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

Abstract

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.

HereditaryC⁎-subalgebra lattices

“…Somewhat analogous type classifications/decompositions have also been obtained for more general C*algebras, for example in [Cun77] and [CP79]. However, it is only recently that completely consistent extensions of the original von Neumann algebra type decomposition have been obtained by utilizing *-annihilators, either explicitly, as in [Bic14a], or implicitly, as in [NW13]. 11 In this section, we outline how to obtain order theoretic type decompositions of A and what algebraic characterizations these types have.…”