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Transactions of the American Mathematical Society

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

 

 

Some applications of direct integral decompositions of $ W\sp{\ast} $-algebras


Author: Edward Sarian
Journal: Trans. Amer. Math. Soc. 279 (1983), 677-689
MSC: Primary 46L45
DOI: https://doi.org/10.1090/S0002-9947-1983-0709576-2
MathSciNet review: 709576
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Abstract: Let $ \mathcal{A}$ be a $ {W^{\ast}}$-algebra and let $ A \in \mathcal{A}$. $ \mathcal{K}(\mathcal{A})$ and $ C(A)$ represent certain convex subsets of $ \mathcal{A}$. We prove the following via direct integral theory:

(1) If $ \mathcal{A}$ is of type $ {{\text{I}}_\infty }$, $ {\text{II}}_\infty $, or III, then $ C(A) = \{ 0\}$ iff $ {\text{A}} \in \mathcal{K}(\mathcal{A})$.

(2) If $ \mathcal{A}$ is of type I or II, then $ \mathcal{K}(\mathcal{A})$ is strongly dense in $ \mathcal{A}$.

(3) If $ \mathcal{A}$ is of type $ {{\text{I}}_\infty }$, $ {\text{II}}_\infty $, or III and $ \mathcal{B}$ is a $ {W^{\ast}}$-subalgebra of $ \mathcal{A}$, we give sufficient conditions for a Schwartz map $ P$ of $ \mathcal{A}$ into $ \mathcal{B}$ to annihilate $ \mathcal{K}(\mathcal{A})$.

Several preliminary lemmas that are useful for direct integral theory are also proved.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0709576-2
Keywords: $ {W^{\ast}}$-algebra, separable Hilbert space, direct integral decomposition, Schwartz map, trace-class, convex hull
Article copyright: © Copyright 1983 American Mathematical Society