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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Author: Shuang Zhang
Journal: Trans. Amer. Math. Soc. 325 (1991), 793-818
MSC: Primary 46L05; Secondary 46L80
MathSciNet review: 998130
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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}}$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-0998130-8
Keywords: $ K$-theory of $ {C^\ast}$-algebras, multiplier algebras, quasidiagonality, projections
Article copyright: © Copyright 1991 American Mathematical Society