$K_ 1$-groups, quasidiagonality, and interpolation by multiplier projections

Abstract:

We relate the following conditions on a $\sigma$-unital ${C^\ast }$-algebra $A$ with the "${\text {FS}}$ property": (a) ${K_1}(A) = 0$; (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 $h = \sum \nolimits _{i = 1}^\infty {{\lambda _i}{p_i} + a}$ for some selfadjoint element $a$ in $A$, some bounded real sequence $\{ {\lambda _i}\}$, and some mutually orthogonal projections $\{ {p_i}\}$ in $A$ with $\sum \nolimits _{i = 1}^\infty {{p_i} = 1}$; (d) $M(A)$ has ${\text {FS}}$; and (e) interpolation by multiplier projections holds: For any closed projections $p$ and $q$ in ${A^{\ast \ast }}$ with $pq = 0$, there is a projection $r$ in $M(A)$ such that $p \leq r \leq 1 - q$. We prove various equivalent versions of (a)-(e), and show that (e) $\Leftrightarrow$ (d) $\Leftrightarrow$ (c) $\Rightarrow$ (b) $\Leftarrow$ (a), and that (a) $\Leftrightarrow$ (b) if, in addition, $A$ is stable. Combining the above results, we obtain counterexamples to the conjecture of G. K. Pedersen "$A$ has $FS \Rightarrow M(A)$ has ${\text {FS}}$" (for example the stabilized Bunce-Deddens algebras). Hence the generalized Weyl-von Neumann theorem does not generally hold in $L({H_A})$ for $\sigma$-unital ${C^\ast }$-algebras with ${\text {FS}}$.