Morita equivalence of quasi-primal algebras and sheaves

In this paper, we show that two quasi-primal algebras are Morita equivalent if and only if their inverse semigroups of inner automorphisms are isomorphic, and if they have the same one-element subalgebras. The proof of this statement uses the representation theory of algebras by sections in sheaves.Presented by H. P. Gumm.     

Construction of stable equivalences of Morita type for finite-dimensional algebras I

In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a non-trivial example is not known. In this paper, we provide two methods to produce new stable equivalences of Morita type from given ones. The main results are Corollary 1.2 and Theorem 1.3. Here the algebras considered are not necessarily self-injective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension , such that one of them is quasi-hereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasi-heredity of algebras. As another by-product, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras and . This leads not only to a general formulation of a result by Linckelmann (1996), but also to a nice correspondence of some torsion pairs in -mod with those in -mod if both and are symmetric algebras. Moreover, under the assumption of symmetric algebras we can get a new stable equivalence of Morita type. Finally, we point out that stable equivalences of Morita type are preserved under separable extensions of ground fields.

     

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.     

Enveloping actions and Takai duality for partial actions

We show that any partial action on a topological space X is the restriction of a suitable global action, called enveloping action, that is essentially unique. In the case of C∗-algebras, we prove that any partial action has a unique enveloping action up to Morita equivalence, and that the corresponding reduced crossed products are Morita equivalent. The study of the enveloping action up to Morita equivalence reveals the form that Takai duality takes for partial actions. By applying our constructions, we prove that the reduced crossed product of the reduced cross-sectional algebra of a Fell bundle by the dual coaction is liminal, postliminal, or nuclear, if and only if so is the unit fiber of the bundle. We also give a non-commutative generalization of the well-known fact that the integral curves of a vector field on a compact manifold are defined on all of .     

Stable isomorphism of dual operator spaces

We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations ? and ψ of X and Y, respectively, and ternary rings of operators M₁, M₂ such that and . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras.     

Contractible subgraphs and Morita equivalence of graph -algebras

In this paper we describe an operation on directed graphs which produces a graph with fewer vertices, such that the -algebra of the new graph is Morita equivalent to that of the original graph. We unify and generalize several related constructions, notably delays and desingularizations of directed graphs.

     

A note on stable equivalences of Morita type

We investigate when an exact functor --Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corollary, we see that the two functors given by tensoring with the bimodules in a stable equivalence of Morita type are right and left adjoints of one another, provided that these bimodules are indecomposable. This fact has many interesting consequences for stable equivalences of Morita type. In particular, we show that a stable equivalence of Morita type induces another stable equivalence of Morita type between certain self-injective algebras associated to the original algebras. We further show that when there exists a stable equivalence of Morita type between Λ and Γ, it is possible to replace Λ by a Morita equivalent k-algebra Δ such that Γ is a subring of Δ and the induction and restriction functors induce inverse stable equivalences.     

On the torsion theories of Morita equivalent rings

We generalize the well-known fact that for a pair of Morita equivalent ringsR andS their maximal rings of quotients are again Morita equivalent: If _n (M) denotes the torsion theory cogenerated by the direct sum of the firstn+1 injective modules forming part of the minimal injective resolution ofM then _n (R)= _n (S) where is the category equivalenceR-ModS-Mod. Consequently the localized ringsR _{n (R)} andS n _(S) are Morita equivalent.     

Mauldin-Williams graphs, Morita equivalence and isomorphisms

We describe a method for associating some non-self-adjoint algebras to Mauldin-Williams graphs and we study the Morita equivalence and isomorphism of these algebras.

We also investigate the relationship between the Morita equivalence and isomorphism class of the -correspondences associated with Mauldin-Williams graphs and the dynamical properties of the Mauldin-Williams graphs.

     

Morita Equivalence of Operator Algebras and Their C*-Envelopes

If two operator algebras A and B are strongly Morita equivalent(in the sense of [5]), then their C*-envelopes C*(A) and C*(B)are strongly Morita equivalent (in the usual C*-algebraic sensedue to Rieffel). Moreover, if Y is an equivalence bimodule fora (strong) Morita equivalence of A and B, then the operation,Y_hA–, of tensoring with Y, gives a bijection between theboundary representations of C*(A) for A and the boundary representationsof C*(B) for B. Thus the ‘noncommutative Choquet boundaries’of Morita equivalent A and B are the same. Other important objectsassociated with an operator algebra are also shown to be preservedby Morita equivalence, such as boundary ideals, the Shilov boundaryideal, Arveson's property of admissability, and the latticeof C*-algebras generated by an operator algebra. 1991 MathematicsSubject Classification 47D25, 46L05, 46M99, 16D90.     

Properties preserved under Morita equivalence of -algebras

We show that important structural properties of -algebras and the multiplicity numbers of representations are preserved under Morita equivalence.

     

On complex and noncommutative tori

The ``noncommutative geometry' of complex algebraic curves is studied. As a first step, we clarify a morphism between elliptic curves, or complex tori, and -algebras , or noncommutative tori. The main result says that under the morphism, isomorphic elliptic curves map to the Morita equivalent noncommutative tori. Our approach is based on the rigidity of the length spectra of Riemann surfaces.

     

An infinite family of non-isomorphic C-algebras with identical -theory

We exhibit a countably infinite family of simple, separable, nuclear, and mutually non-isomorphic C-algebras which agree on -theory and traces. The algebras do not absorb the Jiang-Su algebra tensorially, answering a question of N. C. Phillips. They are also pairwise shape and Morita equivalent, confirming a conjecture from our earlier work. The distinguishing invariant is the radius of comparison, a non-stable invariant of the Cuntz semigroup.

     

Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

It is proved that the filter dimension is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand–Kirillov dimension for all nonzero finitely generated A-modules M. In fact, a stronger result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. A direct consequence of this fact is that an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H _c for integral c: for all nonzero finitely generated H _c -modules M. For these class of algebras, it gives an affirmative answer to a question of Ken Brown about symplectic reflection algebras. Presented by Alain Verschoren.     

Morita equivalences between some blocks for -solvable groups

We prove that any Morita equivalence between some blocks with Abelian defect groups and cyclic inertia quotients for -solvable groups is basic.

     

Analytic norms in Orlicz spaces

It is shown that an Orlicz sequence space admits an equivalent analytic renorming if and only if it is either isomorphic to or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent -Fréchet norm, but no equivalent analytic norm.

     

Sesquilinear Morita Equivalence and Orthogonal Sum of Algebras with Antiautomorphism

We define a notion of Morita equivalence between algebras with antiautomorphisms such that two equivalent algebras have the same category of sesquilinear forms. This generalizes the Morita equivalence of algebras with involutions defined by Fröhlich and Mc Evett [5 Fröhlich , A. , McEvett , A. M. ( 1969 ). Forms over rings with involution . J. Algebra 12 : 79 – 104 .[Crossref], [Web of Science ®] , [Google Scholar]], and their categories of ?-hermitian forms.

For two Morita equivalent algebras with involution, with an additional technical property (which is true for central simple algebras), we define a new algebra with antiautomorphism, called the orthogonal sum, which generalizes the usual notion of orthogonal sum of forms. We explore the invariants of this sum.     

Gruenhage compacta and strictly convex dual norms

We prove that if K is a Gruenhage compact space then admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and , where K is a Gruenhage compact in the w^*-topology and |||||| is equivalent to a coarser, w^*-lower semicontinuous norm on X^*, then X^* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if is a tree, then admits an equivalent, strictly convex dual norm if and only if is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.     

Isomorphisms of row and column finite matrix rings

This paper investigates the ring-theoretic similarities and the categorical dissimilarities between the ring of row finite matrices and the ring of row and column finite matrices. For example, we prove that two rings and are Morita equivalent if and only if the rings and are isomorphic. This resembles the result of V. P. Camillo (1984) for . We also show that the Picard groups of and are isomorphic, even though the rings and are never Morita equivalent.

     

A new invariant of stable equivalences of Morita type

It was proved in an earlier paper by the author that the Hochschild cohomology algebras of self-injective algebras are invariant under stable equivalences of Morita type. In this note we show that the orbit algebra of a self-injective algebra (considered as an --bimodule) is also invariant under stable equivalences of Morita type, where the orbit algebra is the algebra of all stable --bimodule morphisms from the non-negative Auslander-Reiten translations of to .