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.

THE STRONG MORITA EQUIVALENCE FOR INCLUSIONS OF -ALGEBRAS AND CONDITIONAL EXPECTATIONS FOR EQUIVALENCE BIMODULES

We shall introduce the notions of strong Morita equivalence for unital inclusions of unital $C^{\ast }$-algebras and conditional expectations from an equivalence bimodule onto its closed subspace with respect to conditional expectations from unital $C^{\ast }$-algebras onto their unital $C^{\ast }$-subalgebras. Also, we shall study their basic properties.

On the decomposition into Discrete, Type II and Type III C*-algebras

We obtained a “decomposition scheme” of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsidó), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to “classify” C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking “essential extension” and “normal quotient”. Furthermore, there exist the largest discrete finite ideal Ad,1, the largest discrete essentially infinite ideal Ad,∞, the largest type II finite ideal AII,1, the largest type II essentially infinite ideal AII,∞, and the largest type III ideal AIII of any C*-algebra A such that Ad,1 + Ad,∞ + AII,1 + AII,∞ + AIII is an essential ideal of A. This “decomposition” extends the corresponding one for W*-algebras.We also give a closer look at C*-algebras with Hausdorff primitive ideal spaces, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above.

Characterizations of strong Morita equivalence for ordered semigroups with local units

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.