Central sequences in C*-algebras and strongly purely infinite algebras

It is shown that F(A) := (A' boolean AND A(w))/Ann(A, A(w)) is a unital C*-algebra and that A -> F(A) is a stable invariant of separable C*-algebras A with certain local continuity and permanence properties. Here A, means the ultrapower of A. If A is separable, then F(A) is simple, if and only if, either A circle times K congruent to K or A is a simple purely infinite nuclear C*-algebra. In the first case F(A) congruent to C, and in the second case F(A) is purely infinite and A absorbs O-infinity tensorially, i.e. A congruent to A circle times O-infinity. We show that F(Q) = C (.) 1 for the Calkin algebra Q := L/K, in contrast to the separable case. We introduce a "locally semi-projective" invariant cov(B) is an element of N boolean OR {infinity} of unital C*-algebras B with cov(B) <= cov(C) if there is a unital *-homomorphism from C into B. If B is nuclear and has no finite-dimensional quotient then cov (B) <= dr (B) + 1 for the decomposition rank dr(B) of B. (Thus, cov(Z) = 2 for the Jian-Su algebra Z.) Separable (not necessarily simple) C*-algebras A are strongly purely infinite in the sense of [25] if A does not admit a non-trivial lower semi-continuous 2-quasitrace and F(A) contains a simple C*-subalgebra B with cov(B) < infinity and 1 is an element of B. In particular, A circle times Z is strongly purely infinite if A(+) admits no non-trivial lower semi-continuous 2-quasi-trace. Properties of F(A) will be used to show that A is tensorially D-absorbing, (i.e. that A circle times D congruent to A by an isomorphism that is approximately unitarily equivalent to a -> a circle times 1), if A is stable and separable, D is a unital tensorially self-absorbing algebra, and D is unitally contained in F(A). It follows that the class of tensorially D-absorbing separable stable C *-algebras A, is closed under inductive limits and passage to ideals and quotients. The local permanence properties of the functor A -> F(A) imply that this class is also closed under extensions, if and only if, every commutator uvu*v* of unitaries u,v is an element of U(D) is contained in the connected component U-0(D) of 1 in U(D). If this is the case, then the class of (not necessarily stable) D-absorbing separable C *-algebras is also closed under passage to hereditary C *-algebras.