Antisymmetry and the direct integral decomposition of unstarred operator algebras

Abstract

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