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

设K是Hilbert空间，B是K上的von Neumann代数，A是B的弱闭子代数。T=(T_&lt；ij&gt；)表示K&lt；对称&gt；K上的2×2算子矩阵，其中T_&lt；ij&gt；epsilon B(i，j=1，2)，T_&lt；11&lt；大于等于&gt；0，T_&lt；22&gt；&lt；大于或等于&gt；0且T*=T_&lt；12&gt；[B]表示这样的算子矩阵T的集合，[A]_0表示[B]的子集，使得T_t；12^t^A和T^t11^t>=T^t22^t=0。设F是后A的子集，是所有A-不变投影格。我们研究了如果对每个P epsilon F，T epsilon[B]对Pk&lt；对称性&Gt；(1-P)K是正的，那么T+[A]_0中是否存在T^-，它对K&lt；对称性&Gt；K是正的，我们称之为(B，A，F)。当B是可换的von Neumann代数L^“时，如果A是抽象的Hardy空间H^”，则证明了当F很小时，{B=L^“，A=H^”，F}具有提升性质.利用这一结果，我们可以把圆盘代数中的一个Hankel算子范数定理和两个加权范数不等定理推广到一般的一致代数中。当B是非对易的时，如果A和F满足在许多重要例子中满足的两个条件，我们可以认为{B，A，F}具有提升性质。利用Arveson的距离公式或Neharils型定理。我们可以将向量值Hardy空间中的加权范数不等式的两个定理推广到套代数和非交换Hardy空间中的结果。

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.