Isometries of quasitriangular operator algebras

Abstract

Let ( P n ) ({P_n}) be an increasing sequence of finite rank projections on a separable Hilbert space. Assume P n {P_n} converges strongly to the identity operator I. The quasitriangular operator algebra determined by ( P n ) ({P_n}) is defined to be the set of all bounded linear operators T for which \[ lim x → ∞ ‖ ( I − P n ) T P n ‖ = 0. \lim \limits _{x \to \infty } \left \| {(I - {P_n})T{P_n}} \right \| = 0. \] In this note we prove that two quasitriangular algebras are unitarily equivalent if, and only if, there exists a unital linear isometry mapping one algebra onto the other.