Antisymmetry and the direct integral decomposition of unstarred operator algebras

Szymański, Wacław

doi:10.1090/S0002-9939-1986-0822448-6

Antisymmetry and the direct integral decomposition of unstarred operator algebras
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Proc. Amer. Math. Soc. 96 (1986), 497-501 Request permission

Abstract:

It is shown that the direct integral decomposition of a non-self-adjoint operator algebra ${\mathcal A}$ has the diagonal ${\mathcal A} \cap {{\mathcal A}^ * }$ of this algebra as the algebra of diagonalizable operators if and only if almost all direct integrands of ${\mathcal A}$ are antisymmetric algebras. By using the antisymmetric decomposition a direct integral model of a commutative, reflexive algebra is obtained.

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