Compact perturbations of certain von Neumann algebras

Plastiras, Joan K.

doi:10.1090/S0002-9947-1977-0458241-5

Compact perturbations of certain von Neumann algebras
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by Joan K. Plastiras PDF

Trans. Amer. Math. Soc. 234 (1977), 561-577 Request permission

Abstract:

Let $\mathcal {E}$ be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space $\mathcal {H}$. If we denote the commutant of $\mathcal {E}$ by $\mathcal {D}(\mathcal {E})$ and the ideal of compact operators on $\mathcal {H}$ by $\mathcal {C}(\mathcal {H})$, then it is easily verified that $\mathcal {D}(\mathcal {E}) + \mathcal {C}(\mathcal {H}) = \{ T + K:T \in \mathcal {D}(\mathcal {E}),K \in \mathcal {C}(\mathcal {H})\}$ is a ${C^\ast }$-algebra. In this paper we classify all such algebras up to $^\ast$-isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.

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