For any countable graph $E$, we investigate the relationship between the Leavitt path algebra $L_{\C}(E)$ and the graph C*-algebra $C^*(E)$. For graphs $E$ and $F$, we examine ring homomorphisms, ring *-homomorphisms, algebra homomorphisms, and algebra *-homomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$. We prove that in certain situations isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ yield *-isomorphisms between the corresponding C*-algebras $C^*(E)$ and $C^*(F)$. Conversely, we show that *-isomorphisms between $C^*(E)$ and $C^*(F)$ produce isomorphisms between $L_{\C}(E)$ and $L_{\C}(F)$ in specific cases. The relationship between Leavitt path algebras and graph C*-algebras is also explored in the context of Morita equivalence.
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 ${\rm Rep}\,{\cal O}_n$ denote the category of all nondegenerate $^*$ representations of the Cuntz algebra ${\cal O}_n$. For any $2\leq n,m<\infty$, we construct an isomorphism functor $F_{n,m}$ from ${\rm Rep}\,{\cal O}_m$ to ${\rm Rep}\,{\cal O}_n$.
Muhly and Solel developed a notion of Morita equivalence for $C^{*}$- correspondences, which they used to show that if two $C^{*}$-correspondences $E$ and $F$ are Morita equivalent then their tensor algebras $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$ are (strongly) Morita equivalent operator algebras. We give the weak$^{*}$ version of this result by considering (weak) Morita equivalence of $W^{*}$-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of $W^{*}$-correspondences $E$ and…
Any finite algebraic Galois covering corresponds to an algebraic Morita equivalence. Here the $C^*$-algebraic analog of this fact is proven, i.e. any noncommutative finite-fold covering corresponds to a strong Morita equivalence.
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 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 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 take advantage of the correspondence between pseudogroups and inverse quantal frames, and of the recent description of Morita equivalence for inverse quantal frames in terms of biprincipal bisheaves, to define Morita equivalence for pseudogroups and to investigate its applications. In particular, two pseudogroups are Morita equivalent if and only if their corresponding localic étale groupoids are. We explore the clear analogies between our definition of Morita equivalence for pseudogroups and the usual notion of strong Morita equivalence for $C^{\ast}$-algebras and these lead to a number of…
In previous work, we defined and studied $Σ^*$-modules, a class of Hilbert $C^*$-modules over $Σ^*$-algebras (the latter are $C^*$-algebras that are sequentially closed in the weak operator topology). The present work continues this study by developing the appropriate $Σ^*$-algebraic analogue of the notion of strong Morita equivalence for $C^*$-algebras. We define strong $Σ^*$-Morita equivalence, prove a few characterizations, look at the relationship with equivalence of categories of a certain type of Hilbert space representation, study $Σ^*$-versions of the interior and exterior tensor produ…
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.
本文利用Hilbert C ~*-模上的紧算子代数的Morita等价性，引入了Hilbert C ~*-模的Morita等价性的概念.我们研究了新的Morita等价的一些性质，得到了一些结果。然后，我们应用这些结果研究了局部紧群在满Hilbert C*-模上的连续作用.我们也给出了格林定理在Hilbert C*-模中的一个推广。
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 …
We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong $Δ$-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a $σ$-TRO. As an application we show that strong Morita equivalence coinc…
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.
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…
We shall introduce a notion of free coactions of a finite dimensional $C^*$-Hopf algebra on a $C^*$-algebra modifying a notion of free actions of a discrete group on a $C^*$-algebra and we shall study several properties on coactions of a finite dimensional $C^*$-Hopf algebra on $C^*$-algebras, which are relating to strong Morita equivalence for inclusions of $C^*$-algebras.
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…
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…
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 …
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.
We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $Δ$-equivalent if they have completely isometric representations $α$ and $β$ respectively and there exists a ternary ring of operators $M$ such that $α(A)$ (resp. $β(B)$) is equal to the norm closure of the linear span of the set $M^*β(B)M, $ (resp. $Mα(A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $Δ$-equivalent, then the operator algebras $A\otimes \cl K$ and $B\otimes \cl K$ are iso…
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 $f$.
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 …
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 …
Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical orbifold groupoids, a Morita equivalence for the crossed product spectral triples is developed. Noncommutative orbifolds are Morita equivalence classes of the crossed product spectral triples. As a special case of this Morita theory one can study freeness of the $G$-action on the …
