On the structure of multiplier algebras

Shuang Zhang, Purdue University


I. If A is a $\sigma$-unital C*-algebra with FS and M(A) is the multiplier algebra of A, we relate the following conditions by giving various characterizations: (a) $K\sb1(A) = 0$. (b) Every projection in M(A)/A lifts. (c) The 'general Weyl-von Neumann Theorem' holds in M(A): If Q is a self-adjoint element of M(A), then there are mutually orthogonal projections $\{ p\sb{i}\}$ in A, a self-adjoint element a in A and a real bounded sequence $\{\lambda\sb{i}\}$ such that $Q = \sum\sbsp{i = 1}{\infty}\lambda\sb{i}p\sb{i} + a$ and $\sum\sbsp{i = 1}{\infty}p\sb{i} = 1$. (d) M(A) has FS. (e) For any closed projections p and q in A** with pq = 0, there is a projection R in M(A) such that $p\leq R\leq 1-q$. Actually, e $\Longleftrightarrow$ d $\Longleftrightarrow$ c $\Longrightarrow$ b $\Longleftarrow$ a for any $\sigma$-unital C*-algebra A with FS; and a $\Longleftrightarrow$ b if, in addition, A is stable. II. If A has FS, we prove a Riesz decomposition property for D(A),D(M(A)) and sometimes for D(M(A)/A), where D(.) is the local semigroup consisting of equivalence classes of projections in (.). As consequences we describe equivalence classes of projections in M(A) and the ideal structure of M(A). III. We study the ideal structure of L($H\sb{A}$)$\cong M(A\otimes K)$ and prove that $L(H\sb{A})$ is usually not simple, sometimes $L(H\sb{A})$ has a largest nontrivial closed ideal, and sometimes $L(H\sb{A})/K(H\sb{A})$ has a smallest nonzero closed ideal. IV. First, we prove that if A is a $\sigma$-unital C*-algebra with FS,then every hereditary C*-subalgebra of M(A) is the closed linear span of its projections. Second, we prove that if A is a $\sigma$-unital simple C*-algebra with FS, then every nonzero projection in M(A)/A is infinite. Consequently $K\sb0(M/(A)/A)$ and $K\sb1(M(A)/A)$ are described for some $\sigma$-unital C*-algebras with FS. V. We relate the (closed) ideal structure of A to the ideal structure of Corona algebra M(A)/A via a lifting of ideals: $I\mapsto A + M(A,I)\mapsto M(A,I)/I$. We give necessary and sufficient conditions from various perspectives for the lifting of a nontrivial ideal of A to be a nontrivial ideal of M(A)/A. VI. We relate the stable isomorphism of two hereditary C*-subalgebras to the stable equivalence of the corresponding open projections. We prove that if A is completely $\sigma$-unital, then her(p) and her(q) generate the same closed ideal of A iff $p\otimes 1\sim q\otimes 1$ in($A\otimes K$)** iff the central supports of p and q in AA** are the same. If, in addition, $p\perp q$, then the above three equivalent conditions are equivalent to the condition: $p\otimes 1$ and $q\otimes 1$ are in the same path component of open projections in ($A\otimes K$)**.




Brown, Purdue University.

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