Vol. 64, No. 2, 1976

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Quasitriangular operator algebras

Joan Kathryn Plastiras

Vol. 64 (1976), No. 2, 543–549

Fix a sequence 𝒫 = {Pn}n=1 of finite dimensional projections increasing to the identity on a separable Hilbert space and let () denote the algebra of all bounded operators on . The quasitriangular algebra associated with 𝒫 and denoted as 𝒬𝒯 (𝒫) is defined to be the set of those operators T in () for which PnTPn∥→ 0.

In this paper we will examine the structure of the 𝒬𝒯 (𝒫) algebras. Specifically, if = {Rn}n=1 is another sequence of finite dimensional projections increasing to the identity on the same Hilbert space, when is 𝒬𝒯 () equal to 𝒬𝒯 (𝒫)? By an algebraic isomorphism between two algebras we shall mean a bijection which preserves algebraic structure: that is to say — addition, scalar multiplication, multiplication, but we do not impose any topological condition. When are two quasitriangular algebras isomorphic?

Mathematical Subject Classification 2000
Primary: 46L15, 46L15
Secondary: 47A65
Received: 23 February 1976
Published: 1 June 1976
Joan Kathryn Plastiras