Taking advantage of the quantale-theoretic description \'etale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms modules on inverse quantal frames. The Hilbert module quantale sheaves leads naturally to a formulation bimodules that resemble imprimitivity C*-algebras.

It is about a decade since Broué made his celebrated conjecture [2] on equivalences of derived categories in block theory: that the module categories of a block algebra A of a finite group algebra and its Brauer correspondent B should have equivalent derived categories if their defect group is abelian. Since then, character-theoretic evidence for the conjecture has accumulated rapidly, but unti...

In this work we study a new equivalence relation between w∗ closed algebras of operators on Hilbert spaces. The algebras A and B are called TRO equivalent if there exists a ternary ring of operatorsM (i.e. MM∗M ⊂ M) such that A = [M∗BM]−w ∗ and B = [MAM∗]−w ∗ . We prove that two reflexive algebras are TRO equivalent if and only if there exists a ∗ isomorphism between the commutants of their dia...

We classify derived-discrete algebras over the real numbers up to Morita equivalence, using classification of complex in D. Vossieck, (J. Algebra, 243, 168–176 2001). To this end, we investigate quiver presentation complexified algebra a given by modulated and an admissible ideal.

We prove that two dual operator algebras are weak Morita equivalent in the sense of [4] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weakcontinuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case we can characterize such functors as the module normal Haagerup tensor product ...

In this paper, we give the properties of tensor product and study the relationship between Hom functors and left (right) exact sequences in FS-Act. Also, we get some necessary conditions for equivalence of two fuzzy Sacts category. Moreover, we prove that two monoids S and T are Morita equivalent if and only if FS-Act and FT -Act are equivalent.

It is shown that a singular equivalence induced by tensoring with suitable complex of bimodules defines Morita type level, in the sense Wang. This result applied to homological ideals and idempotents produce new reduction techniques for testing properties syzygy-finite injectives generation finite dimensional algebras over field.