Fix a sequence 𝒫 = {Pn}n=1∞
of finite dimensional projections increasing to the identity on a separable Hilbert
space ℋ and let ℒ(ℋ) denote the algebra of all bounded operators on ℋ. The
quasitriangular algebra associated with 𝒫 and denoted as 𝒬𝒯 (𝒫) is defined to be the
set of those operators T in ℒ(ℋ) for which ∥Pn⊥TPn∥→ 0.
In this paper we will examine the structure of the 𝒬𝒯 (𝒫) algebras. Specifically, if
ℛ = {Rn}n=1∞ is another sequence of finite dimensional projections increasing to
the identity on the same Hilbert space, when is 𝒬𝒯 (ℛ) 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?
