Let be a maximal subdiagonal algebra of a von Neumann algebra M with respect to a faithful normal expectation ф. Then we show that if is a faithful normal state of M such thatф, then Aist-invariant, where {t}t2R is the modular automorphism group associated with. As an application, we prove that every -weakly closed subdiagonalalgebra of B(H) is a nest algebra with an atomic nest.

Let M be a-finite von Neumann algebra and let A be a maximal subdiagonal algebra of M with respect to a faithful normal conditional expectation 8. We show that if S is an invertible operator in M, then there exists an isometry W in M such that both S & W and W * S belong to A. We also give several characterizations of a maximal subdiagonal algebra A such that every invertible operator S in M can be factored as UA, where U is a unitary operator in M and both A and A&1 are in A.

Let A be a maximal subdiagonal algebra of a von Neumann algebra M with respect to a faithful normal expectation. Then we show that if is a faithful normal state of M such that=then A is -invariant, where2R is the modular automorphism group associated with. As an application, we prove that every -weakly closed subdiagonal algebra of B(H) is a nest algebra with an atomic nest.

We determine kernels of similarity-preserving bounded linear maps on B(H) and give characterizations for elementary operators of length 1 to be similarity-preserving.

It is proved that a bounded linear surjective map on B(H) which is asymptotic similaritypreserving is nonzero scalar multiple of either an automorphism or an anti-automorphism.