Strongly Morita Equivalent Research Articles

Strong Morita Equivalence for Completely Positive Linear Maps on $$C^*$$-Algebras

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 two classes of the completely positive linear maps are also strongly Morita equivalent.

Strong morita equivalence for conditional expectations

AbstractWe 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 the basic properties about the isometric isomorphism. And, using this isometric isomorphism, we define the Picard group for a bimodule linear map and discuss the Picard group for a bimodule linear map.

Strong Morita equivalence for inclusions of $C^*$-algebras induced by twisted actions of a countable discrete group

We consider two twisted actions of a countable discrete group on $\sigma$-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 of $C^*$-algebras is irreducible, that is, the relative commutant of one of the $\sigma$-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.

Coactions of a finite-dimensional $C^*$-Hopf algebra on unital $C^*$-algebras, unital inclusions of unital $C^*$-algebras and strong Morita equivalence

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 $(\rho, u)$ and $(\sigma, v)$ 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 $A\subset A\rtimes_{\rho, u}H$ and $B\subset B\rtimes_{\sigma, v}H$ are strongly Morita equivalent. If $A'\cap (A\rtimes_{\rho, u}H)=\BC1$, then there is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that the twisted coaction $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $((\id_B \otimes\lambda^0 )\circ\sigma \, , \, (\id_B \otimes\lambda^0 \otimes\lambda^0 )(v))$ induced by $(\sigma, v)$ and $\lambda^0$.

Morita equivalence of C*-correspondences passes to the related operator algebras

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 coincides with strong Δ-equivalence for tensor algebras of aperiodic C*-correspondences.

Characterizations of strong Morita equivalence for ordered semigroups with local units

Abstract We prove that partially ordered semigroups S and T with local units are strongly Morita equivalent if and only if there exists a surjective strict local isomorphism to T from a factorizable Rees matrix posemigroup over S. We also provide two similar descriptions which use Cauchy completions and Morita posemigroups instead.

Strong Morita equivalence for ordered semigroups with local units

We prove that partially ordered semigroups with local units are strongly Morita equivalent if and only if they have a joint enlargement, which in turn happens if and only if the Cauchy completions of the semigroups are equivalent.

Morita invariants for partially ordered semigroups with local units

We study Morita invariants for strongly Morita equivalent partially ordered semigroups with several types of local units. These include the greatest commutative images, satisfying a given inequality and the fact that strong Morita equivalence preserves various sublattices of the lattice of ideals.

Conditions for strong Morita equivalence of partially ordered semigroups

Abstract We investigate when a partially ordered semigroup (with various types of local units) is strongly Morita equivalent to a posemigroup from a given class of partially ordered semigroups. Necessary and sufficient conditions for such equivalence are obtained for a series of well-known classes of posemigroups. A number of sufficient conditions for several classes of naturally ordered posemigroups are also provided.

Strong Morita equivalence of semigroups with local units

In this paper we study Morita contexts for semigroups. We prove a Rees matrix cover connection between strongly Morita equivalent semigroups and investigate how the existence of a unitary Morita semigroup over a given semigroup is related to the existence of a ‘good’ Rees matrix cover of this semigroup.

Morita invariants for semigroups with local units

In this paper we study Morita invariants for strongly Morita equivalent semigroups with local units of various kinds. Among others we prove that, under a certain condition of this kind, congruence lattices are preserved by strong Morita equivalence.

Context equivalence of semigroups

Two semigroups are called strongly Morita equivalent if they are contained in a Morita context with unitary bi-acts and surjective mappings. We consider the notion of context equivalence which is obtained from the notion of strong Morita equivalence by dropping the requirement of unitariness. We show that context equivalence is an equivalence relation on the class of factorisable semigroups and describe factorisable semigroups that are context equivalent to monoids or groups, and semigroups with weak local units that are context equivalent to inverse semigroups, orthodox semigroups or semilattices.

Crossed Products of pro-C*-Algebras and Morita Equivalence

We introduce the notion of strong Morita equivalence for group actions on pro-C* -algebras and prove that the crossed products associated with two strongly Morita equivalent continuous inverse limit actions of a locally compact group G on the pro-C* -algebras A and B are strongly Morita equivalent. This generalizes a result of F. Combes [2] and R.E. Curto, P.S. Muhly, D.P. Williams [3].

On Morita's fundamental theorem for $C^*$-algebras

We give a solution, via operator spaces, of an old problem in the Morita equivalence of $C^*$-algebras. Namely, we show that $C^*$-algebras are strongly Morita equivalent in the sense of Rieffel if and only if their categories of left operator modules are isomorphic 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. An operator module over a $C^*$-algebra $\mathcal A$ is a closed subspace of some B(H) which is left invariant under multiplication by $\pi(\mathcal\ A)$, where $\pi$ is a*-representation of $\mathcal A$ on $H$. The category $_{\mathcal{AHMOD}}$ of *-representations of $\mathcal A$ on Hilbert space is a full subcategory of the category $_{\mathcal{AOMOD}}$ of operator modules. Our main result remains true with respect to subcategories of $OMOD$ which contain $HMOD$ and the $C^*$-algebra itself. It does not seem possible to remove the operator space framework; in the very simplest cases there may exist no bounded equivalence functors on categories with bounded module maps as morphisms (as opposed to completely bounded ones). Our proof involves operator space techniques, together with a $C^*$-algebra argument using compactness of the quasistate space of a $C^*$-algebra, and lowersemicontinuity in the enveloping von Neumann algebra.

A Morita theorem for algebras of operators on Hilbert space

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.

Morita Equivalence of Twisted Crossed Products

We introduce a natural notion of strong Morita equivalence of twisted actions of a locally compact group on ${C^ \ast }$-algebras, and then show that the corresponding twisted crossed products are strongly Morita equivalent. This result is a generalization of the result of Curto, Muhly and Williams concerning strong Morita equivalence of crossed products by actions.

Morita equivalence of twisted crossed products

We introduce a natural notion of strong Morita equivalence of twisted actions of a locally compact group on C ∗ {C^ \ast } -algebras, and then show that the corresponding twisted crossed products are strongly Morita equivalent. This result is a generalization of the result of Curto, Muhly and Williams concerning strong Morita equivalence of crossed products by actions.

Morita Equivalence of Twisted Crossed Products by Coactions

We introduce a natural notion of strong Morita equivalence of twisted coactions on C*-algebras and then show that the corresponding twisted crossed product C*-algebras are strongly Morita equivalent. This is an analogue of the notion of strong Morita equivalence of Green′s twisted actions. We also present criteria for strong Morita equivalence of crossed products by coactions and produce a family of twisted coactions on a groupoid C*-algebra.

Cross products of strongly Morita equivalent 𝐶*-algebras

Suppose that a locally compact group G G acts on strongly Morita equivalent C ∗ {C^ * } -algebras A A and B B and let A ⋊ G A \rtimes G and B ⋊ G B \rtimes G denote the corresponding crossed products. We present conditions which imply that A ⋊ G A \rtimes G and B ⋊ G B \rtimes G are also strongly Morita equivalent and we apply our result to improve upon known theorems concerning strong Morita equivalence between certain transformation group C ∗ {C^ * } -algebras.

Strong morita equivalence of certain transformation groupC*-algebras

In the course of extending certain results of Takesaki [11], Nielsen obtained [6] a generalization of a theorem of Mackey [5] and Blattner [1] concerning intertwining operators for induced representations. Nielsen's generalization concerns intertwining operators for the restrictions of induced representations to a subgroup. Now in [7] it was shown that the theorem of Mackey and Blattner is a large part of the statement that the inducing process establishes a Morita equivalence, that is, an equivalence between the representation theories [8], for certain C*-algebras related to induced representations. In fact, the equivalence is a strong Morita equivalence in the sense that it is implemented by what is called in [7] an "imprimitivity bimodule". All this suggests that perhaps Nielsen's theorem is a reflection of the fact that certain algebras associated with the situation he considers are strongly Morita equivalent. We show here that this is in fact the case. Specifically, let G be a locally compact group, and let H and K be closed subgroups of G. Let K act on the left on G/H and let C*(K, G/H) be the corresponding transformation group C*-algebra [4]. Similarly, let H act on the right on K\G and let C*(H,K\G) be the corresponding C*-algebra. We show that there is a natural imprimitivity bimodule establishing a strong Morita equivalence between these two C*-algebras. tn particular, it follows that these two algebras have equivalent representation theories. We then show that Nielsen's theorem follows from this. Our proofs are essentially algebraic -- requiring little more measure theory than those facts about Haar measure needed to define transformation group algebras and their representations. In particular, our results give a proof of Nielsen's theorem which avoids the measure-theoretic details involving liftings of measures in which his proof becomes embroiled. Some of the formulas developed here can be used to motivate a considerably more elementary proof of Takesaki's results [11] concerning generalized commutation relations (as extended by Nielsen). But the theorems obtained here are not needed for that purpose, and so this matter will be treated elsewhere [9].