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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 A d,1, the largest discrete essentially infinite ideal A d,∞, the largest type II finite ideal A II,1, the largest type II essentially infinite ideal A II,∞, and the largest type III ideal A III of any C *-algebra A such that A d,1 + A d,∞ + A II,1 + A II,∞ + A III 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. [ABSTRACT FROM AUTHOR]

In this paper, we extend the definition of property T and strong property T to general C*-algebras (not necessarily unital). We show that if an inclusion pair of locally compact groups (G,H) has property T, then (C*(G), C*(H)) has property T. As a partial converse, if T is abelian and C*(G) has property T, then T is compact. We also show that if Ω is a first countable locally compact Hausdorff space, then C0(Ω) has (strong) property T if and only if Ω is discrete. Furthermore, the non-unital C*-algebra $c_0(\mathbb{Z}^n)\rtimes SL_n(\mathbb{Z})$ has strong property T when n ≥ 3. We also give some equivalent forms of strong property T, which are new even in the unital case. [ABSTRACT FROM PUBLISHER]