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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.


References [Enhancements On Off] (What's this?)

  • [1] J. B. Conway, The numerical range and a certain convex set in an infinite factor, J. Funct. Anal. 5 (1970), 428-435. MR 0262839 (41:7444)
  • [2] A. DeKorvin, Stable maps and Schwartz maps, Trans. Amer. Math. Soc. 148 (1970), 283-291. MR 0264412 (41:9007)
  • [3] N. Dunford and J. T. Schwartz, Linear operators, Vol. 1, Interscience, New York, 1958; Vol. 2, 1963. MR 0188745 (32:6181)
  • [4] J. T. Schwartz, $ {W^{\ast}}$-algebras, Gordon & Breach, New York, 1967. MR 0232221 (38:547)
  • [5] J. von Neumann. On rings of operators, Reduction theory, Collected Works, Vol. 3, Pergamon Press, New York, 1961, pp. 400-484.
  • [6] P. Willig, Trace norms, global properties and direct integral decompositions of $ {W^{\ast}}$-algebras, Comm. Pure Appl. Math. 22 (1969), 839-862. MR 0270170 (42:5063)
  • [7] -, Property $ P$ and direct integral decompositions of $ {W^{\ast}}$-algebras, Proc. Amer. Math. Soc. 29 (1971), 494-498. MR 0279600 (43:5321)

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Additional Information

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

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