Research of a lifting theorem of uniform algebras and analytic operator algebras

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Let K be a Hilbert space, B a von Neumann algebra on K and A a weakly closed subalgebra of B. T = (T_<ij>) denotes 2 x 2 operator matrices on K <symmetry> K where T_<ij> epsilon B (i, j = 1, 2), T_<11> <greater than or equal> 0, T_<22> <greater than or equal> 0 and T^*_ = T_<12>. [B] denotes the set of such operator matrices T and [A]_0 denotes the subset of [B] such that T_<12> epsilon A and T_<11> = T_<22> = 0. Let F be the subset of lat A, the lattice of all A-invariant projections. We studied whether if T epsilon [B] is positive on PK <symmetry> (1 - P)K for every P epsilon F then there exists T^^- in T + [A]_0 which is positive on K <symmetry> K. We say that (B, A, F). has a lifting property when there exists such a T^^-.When B is a commutative von Neumann algebra L^", if A is an abstract Hardy space H^", we could show that {B = L^", A = H^", F} has a lifting property even if F is very small. Using this result we could generalize a theorem of norms of Hankel operators and two theorems of weighted norm inequalities in the disc algebra to those in a general uniform algebra. When B is non-commutative, if A and F satisfy two conditions which are satisfied in many important examples, we could that {B, A, F} has a lifting property. Using the distanck formula of Arveson or the theorem of Neharils type. We could generalize two theorems of weighted norm inequalities in the vector-valued Hardy space to those in nest algebras and non-commutative Hardy spaces.