TOEPLITZ AND HANKEL OPERATORS ASSOCIATED WITH SUBDIAGONAL ALGEBRAS

Let M be a sigma-finite von Neumann algebra and let A subset of M be a maximal subdiagonal algebra with respect to some faithful normal expectation epsilon on M. Let phi be a normal faithful epsilon-invariant state on M, let L-2(M, phi) be the non-commutative Lebesgue space in the sense of U. Haagerup, and consider the Hardy space H-2(A, phi) subset of L-2(M, phi) associated with the pair (A, phi). For each x is an element of M, the Toeplitz operator T-x is an element of B(H-2(A, phi)) and the Hankel operator H-x is an element of B(H-2(A, phi), H-2(A, phi)(perpendicular to)) are defined as in the classical case of the unit circle. We show that the mapping x -> T-x is completely isometric on M and therefore sigma(x) subset of sigma(T-x) for all x is an element of M. We also show that parallel to H-x parallel to = dist(x, A) for every x is an element of M.