Morita Invariance and Maximal Left Quotient Rings

Abstract

We prove that under conditions of regularity the maximal left quotient ring of a corner of a ring is the corner of the maximal left quotient ring. We show that if R and S are two non-unital Morita equivalent rings then their maximal left quotient rings are not necessarily Morita equivalent. This situation contrasts with the unital case. However we prove that the ideals generated by two Morita equivalent idempotent rings inside their own maximal left quotient rings are Morita equivalent.     

Derived equivalences between associative deformations

We prove that if two associative deformations (parameterized by the same complete local ring) are derived Morita equivalent, then they are Morita equivalent (in the classical sense).     

Morita theory in deformation quantization

Various aspects of Morita theory of deformed algebras and in particular of star product algebras on general Poisson manifolds are discussed. We relate the three flavours ring-theoretic Morita equivalence, *-Morita equivalence, and strong Morita equivalence and exemplify their properties for star product algebras. The complete classification of Morita equivalent star products on general Poisson manifolds is discussed as well as the complete classification of covariantly Morita equivalent star products on a symplectic manifold with respect to some Lie algebra action preserving a connection.     

Morita duality for monoids

In this paper Morita duality for monoids is introduced. Necessary and sufficient conditions for two monoids S and T to be Morita dual are given. Moreover, it is shown that if S and T are Morita dual monoids, then S and U are Morita dual if and only if T and U are Morita equivalent. In addition, every finite monoid having Morita duality is selfdual and even reflexive.     

INVARIANTS UNDER STABLE EQUIVALENCES OF MORITA TYPE

The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type.First of all,we show that,if two finite-dimensional selfinjective k-algebras are sta...     

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.     

可分解半群的Morita等价

设S,R是可分解半群.记US-FAct={sM∈S-Act|SM=M且SHoms(S,M)≌M],给出了范畴US-FAct与UR-FAct等价的刻划;S分别强Morita等价于一个夹层半群、局部单位半群、幺半群和群的条件;S是完全单半群当且仅当S强Morita等价于一个群且对任何指标集I,S SHoms(S,i∈I S)→i∈I S,s t·f→(st)f,是同构.     

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.     

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse 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.     

辛群胚的弱Morita等价与orbifold基本群

本文研究辛orbifold群胚的弱Morita等价,证明了两个辛orbifold群胚弱Morita等价当且仅当其orbifold基本群同构.     

Kac-系统与Morita等价

本文讨论与Hilbert C~*-模相关的Kac-系统在C~*-代数上的作用的性质,给出了两个Morita等价的C~*-代数(在Kac-系统的作用下得到的C~*-代数的余交叉积是Morita等价的)。从而,Baaj和Skandalis等人的结论是本文定理的特殊情况。     

Monoidal Morita invariants for finite group algebras

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer n, the number of elements of order n is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.     

On the Morita Equivalence of Tensor Algebras

We develop a notion of Morita equivalence for general C^*-correspondencesover C^*-algebras. We show that if two correspondences are Moritaequivalent, then the tensor algebras built from them are stronglyMorita equivalent in the sense developed by Blecher, Muhly andPaulsen. Also, the Toeplitz algebras are strongly Morita equivalentin the sense of Rieffel, as are the Cuntz–Pimsner algebras.Conversely, if the tensor algebras are strongly Morita equivalent,and if the correspondences are aperiodic in a fashion that generalizesthe notion of aperiodicity for automorphisms of C^*-algebras,then the correspondences are Morita equivalent. This generalizesa venerated theorem of Arveson on algebraic conjugacy invariantsfor ergodic, measure-preserving transformations. The notionof aperiodicity, which also generalizes the concept of fullConnes spectrum for automorphisms, is explored; its role inthe ideal theory of tensor algebras and in the theory of theirautomorphisms is investigated. 1991 Mathematics Subject Classification:46H10, 46H20, 46H99, 46M99, 47D15, 47D25.     

Functoriality of the Bimodule Associated to a Hilsum–Skandalis Map

We extend the functoriality of the Connes convolution algebra to the category of Hilsum–Skandalis maps between separated smooth étale groupoids. Thereby we establish that Morita equivalent separated smooth étale groupoids have Morita equivalent convolution algebras, analogously to the results of Muhly, Renault and Williams, and others, on the C^*algebras of Morita equivalent groupoids.     

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]. This research was supported by CEEX grant-code PR-D11-PT00-48/2005 from The Romanian Ministry of Education and Research.     

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 equivalence and Pedersen ideals

We show that two -algebras are strongly Morita equivalent if and only if their Pedersen ideals are Morita equivalent as rings with involution.

     

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.     

Morita equivalence for infinite matrix rings

In this paper, weak distinguished subcategory and distinguished subcategory of modules are introduced. Left(right) local unital rings are particularly considered. Also, representable equivalent functors between categories. By using the replacement techniques of modules, a general theory of Morita equivalence for infinite matrix rings is established. This theory not only extends the classical Morita theory of equivalence from finite matrix rings to infinite matrix rings and also contains some new results which are useful in studying the algebraic structures for infinite matrix rings. Some results of classical Morita theory are included as its special cases.