-Pimsner algebras associated with a Hilbert A-bimodule M, where A is a separable and nuclear C*-algebra. We assume that the left A-action on the bimodule M is given in terms of compact module operators and that M is direct summand of the standard Hilbert module over A.
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