Isometries of quasitriangular operator algebras

Abstract:

Let $({P_n})$ be an increasing sequence of finite rank projections on a separable Hilbert space. Assume ${P_n}$ converges strongly to the identity operator I. The quasitriangular operator algebra determined by $({P_n})$ is defined to be the set of all bounded linear operators T for which \[ \lim \limits _{x \to \infty } \left \| {(I - {P_n})T{P_n}} \right \| = 0.\] 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.