Roughly speaking, two dynamical systems are transversally equivalent if they have the same space of orbits; a property is transverse if it is preserved under transverse equivalence. Various notions of transverse equivalence have been defined: among them, similarity of measured groupoids, Morita equivalence of locally compact groupoids, stable orbit equivalence of measure equivalence relations. ...

In this article we review recent developments on Morita equivalence of star products and their Picard groups. We point out the relations between noncommutative field theories and deformed vector bundles which give the Morita equivalence bimodules.

In a recent paper [5], we introduced a classes of derived equivalences called almost ν-stable derived equivalences. The most important property is that an almost ν-stable derived equivalence always induces a stable equivalence of Morita type, which generalizes a well-known result of Rickard: derived-equivalent self-injective algebras are stably equivalent of Morita type. In this paper, we shall...

We develop a general framework for the study of strong Morita equivalence in which C∗algebras and hermitian star products on Poisson manifolds are treated in equal footing. We compare strong and ring-theoretic Morita equivalences in terms of their Picard groupoids for a certain class of unital ∗-algebras encompassing both examples. Within this class, we show that both notions of Morita equivale...

This paper is a survey on the problem of classifying non-commutative tori up to Morita equivalence and will review the necessary background and discuss some results concerning this question (see [28],[29] and [22]). The concept of Morita equivalence was first introduced in operator algebras by M.Rieffel in the 1970’s, in connection with the problem of characterizing representations of locally c...

We study the Morita equivalence for fermion theories on noncommutative two-tori. For rational values of the θ parameter (in appropriate units) we show the equivalence between an abelian noncommutative fermion theory and a nonabelian theory of twisted fermions on ordinary space. We study the chiral anomaly and compute the determinant of the Dirac operator in the dual theories showing that the Mo...

We study the Morita equivalence for fermion theories on noncom-mutative two-tori. For rational values of the θ parameter (in appropriate units) we show the equivalence between an abelian noncommu-tative fermion theory and a nonabelian theory of twisted fermions on ordinary space. We study the chiral anomaly and compute the determinant of the Dirac operator in the dual theories showing that the ...