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We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of its support, and study these concepts from different sides : hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions as Regularity, $omega$-comparison, Corona Factorization Property, property R, etc.. are equivalent under mild assumptions.
We consider a Banach algebra $A$ with the property that, roughly speaking, sufficiently many irreducible representations of $A$ on nontrivial Banach spaces do not vanish on all square zero elements. The class of Banach algebras with this property tur
Let $A$ be a unital separable simple ${cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $uin U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $epsilon>0,$ there exists a self-adjoint ele
We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^*$-valued inner product struct
In this thesis, we investigate the proof of the Baum-Connes Conjecture with Coefficients for a-$T$-menable groups. We will mostly and essentially follow the argument employed by N. Higson and G. Kasparov in the paper [Nigel Higson and Gennadi Kasparo
Working within the framework of free actions of countable amenable groups on compact metrizable spaces, we show that the small boundary property is equivalent to a density version of almost finiteness, which we call almost finiteness in measure, and