$K\sb 1$-groups, quasidiagonality, and interpolation by multiplier projections

Abstract

We relate the following conditions on a a-unital C*-algebra A with the property: (a) K1 (A) = 0; (b) every projection in M(A)/A lifts; (c) the general Weyl-von Neumann theorem holds in M(A): Any selfadjoint element h in M(A) can be written as h = A? )LPi + a for some selfadjoint element a in A, some bounded real sequence {Ai}, and some mutually orthogonal projections {pi} in A with EZ1 pi = 1; (d) M(A) has FS; and (e) interpolation by multiplier projections holds: For any closed projections p and q in A** with pq = O, there is a projection r in M(A) such that p (b) 4 (a), and that (a) X* (b) if, in addition, A is stable. Combining the above results, we obtain counterexamples to the conjecture of G. K. Pedersen A has ?M(A) has FS (for example the stabilized BunceDeddens algebras). Hence the generalized Weyl-von Neumann theorem does not generally hold in L(HA) for a-unital C*-algebras with FS.