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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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$K_ 1$-groups, quasidiagonality, and interpolation by multiplier projections
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by Shuang Zhang PDF
Trans. Amer. Math. Soc. 325 (1991), 793-818 Request permission

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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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 325 (1991), 793-818
  • MSC: Primary 46L05; Secondary 46L80
  • DOI: https://doi.org/10.1090/S0002-9947-1991-0998130-8
  • MathSciNet review: 998130