Some dominated convergence theorems in a von Neumann algebra
Appaswamy R. Padmanabhan · Proceedings of the Japan Academy Series A Mathematical Sciences · 1966
In his fundamental paper 2-, Stinespring proved several dominated convergence theorems for operators measurable w.r.t, a gage space.In this paper we state and prove some dominated convergence theorems.One of our theorems is a generalisation of a theorem of Stinespring 2. In the others, we obtain some results under assumptions, which are weaker than those of Stinespring.Throughout this paper, the notation and terminology will be the same as those of Segal 1 and Stinespring 2. Let (H,/2, m) be a regular gage space, namely, H is a complex Hilbert Space, /2 a ring of operators (-von Neumann Algebra) acting on H and m, a gage on 2, such that for any projection P, m(P)-O implies P:0.Denote the L and L-space of the gage space by L(H, 9, m)-L(/2, m) and L(H, 9, m)-L(9, m), respectively.A sequence of measurable operators (measurable w.r.t. 2 in the sense of 1), is said to converge grossly to a measurable operator A, 2, p. 26, if, for every T in L(;2, m), and for every e>0, there exists a positive integer N such that for all n>=N, there exists a projection P with the property that II(A.-A)PII<e and Im(TQ) l< for any projection Q