Let $A \subset C$ and $B \subset D$ be unital inclusions of unital $C^*$-algebras. Let ${}_A \mathbf{B}_A (C, A)$ (resp. ${}_B \mathbf{B}_B (D, B)$) be the space of all bounded $A$-bimodule (resp. $B$-bimodule) linear maps from $C$ (resp. $D$) to $A$ (resp. $B$). We suppose that $A \subset C$ and $B \subset D$ are strongly Morita equivalent. We shall show that there is an isometric isomorphism $f$ of ${}_A \mathbf{B}_A (C, A)$ onto ${}_B \mathbf{B}_B (D, B)$ and we shall study on basic properties about $f$.
We shall introduce the notions of the strong Morita equivalence for unital inclusions of unital $C^*$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^*$-algebras onto their unital $C^*$-subalgebras. Also, we shall study their basic properties.
We define an equivalence relation between bimodules over maximal abelian selfadjoint algebras (masa bimodules) which we call spatial Morita equivalence. We prove that two reflexive masa bimodules are spatially Morita equivalent iff their (essential) bilattices are isomorphic. We also prove that if S^1, S^2 are bilattices which correspond to reflexive masa bimodules U_1, U_2 and f: S^1\rightarrow S^2 is an onto bilattice homomorphism, then: (i) If U_1 is synthetic, then U_2 is synthetic. (ii) If U_2 contains a nonzero compact (or a finite or a rank 1) operator, then U_1 also contains a nonzero …
Let $A$ be a unital $C^*$-algebra and $X$ an invulutive $A-A$-equivalence bimodule. Let $A\subset C_X$ be the unital inclusion of unital $C^*$-algebras induced by $X$. We suppose that $A' \cap C_X =\mathbf{C} 1$. We shall compute the Picard group of the unital inclusion $A\subset C_X$.
Let $A$ and $B$ be unital $C^*$-algebras and let $H$ be a finite dimensional $C^*$-Hopf algebra. Let $H^0$ be its dual $C^*$-Hopf algebra. Let $(ρ, u)$ and $(σ, v)$ be twisted coactions of $H^0$ on $A$ and $B$, respectively. In this paper, we shall show the following theorem: We suppose that the unital inclusions $A\subset A\rtimes_{ρ, u}H$ and $B\subset B\rtimes_{σ, v}H$ are strongly Morita equivalent. If $A'\cap (A\rtimes_{ρ, u}H)=\BC1$, then there is a $C^*$-Hopf algebra automorphism $λ^0$ of $H^0$ such that the twisted coaction $(ρ, u)$ is strongly Morita equivalent to the twisted coaction…
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 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.
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.
In this paper, we answer a question of Blecher-Muhly-Paulsen pertaining to identifying topological invariants for completely bounded Morita equivalences of holomorphic cross-section algebras. Given a certain natural subcontext of a strong Morita context of $n$-homogeneous $C^*$-algebras whose spectrum $T$ is an annulus, Blecher-Muhly-Paulsen are able to estimate the norm of a lifting of the identity of a holomorphic subalgebra by a conformal invariant of the annulus and a property of the associated matrix bundle. We give a generalization of the above example in which $T$ is a bordered Riemann …
We consider two inclusions of $C^*$-algebras whose small $C^*$-algebras have approximate units of the large $C^*$-algebras and their two spaces of all bounded bimodule linear maps. We suppose that the two inclusions of $C^*$-algebras are strongly Morita equivalent. In this paper, we shall show that there exists an isometric isomorphism from one of the spaces of all bounded bimodule linear maps to the other space and we shall study on basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss on the P…
Consider a unital C*-algebra A, a von Neumann algebra M, a unital sub-C*-algebra C of A and a unital *-homomorphism $π$ from C to M. Let u: A --> M be a decomposable map (i.e. a linear combination of completely positive maps) which is a C-bimodule map with respect to $π$. We show that u is a linear combination of C-bimodule completely positive maps if and only if there exists a projection e in the commutant of $π(C)$ such that u is valued in eMe and $eπ(.)e$ has a completely positive extension A --> eMe.
We will introduce the notion of strong Morita equivalence for completely positive linear maps and study its basic properties. Also, we will discuss the relation between strong Morita equivalence for bounded $C^*$-bimodule linear maps and strong Morita equivalence for completely positive linear maps. Furthermore, we will show that if two unital $C^*$-algebras are strongly Morita equivalent, then there is a $1-1$ correspondence between the two sets of all strong Morita equivalence classes of completely positive linear maps on the two unital $C^*$-algebras and we will show that the corresponding …
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.
We revisit and generalize the application of a method introduced by Latrémolière and Packer for constructing finitely generated projective modules over the noncommutative solenoid C*-algebras. By realizing them as direct limits of rotation algebras, the method constructs directed systems of equivalence bimodules between rotation algebras that satisfy the necessary compatibility conditions to build Morita equivalence bimodules between the direct limit C*-algebras. In the irrational case, we use a fixed projection in a matrix algebra over the rotation algebra satisfying a key condition to build …
We shall generalize the notion of the strong Morita equivalence for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algebra and define the Picard groups with respect to the generalized strong Morita equivalence. We call them the generalized Picard groups for coactions of a finite dimensional $C^*$-Hopf algebra on a unital $C^*$-algeba. We shall investigate basic properties of the generalized Picard groups and clarify the relation between the generalized Picard groups and the Picard groups for unital inclusions of unital $C^*$-algebras.
In this paper, we will introduce a notion of basis related Morita equivalence in the Cuntz--Krieger algebras $({\mathcal{O}_A}, \{S_a\}_{a \in E_A})$ with the canonical right finite basis $\{S_a\}_{a \in E_A}$ as Hilbert $C^*$-bimodule, and prove that two nonnegative irreducible matrices $A$ and $B$ are elementary equivalent, that is, $A = CD, B = DC$ for some nonnegative rectangular matrices $C, D$, if and only if the Cuntz--Krieger algebras $({\mathcal{O}_A}, \{S_a\}_{a \in E_A})$ and $({\mathcal{O}_B}, \{ S_b\}_{b\in E_B})$ with the canonical right finite bases are basis relatedly elementar…
We will consider the set of all completely positive linear maps from a unital $C^*$-algebra to the $C^*$-algebra of all (bounded) adjointable right Hilbert $C^*$-module maps, which are automatically bounded, on a right Hilbert $C^*$-module and we will introduce strong Morita equivalence for elements in this set. In this paper, we will give the following result: If two classes of two unital $C^*$-algebras are strongly Morita equivalent, respectively, then we can construct a bijective correspondence between two sets of all strong Morita equivalence classes of completely positive linear maps give…
We characterize the pairs of operator spaces which occur as pairs of Morita equivalence bimodules between non-selfadjoint operator algebras in terms of the mutual relation between the spaces. We obtain a characterization of the operator spaces which are completely isometrically isomorphic to imprimitivity bimodules between some strongly Morita equivalent (in the sense of Rieffel) C*-algebras. As corollaries, we give representation results for such operator spaces.
We consider two twisted actions of a countable discrete group on $σ$-unital $C^*$-algebras. Then by taking the reduced crossed products, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions is irreducible, that is, the relative commutant of one of the $σ$-unital $C^*$-algebra in the multiplier $C^*$-algebra of the reduced twisted crossed product is trivial. We show that the two actions are then strongly Morita equivalent up to some automorphism of the group.
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 classical Morita Theorem for rings established the equivalence of three statements, involving categorical equivalences, isomorphisms between corners of finite matrix rings, and bimodule homomorphisms. A fourth equivalent statement (established later) involves an isomorphism between infinite matrix rings. In our main result, we establish the equivalence of analogous statements involving graded categorical equivalences, graded isomorphisms between corners of finite matrix rings, graded bimodule homomorphisms, and graded isomorphisms between infinite matrix rings.
Let $\mathcal{A}= \{A_t \}_{t \in G}$ and $\mathcal{B}= \{B_t \}_{t\in G}$ be $C^*$-algebraic bundles over a finite group $G$. Let $C=\oplus_{t \in G}A_t$ and $D=\oplus_{t\in G}B_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit element in $G$. We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$, respectively. In this paper, we shall show that if there is an equivalence $\mathcal{A}-\mathcal{B}$-bundle over $G$ with some properties, then the unital inclusions of unital $C^*$-algebras $A \subset C$ and $B \subset D$ induced by $\mathcal{A}$ and $\…
Let A be a unital C*-algebra. We shall introduce involutive A-A-equivalence bimodules and prove that any C*-algebra containing A with Watatani index 2 is constructed by an involutive A-A-equivalence bimodule.
We generalize the classification result of Restorff on Cuntz-Krieger algebras to cover all unital graph C*-algebras with real rank zero, showing that Morita equivalence in this case is determined by ordered, filtered K-theory as conjectured by three of the authors. The classification result is geometric in the sense that it establishes that any Morita equivalence between C*(E) and C*(F) in this class can be realized by a sequence of moves leading from E to F in a way resembling the role of Reidemeister moves on knots. As a key technical step, we prove that the so-called Cuntz splice leaves uni…
We consider two saturated Fell bundles over a countable discrete group, whose unit fibers are $σ$-unital $C^*$-algebras. Then by taking the reduced cross-sectional $C^*$-algebras, we get two inclusions of $C^*$-algebras. We suppose that they are strongly Morita equivalent as inclusions of $C^*$-algebras. Also, we suppose that one of the inclusions of $C^*$-algebras is irreducible, that is, the relative commutant of one of the unit fiber algebras, which is a $σ$-unital $C^*$-algebra, in the multiplier $C^*$-algebra of the reduced cross-sectional $C^*$-algebra is trivial. We show that the two sa…
The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level an…
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.
Rings form a bicategory [Rings], with classes of bimodules as horizontal arrows, and bimodule maps as vertical arrows. The notion of Morita equivalence for rings can be translated in terms of bicategories in the following way. Two rings are Morita equivalent if and only if they are isomorphic objects in the bicategory. We repeat this construction for von Neumann algebras. Von Neumann algebras form a bicategory [W*], with classes of correspondences as horizontal arrows, and intertwiners as vertical arrows. Two von Neumann algebras are Morita equivalent if and only if they are isomorphic objects…
Let $p\in \mathbb{N}$ be prime, and let $θ$ be irrational. The authors have previously shown that the noncommutative $p$-solenoid corresponding to the multiplier of the group $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$ parametrized by $α=(θ+1, (θ+1)/p,\cdots, (θ+1)/p^j,\cdots )$ is strongly Morita equivalent to the noncommutative solenoid on $\left(\mathbb{Z}\left[\frac{1}{p}\right]\right)^2$ coming from the multiplier $β= (1-\frac{θ+1}θ,1-\frac{θ+1}{pθ}, \cdots, 1-\frac{θ+1}{p^jθ}, \cdots )$ . The method used a construction of Rieffel referred to as the "Heisenberg bimodule" in which …
We introduce the notion of $Δ$ and $σ\,Δ-$ pairs for operator algebras and characterise $Δ-$ pairs through their categories of left operator modules over these algebras. Furthermore, we introduce the notion of $Δ$-Morita equivalent operator spaces and prove a similar theorem about their algebraic extensions. We prove that $σΔ$-Morita equivalent operator spaces are stably isomorphic and vice versa. Finally, we study unital operator spaces, emphasising their left (resp. right) multiplier algebras, and prove theorems that refer to $Δ$-Morita equivalence of their algebraic extensions.
The $w^{\star}$-rigged modules over dual operator algebras were introduced by Blecher and Kashyap as a generalization of $W^{\star}$-modules. In this paper, we introduce two new types of Morita equivalence between right $w^{\star}$-rigged modules over unital dual operator algebras and we examine whether these notions imply stable isomorphism between the corresponding modules. Furthermore we investigate them in detail for the class of right $w^{\star}$-rigged modules over nest algebras, that was characterised by G.K.Eleftherakis.
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 generalize the main theorem of Rieffel for Morita equivalence of W*-algebras to the case of unital dual operator algebras: two unital dual operator algebras A and B have completely isometric normal representations alpha, beta such that alpha(A) is the w*-closed span of M*beta(B)M and beta(B) is the w*-closed span of Malpha(A)M* for a ternary ring of operators M (i.e. a linear space M such that MM*M \subset M if and only if there exists an equivalence functor $F:_{A}M\to_{B}M$ which "extends" to a *-functor implementing an equivalence between the categories $_{A}DM$ and $_{B}DM.$ By $_{A}M$ …
We give a dynamical characterization of categorical Morita equivalence between compact quantum groups. More precisely, by a Tannaka-Krein type duality, a unital C*-algebra endowed with commuting actions of two compact quantum groups corresponds to a bimodule category over their representation categories. We show that this bimodule category is invertible if and only if the actions are free, with finite dimensional fixed point algebras, which are in duality as Frobenius algebras in an appropriate sense. This extends the well-known characterization of monoidal equivalence in terms of bi-Hopf-Galo…
We provide a complete classification of the class of unital graph $C^*$-algebras - prominently containing the full family of Cuntz-Krieger algebras - showing that Morita equivalence in this case is determined by ordered, filtered $K$-theory. The classification result is geometric in the sense that it establishes that any Morita equivalence between $C^*(E)$ and $C^*(F)$ in this class can be realized by a sequence of moves leading from $E$ to $F$, in a way resembling the role of Reidemeister moves on knots. As a key ingredient, we introduce a new class of such moves, establish that they leave th…
A $*$-bimodule for a unital $*$-algebra $A$ is an $A$-bimodule $X$ which is a vector space with involution $x\mapsto x^+$ satisfying $(a\cdot x\cdot b)^+=b^+\cdot x^+\cdot b^+$ for $x\in X$ and $a,b\in A$. An algebraic model for $*$-bimodules is given. Hilbert space representations of $*$-bimodules are defined and studied. A GNS-like representation theorem is obtained.
We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such functor is completely isometrically isomorphic to the Haagerup tensor product (= interior tensor product) with a strong Morita equivalence bimodule.
Let $\mathcal R$ be a ring, $\mathcal{M}$ be a $\mathcal R$-bimodule and $m,n$ be two fixed nonnegative integers with $m+n\neq0$. An additive mapping $δ$ from $\mathcal R$ into $\mathcal{M}$ is called an \emph{$(m,n)$-Jordan derivation} if $(m+n)δ(A^{2})=2mAδ(A)+2nδ(A)A$ for every $A$ in $\mathcal R$. In this paper, we prove that every $(m,n)$-Jordan derivation from a $C^{*}$-algebra into its Banach bimodule is zero. An additive mapping $δ$ from $\mathcal R$ into $\mathcal{M}$ is called a $(m,n)$-Jordan derivable mapping at $W$ in $\mathcal R$ if $(m+n)δ(AB+BA)=2mδ(A)B+2mδ(B)A+2nAδ(B)+2nBδ(A)$…
We prove that every 2-local derivation from the algebra $M_n(\mathcal{A})(n>2)$ into its bimodule $M_n(\mathcal{M})$ is a derivation, where $\mathcal{A}$ is a unital Banach algebra and $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule such that each Jordan derivation from $\mathcal{A}$ into $\mathcal{M}$ is an inner derivation, and that every 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, Jiang-Su algebra and UHF algebras.
We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra $A$ as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product $A\otimes_h A$ labelled by the essential closed right ideals of $A$ into $A$. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.
We answer the title question for sigma-unital C*-algebras. The answer is that the algebra must be the direct sum of a dual C*-algebra and a C*-algebra satisfying a certain local unitality condition. We also discuss similar problems in the context of Hilbert C*-bimodules and imprimitivity bimodules and in the context of centralizers of Pedersen's ideal.
We introduce and examine the notions of strong $Δ$-equivalence and strong TRO equivalence for operator spaces. We show that they behave in an analogous way to how strong Morita equivalence does for the category of C*-algebras. In particular, we prove that strong $Δ$-equivalence coincides with stable isomorphism under the expected countability hypothesis, and that strongly TRO equivalent operator spaces admit a correspondence between particular representations. Furthermore we show that strongly $Δ$-equivalent operator spaces have stably isomorphic second duals and strongly $Δ$-equivalent TRO en…
Given C*-algebras A and B and an imprimitivity A-B-bimodule X, we construct an explicit isomorphism X_* : K_i(A) --> K_i(B) where K_i denote the complex K-theory functors for i=0, 1. Our techniques do not require separability nor existence of countable approximate identities. We thus extend, to general C*-algebras, the result of Brown, Green and Rieffel according to which strongly Morita equivalent C*-algebras have isomorphic K-groups. The method employed includes a study of Fredholm operators on Hilbert modules.
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 introduce a notion of Morita equivalence for non-selfadjoint operator algebras equipped with a completely isometric involution (operator *-algebras). We then show that the unbounded Kasparov product by a Morita equivalence bimodule induces an isomorphism between equivalence classes of twisted spectral triples over Morita equivalent operator *-algebras. This leads to a tentative definition of unbounded bivariant K-theory and we prove that this bivariant theory is related to Kasparov's bivariant K-theory via the Baaj-Julg bounded transform. Moreover, the unbounded Kasparov product provides a …
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…
Let $G$ and $H$ be Hausdorff ample groupoids and let $R$ be a commutative unital ring. We show that if $G$ and $H$ are equivalent in the sense of Muhly-Renault-Williams, then the associated Steinberg algebras of locally constant $R$-valued functions with compact support are Morita equivalent. We deduce that collapsing a ``collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path $R$-algebra, and therefore a number of graphical constructions which yield Morita equivalent $C^*$-algebras also yield Morita eq…
Given a conditionally completely positive map $\mathcal L$ on a unital $\ast$-algebra $\A$, we find an interesting connection between the second Hochschild cohomology of $\A$ with coefficients in the bimodule $E_{\mathcal L}=\B^a(\A \oplus M)$ of adjointable maps, where $M$ is the GNS bimodule of $\mathcal L$, and the possibility of constructing a quantum random walk (in the sense of \cite{AP,LP,L,KBS}) corresponding to $\mathcal L$.
We revisit the characterisation of modules over non-unital $C^*$-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical index and finite Watatani index.
We prove that two dual operator algebras are weak$^*$ Morita equivalent if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak$^*$-continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak$^*$ Morita equivalence bimodule. We also develop the theory of the $W^*$-dilation, which connects the non-selfadjoint dual operator algebra with the $W^*$-algebraic framework. In the case of w…
We introduce the notion of finite right (respectively left) numerical index on a bimodule $X$ over C*-algebras A and B with a bi-Hilbertian structure. This notion is based on a Pimsner-Popa type inequality. The right (respectively left) index element of X can be constructed in the centre of the enveloping von Neumann algebra of A (respectively B). X is called of finite right index if the right index element lies in the multiplier algebra of A. In this case we can perform the Jones basic construction. Furthermore the C*--algebra of bimodule mappings with a right adjoint is a continuous field of…
We solve two problems in the theory of correspondences that have important implications in the theory of product systems. The first problem is the question whether every correspondence is the correspondence associated (by the representation theory) with a unital endomorphism of the algebra of all adjointable operators on a Hilbert module. The second problem is the question whether every correspondence allows for a nondegenerate faithful representation on a Hilbert space. We also resolve an extension problem for representations of correspondences and we provide new efficient proofs of several w…
The curvature of the noncommutative torus $T^2_θ$ ($θ$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the $C^*$-algebra $A_θ= C(T^2_θ)$ with other toric algebras $A_{θ'}$. In the enlarged context the conformal metric on $T^2_θ$ is exchanged with an arbitrary Hermitian metric…
We point out a connection between Gabor analysis and noncommutative analysis. Especially, the strong Morita equivalence of noncommutative tori appears as underlying setting for Gabor analysis, since the construction of equivalence bimodules for noncommutative tori has a natural formulation in the notions of Gabor analysis. As an application we show that Feichtinger's algebra is such an equivalence bimodule. Furthermore, we present Connes's construction of projective modules for noncommutative tori and the relevance of a generalization of Wiener's lemma for twisted convolution by Gröchenig and …
Any etale Lie groupoid G is completely determined by its associated convolution algebra C_c(G) equipped with the natural Hopf algebroid structure. We extend this result to the generalized morphisms between etale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c(G)-C_c(H)-bimodule C_c(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c gives an equivalence between the Morita category of etale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.
We give a notion of equivalence for Fell bundles over groups, not necessarily saturated nor separable, and show that equivalent Fell bundles have Morita-Rieffel equivalent cross-sectional $C^*$-algebras. Our notion is originated in the context of partial actions and their enveloping actions. The equivalence between two Fell bundles is implemented by a bundle of Hilbert bimodules with some extra structure. Suitable cross-sectional spaces of such a bundle turn out to be imprimitivity bimodules for the cross-sectional $C^*$-algebras of the involved Fell bundles. We show that amenability is preser…
In this paper, we will introduce notions of relative version of imprimitivity bimodules and relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal{A}, \mathcal{D})$ such that $\mathcal{D}$ is a $C^*$-subalgebra of $\mathcal{A}$ with certain conditions. We will then prove that two pairs $(\mathcal{A}_1, \mathcal{D}_1)$ and $(\mathcal{A}_2, \mathcal{D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particularly, for two pairs $(\mathcal{O}_A, \mathcal{D}_A)$ and $(\mathcal{O}_B, \mathcal{D}_B)$ of Cuntz--Krieger …
