Beatriz Abadie
We discuss the crossed product by the dual action of the circle on the crossed product of a C*-algebra A by a Hilbert C*-bimodule X. When X is an A-A Morita equivalence bimodule, the double crossed product is shown to be Morita equivalent to the C*-algebra A.
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Takai duality and crossed product hosts in C*-actions
本文研究奇异 C*-作用下的交叉积宿主结构，验证 Takai 型对偶的有效性，通过 Landstad 代数建立与常规交叉积的联系，并给出基态存在性的充要条件及顺从群情形下完全与约化交叉积宿主的一致性。
The aim of this paper is to study a problem raised by N. C. Phillips concerning the existence of Takai duality for $L^p$ operator crossed products $F^{p}(G,A,α)$, where $G$ is a locally compact Abelian group, $A$ is an $L^{p}$ operator algebra and $α$ is an isometric action of $G$ on $A$. Inspired by D. Williams' proof for the Takai duality theorem for crossed products of $C^*$-algebras, we construct a homomorphism $Φ$ from $F^{p}(\hat{G},F^p(G,A,α),\hatα)$ to $\mathcal{K}(l^{p}(G))\otimes_{p}A$ which is a natural $L^p$-analog of D. Williams' map. For countable discrete Abelian groups $G$ and …
Let $(\mathcal G, Σ)$ be an ordered abelian group with Haar measure $μ$, let $(\mathcal A, \mathcal G, α)$ be a dynamical system and let $\mathcal A\rtimes_α Σ$ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism \[ \mathcal A\rtimes_α Σ\sim_{s} \big(\mathcal A \otimes \mathcal K(\mathcal G, Σ, μ)\big)\rtimes_{α\otimes {\rm Ad}\: ρ} \mathcal G, \] where $\mathcal K(\mathcal G, Σ, μ)$ denotes the compact operators in the CSL algebra ${\rm Alg}\:\mathcal L(\mathcal G, Σ, μ)$ and $ρ$ denotes the right regular representation of $\mathcal G$. We also show th…
We decompose the crossed product functor for actions of crossed modules of locally compact groups on C*-algebras into more elementary constructions: taking crossed products by group actions and fibres in C*-algebras over topological spaces. For this, we extend the theory of partial crossed products from groups to crossed modules; extend Takesaki-Takai duality to Abelian crossed modules; show that equivalent crossed modules have equivalent categories of actions on C*-algebras; and show that certain crossed modules are automatically equivalent to Abelian crossed modules.
For partial automorphisms of $C^*$-algebras, Takai-Takesaki crossed product duality tends to fail, in proportion to the extent to which the partial automorphism is not an automorphism.
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.
The crossed products of locally C*-algebras are defined and a Takai duality theorem for inverse limit actions of a locally compact group on a locally C*-algebra is proved.
We show that if $(X.A)$ and $(Y,B)$ are two isomorphic Hilbert pro-$C^{\ast} $-bimodules, then the crossed product $A\times_{X}\mathbb{Z}$ of $A$ by $X$ and the crossed product $B\times_{Y}\mathbb{Z}$ of $B$ by $Y$ are isomorphic as pro-$C^{\ast}$-algebras. We also prove a property of "associativity" between " $\otimes_{\min}$" and "$\times_{X}$" $\ $as well as " $\otimes_{\max}$" and "$\times_{X}$". As an application of these results we show that the crossed product of a nuclear pro-$C^{\ast}$ -algebra $A$ by a full Hilbert pro-$C^{\ast}$-bimodule $X$ is a nuclear pro-$C^{\ast}$-algebra.
Green's theorem gives a Morita equivalence $C_0(G/H,A)\rtimes G\sim A\rtimes H$ for a closed subgroup $H$ of a locally compact group $G$ acting on a $C^*$-algebra $A$. We prove an analogue of Green's theorem in the case $G=\mathbb{Z}$, where the automorphism generating the action is replaced by a Hilbert $C^*$-bimodule.
A Comparison of Takai and Treumann Dualities
本文证明了Takai对偶与Treumann对偶之间的比较结果，通过交叉积函子和特定双模张量建立等变Kasparov范畴与稳定无穷范畴的等价关系。
In this paper, we study bimodules over a von Neumann algebra $M$ in two related contexts. The first is an inclusion $M \subseteq M \rtimes_αG$, where $G$ is a discrete group acting on a factor $M$ by outer automorphisms. The second is a regular inclusion $M \subseteq N$ of finite factors. In the case of crossed products, we characterize the $M$-bimodules $X$ that lie between $M$ and $M \rtimes_αG$ and are closed in the Bures topology, in terms of the subsets of $G$. We show that this characterization also holds for $w^*$-closed bimodules when $G$ has the approximation property ($AP$), a class …
Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer $*$-automorphisms. In this paper we study the containment $M \subseteq M\rtimes_αG$ of $M$ inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the $M$-bimodules that are closed in the Bures topology and which coincide with the $w^*$-closed ones under a mild hypothesis on $G$. We use these results to obtain a general version of Mercer's theorem concerning the extension of certain isometric $w^*$-continuous …
In this paper we introduce the notion of multiplier of a Hilbert pro-$C^{\ast }$-bimodule and we investigate the structure of the multiplier bimodule of a Hilbert pro-$C^{\ast}$-bimodule. We also investigate the relationship between the crossed product $A\times _{X}\mathbb{Z}$ of a pro-$% C^{\ast }$-algebra $A$ by a Hilbert pro-$C^{\ast }$-bimodule $X$ over $A$, the crossed product $M(A)\times _{M(X)}\mathbb{Z}$ of the multiplier algebra $M(A)$ of $A$ by the multiplier bimodule $M(X)$ of $X$ and the multiplier algebra $M(A\times _{X}\mathbb{Z})$ of $A\times _{X}\mathbb{Z}$.
We study crossed products of arbitrary operator algebras by locally compact groups of completely isometric automorphisms. We develop an abstract theory that allows for generalizations of many of the fundamental results from the selfadjoint theory to our context. We complement our generic results with the detailed study of many important special cases. In particular we study crossed products of tensor algebras, triangular AF algebras and various associated C*-algebras. We make contributions to the study of C*-envelopes, semisimplicity, the semi-Dirichlet property, Takai duality and the Hao-Ng i…
Let B be a C*-algebra with a maximal coaction of a locally compact group G, and let N and H be closed normal subgroups of G with N contained in H. We show that the process Ind_(G/H)^G which uses Mansfield's bimodule to induce representations of the crossed product of B by G from those of the restricted crossed product of B by (G/H) is equivalent to the two-stage induction process: Ind_(G/N)^G composed with Ind_(G/H)^(G/N). The proof involves a calculus of symmetric imprimitivity bimodules which relates the bimodule tensor product to the fibred product of the underlying spaces.
For any maximal coaction (A, G, delta) and any closed normal subgroup N of G, there exists an imprimitivity bimodule Y between the full crossed product A x G x N and A x G/N, together with a compatible coaction delta_Y of G. The assignment (A, delta) -> (Y, delta_Y) implements a natural equivalence between the crossed-product functors "x G x N" and "x G/N", in the category whose objects are maximal coactions of G and whose morphisms are isomorphism classes of right-Hilbert bimodule coactions of G.
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…
We study coactions of concrete Hopf $C^{*}$-bimodules in the framework of (weak) $C^{*}$-pseudo-Kac systems, define reduced crossed products and dual coactions, and prove an analogue of Baaj-Skandalis duality.
The purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces. In particular we shall put emphasis to reveal the full power of the universal properties of the the universal crossed products. It turns out that to make things consistent, it seems useful to perform our constructions on some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non-discrete groups. In the case …
We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that it becomes induced by a Hilbert C(X)-bimodule. Furthermore we introduce the notion of C(X)-category, and discuss relationships with crossed products and duality for compact groups.
A Hilbert bimodule is a right Hilbert module X over a C*-algebra A together with a left action of A as adjointable operators on X. We consider families X = {X_s :s\in P} of Hilbert bimodules, indexed by a semigroup P, which are endowed with a multiplication which implements isomorphisms X_s\otimes_A X_t \to X_{st}; such a family is a called a product system. We define a generalized Cuntz- Pimsner algebra O_X, and we show that every twisted crossed product of A by P can be realized as O_X for a suitable product system X. Assuming P is quasi- lattice ordered in the sense of Nica, we analyze a ce…
It is shown that topological freeness of Rieffel's induced representation functor implies that any $C^*$-algebra generated by a faithful covariant representation of a Hilbert bimodule $X$ over a $C^*$-algebra $A$ is canonically isomorphic to the crossed product $A\rtimes_X \mathbb{Z}$. An ideal lattice description and a simplicity criterion for $A\rtimes_X \mathbb{Z}$ are established.
Let X be a Hilbert bimodule over a C*-algebra A and $O_X= A \rtimes_X \Z$. Using a finite section method we construct a sequence of completely positive contractions factoring through matrix algebras over A which act on $s_ξ s_η^*$ as Schur multipliers converging to the identity. This shows immediately that for a finitely generated X the algebra $O_X$ inherits any standard approximation property such as nuclearity, exactness, CBAP or OAP from A. We generalise this to certain general Pimsner algebras by proving semi-splitness of the Toeplitz extension under certain conditions and discuss some ex…
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.
It is shown that a C*-algebra generated by any faithful covariant representation of a Hilbert bimodule X is canonically isomorphic to the crossed product associated to X provided that Rieffel's induced representation functor X-ind is topologically free. It is discussed how this result could be applied to universal C*-algebras generated by relations with a circle gauge action. In particular, it leads to generalizations of isomorphism theorems for various crossed products, and is shown to be equivalent to Cuntz-Krieger uniqueness theorem for finite graph C*-algebras (on that occasion an intrigui…
C*-endomorphisms arising from superselection structures with non-trivial centre 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 and first Chern class, and can be used to construct a duality for abstract (nonsymmetric) tensor categories vs. 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 …
Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product C*-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Green's Imprimitivity Theorem for actions of groups, and Mansfield's Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have C*-algebras with actions or coactions (or both) of a fixed locally compact group G as their objects, and equivariant equivale…
We propose a generalisation 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.
The Takai duality of $L^p$-operator algebras with incompressibility
本文研究具有不可压缩性的$L^p$算子代数的高井对偶性，证明$C^*$代数的该对偶性无法推广到$p\neq 2$的$L^p$算子代数，并以否定方式解决了N. C. Phillips提出的问题。
Let $A$ and $B$ be arbitrary $C^*$-algebras, we prove that the existence of a Hilbert $A$-$B$-bimodule of finite index ensures that the WEP, QWEP, and LLP along with other finite-dimensional approximation properties such as CBAP and (S)OAP are shared by $A$ and $B$. For this, we first study the stability of the WEP, QWEP and LLP under Morita equivalence of $C^*$-algebras. We present examples of Hilbert $A$-$B$-bimodules which are not of finite index, while such properties are shared between $A$ and $B$. To this end, we study twisted crossed products by amenable discrete groups.
We provide an exposition and proof of Renault's equivalence theorem for crossed products by locally Hausdorff, locally compact groupoids. Our approach stresses the bundle approach, concrete imprimitivity bimodules and is a preamble to a detailed treatment of the Brauer semigroup for a locally Hausdorff, locally compact groupoid.
Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger algebras O_B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O_\infty.
Let P be a left LCM semigroup, and $α$ an action of $P$ by endomorphisms of a $C^{*}$-algebra $A$. We study a semigroup crossed product $C^{*}$-algebra in which the action $α$ is implemented by partial isometries. This crossed product gives a model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules (associated with semigroup dynamical systems) studied first by Fowler, for which we provide a structure theorem as it behaves well under short exact sequences and tensor products.
Let $Γ^+$ be the positive cone in a totally ordered abelian group $Γ$, and let $α$ be an action of $Γ^+$ by endomorphisms of a $C^*$-algebra $A$. We consider a new kind of crossed-product $C^*$-algebra $A\times_αΓ^+$, which is generated by a faithful copy of $A$ and a representation of $Γ^+$ as partial isometries. We claim that these crossed products provide a rich and tractable family of Toeplitz algebras for product systems of Hilbert bimodules, as recently studied by Fowler, and we illustrate this by proving detailed structure theorems for actions by forward and backward shifts.
We associate a pro-C*-algebra to a pro-C*-correspondence and show that this construction generalizes the construction of crossed products by Hilbert pro-C*-bimodules and the construction of pro-C*-crossed products by strong bounded automorphisms.
We give a construction allowing to lift spectral triples to crossed products by Hilbert bimodules. The spectral triple one obtains is a concrete unbounded representative of the Kasparov product of the spectral triple and the Pimsner-Toeplitz extension associated to the crossed product by the Hilbert bimodule. To prove that the lifted spectral triple is the above-mentioned Kasparov product, we rely on operator-$*$-algebras and connexions.
We consider a class of proper actions of locally compact groups on imprimitivity bimodules over C*-algebras which behave like the proper actions on C*-algebras introduced by Rieffel in 1988. We prove that every such action gives rise to a Morita equivalence between a crossed product and a generalized fixed-point algebra, and in doing so make several innovations which improve the applicability of Rieffel's theory. We then show how our construction can be used to obtain canonical tensor-product decompositions of important Morita equivalences. Our results show, for example, that the different pro…
We have introduced a notion of $C^*$-symbolic dynamical system in [K. Matsumoto: Actions of symbolic dynamical systems on $C^*$-algebras, to appear in J. Reine Angew. Math.], that is a finite family of endomorphisms of a $C^*$-algebra with some conditions. The endomorphisms are indexed by symbols and yield both a subshift and a $C^*$-algebra of a Hilbert $C^*$-bimodule. The associated $C^*$-algebra with the $C^*$-symbolic dynamical system is regarded as a crossed product by the subshift. We will study a simplicity condition of the $C^*$-algebras of the $C^*$-symbolic dynamical systems. Some ex…
Let U, V, and W be multiplicative unitaries coming from discrete Kac systems such that W is an amenable normal submultiplicative unitary of V with quotient U. We define notions for right-Hilbert bimodules of coactions of S_V and (S_V)^, their restrictions to S_W and (S_U)^, their dual coactions, and their full and reduced crossed products. If N(A) denotes the imprimitivity bimodule associated to any coaction of S_V on a C*-algebra A by Ng's imprimitivity theorem, then for any suitably nondegenerate injective coaction of S_V on a right-Hilbert A - B bimodule X we establish an isomorphism betwee…
We study the ideal structure of $C^*$-algebras arising from $C^*$-correspondences. We prove that gauge-invariant ideals of our $C^*$-algebras are parameterized by certain pairs of ideals of original $C^*$-algebras. We show that our $C^*$-algebras have a nice property which should be possessed by generalization of crossed products. Applications to crossed products by Hilbert $C^*$-bimodules and relative Cuntz-Pimsner algebras are also discussed.
Let $A$ and $B$ be $σ$-unital $C^*$-algebras and $X$ and $Y$ an $A-A$-equivalence bimodule and a $B-B$-equivalence bimodule, respectively. Also, let $A\rtimes_X \mathbb{Z}$ and $B\rtimes_Y \mathbb{Z}$ be the crossed products of $A$ and $B$ by $X$ and $Y$, respectively. Furthermore, let $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_Y \mathbb{Z}$ be the inclusions of $C^*$-algebras induced by $X$ and $Y$, respectively. We suppose that $A' \cap M(A\rtimes_X \mathbb{Z})=\mathbb{C} 1$. In this paper we shall show that the inclusions $A\subset A\rtimes_X \mathbb{Z}$ and $B\subset B\rtimes_…
We study simplicity and pure infiniteness criteria for C*-algebras associated to inverse semigroup actions by Hilbert bimodules and to Fell bundles over etale not necessarily Hausdorff groupoids. Inspired by recent work of Exel and Pitts, we introduce essential crossed products for which there are such criteria. In our approach the major role is played by a generalised expectation with values in the local multiplier algebra. We give a long list of equivalent conditions characterising when the essential and reduced C*-algebras coincide. Our most general simplicity and pure infiniteness criteria…
We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that the endomorphism itself becomes induced by the bimodule of continuous sections of a vector bundle. Some motivating examples for such a construction are given. Furthermore, we study the C*-algebra of G-invariant elements of the Cuntz-Pimsner algebra associated with a G-vector bundle, where G is a (noncompact, in general) group. In particular, the C*-algebra of invariant elements w.r.t. the action of the group of special unitaries of the given vector bundle is a crossed product in the above sense. We also s…
For an action $α$ of a locally compact group $G$ on a dual operator space $X$ by w*-continuous completely isometric isomorphisms one can define two generally different notions of crossed products, namely the Fubini crossed product $X\rtimes_α^{F}G$ and the spatial crossed product $X\overline{\rtimes}_α G$. It is shown that $X\rtimes_α^{F}G=X\overline{\rtimes}_α G$ if and only if the dual comodule action $\widehatα$ of the group von Neumann algebra $L(G)$ on the Fubini crossed product of $X\rtimes_α^{F}G$ is non-degenerate. As an application, this yields an alternative proof of the result of Cr…
In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be formulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomorphisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed-product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an …
The quantum Heisenberg manifolds are noncommutive manifolds constructed by M. Rieffel as strict deformation quantizations of Heisenberg manifolds and have been studied by various authors. Rieffel constructed the quantum Heisenberg manifolds as the generalized fixed-point algebras of certain crossed product $C^*$-algebras, and they also can be realized as crossed products of $C(\mathbb{T}^2)$ by Hilbert $C^*$-bimodules in the sense of Abadie et al. In this paper, we describe how the quantum Heisenberg manifolds can also be realized as twisted groupoid $C^*$-algebras.
We discuss Morita equivalence within the family of quantum Heisenberg manifolds. The main tool employed is the generalization of a result of P. Green and M. Rieffel about Morita equivalence of transformation groups to crossed products by Hilbert bimodules.
Coactions of Hopf C*-bimodules simultaneously generalize coactions of Hopf C*-algebras and actions of groupoids. Following an approach of Baaj and Skandalis, we construct reduced crossed products and establish a duality for fine coactions. Examples of coactions arise from Fell bundles on groupoids and actions of a groupoid on bundles of C*-algebras. Continuous Fell bundles on an etale groupoid correspond to coactions of the reduced groupoid algebra, and actions of a groupoid on a continuous bundle of C*-algebras correspond to coactions of the function algebra.
Heisenberg-Weyl Representations and Morita equivalence for crossed products of Noncommutative solenoids
研究非交换螺线管交叉积的强莫里塔等价，构造不变乘子，建立海森堡 - 外尔框架，通过酉算子扩展等价双模，得到无限和有限循环子群交叉积的强莫里塔等价。
We define and systematically study nonassociative C*-algebras as C*-algebras internal to a topological tensor category. We also offer a concrete approach to these C*-algebras, as G-invariant, norm closed *-subalgebras of bounded operators on a G-Hilbert space, with deformed composition product. Our central results are those of stabilization and Takai duality for (twisted) crossed products in this context.
Ideal structure and pure infiniteness of inverse semigroup crossed products
本文研究 $C^*$-包含 $A\subseteq B$ 的理想分离与纯无限性判定，并将其应用于逆半群本质交叉积及非 Hausdorff étale 群胚上 Fell 丛截面代数。
For $A$ a $C^*$-algebra, $E_1, E_2$ two Hilbert bimodules over $A$, and a fixed isomorphism $χ: E_1\otimes_AE_2\to E_2\otimes_AE_1$, we consider the problem of computing the $K$-theory of the Cuntz-Pimsner algebra ${\mathcal O}_{E_2\otimes_A{\mathcal O}_{E_1}}$ obtained by extending the scalars and by iterating the Pimsner construction. The motivating examples are a commutative diagram of Douglas and Howe for the Toeplitz operators on the quarter plane, and the Toeplitz extensions associated by Pimsner and Voiculescu to compute the $K$-theory of a crossed product. The applications are for Hilb…
Mansfield showed how to induce representations of crossed products of C*-algebras by coactions from crossed products by quotient groups and proved an imprimitivity theorem characterising these induced representations. We give an alternative construction of his bimodule in the case of dual coactions, based on the symmetric imprimitivity theorem of the third author; this provides a more workable way of inducing representations of crossed products of C*-algebras by dual coactions. The construction works for homogeneous spaces as well as quotient groups, and we prove an imprimitivity theorem for t…
Let $θ, θ'$ be irrational numbers and $A, B$ be matrices in $SL_2(\mathbb{Z})$ of infinite order. We compute the $K$-theory of the crossed product $\mathcal{A}_θ\rtimes_A \mathbb{Z}$ and show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are $*$-isomorphic if and only if $θ= \pmθ' \pmod{\mathbb{Z}}$ and $I-A^{-1}$ is matrix equivalent to $I-B^{-1}$. Combining this result and an explicit construction of equivariant bimodules, we show that $\mathcal{A}_θ \rtimes_A\mathbb{Z}$ and $\mathcal{A}_{θ'} \rtimes_B \mathbb{Z}$ are Morita equivalent if and only if $θ…
In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning actions of locally compact quantum groups on C*-algebras [S. Baaj, G. Skandalis and S. Vaes, 2003]. Let $\cal G$ be a measured quantum groupoid on a finite basis. We prove that if $\cal G$ is regular, then any weakly continuous action of $\cal G$ on a C*-algebra is necessarily strongly continuous. Following [S. Baaj and G. Skandalis, 1989], we introduce and investigate a notion of $\cal G$-equivariant Hilbert C$^*$-modules. By applying the previous results and a version …
This partly expository paper first supplies the details of a method of factoring a stable C*-algebra A as B \otimes K in a canonical way. Then it is shown that this method can be put into a categorical framework, much like the crossed-product dualities, and that stabilization gives rise to an equivalence between the nondegenerate category of C*-algebras and a category of "K-algebras". We consider this equivalence as "inverting" the stabilization process, that is, a "destabilization". Furthermore, the method of factoring stable C*-algebras generalizes to Hilbert bimodules, and an analogous cate…
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。
We study the 2-adic version of the ring $C^*$-algebra of the integers. First, we work out the precise relation between the Cuntz algebra $\cO_2$ and our 2-adic ring $C^*$-algebra in terms of representations. Secondly, we prove a 2-adic duality theorem identifying the crossed product arising from 2-adic affine transformations on the 2-adic numbers with the analogous crossed product algebra over the real numbers. And finally, as an outcome of this duality result, we construct an explicit imprimitivity bimodule and prove that it transports one canonical representation into the other.
We introduce {\it covariant structures} $\left\{(\A,\k),(\a,å),\(\ha,\haa\)\right\}$ formed of a separable $C^*$-algebra $\A$, a measurable twisted action $(\a,å)$ of the second-countable locally compact group $\G$\,, a measurable twisted action $(\ha,\haa)$ of another second-countable locally compact group $\hG$ and a strictly continuous function $\k:\G\times\hG\to\U\M(\A)$ suitably connected with $(\a,å)$ and $\(\ha,\haa\)$\,. Natural notions of covariant morphisms and representations are considered, leading to a sort of twisted crossed product construction. Various $C^*$-algebras emerge by …
Following Jansen and Waldmann, and Kajiwara and Watatani, we shall introduce notions of coactions of a finite dimensional $C^*$-Hopf algebra on a Hilbert $C^*$-bimodule of finite type in the sense of Kajiwara and Watatani and define their crossed product. We shall investigate their basic properties and show that the strong Morita equivalence for coactions preserves the Rohlin property for coactions of a finite dimensional $C^*$-Hopf algebra on unital $C^*$-algebras.
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