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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]

In this paper, we will define the reduced cross-sectional C*-algebras of C*-algebraic bundles over locally compact groups and show that if a C*-algebraic bundle has the approximation property (defined similarly as in the discrete case), then the full cross-sectional C*-algebra and the reduced one coincide. Moreover, if a semi-direct product bundle has the approximation property and the underlying C*-algebra is nuclear, then the cross-sectional C*-algebra is also nuclear. We will also compare the approximation property with the amenability of Anantharaman-Delaroche in the case of discrete groups. [ABSTRACT FROM PUBLISHER]

In this paper, we will prove that if A is a C*-algebra with an effective coaction by a compact quantum group, then the fixed point algebra and the reduced crossed product are Morita equivalent. As an application, we prove an imprimitivity type theorem for crossed products of coactions by discrete Kac C*-algebras. [ABSTRACT FROM AUTHOR]