On Extending an Unbounded Orthogonal Vector Measure in a Semifinite Von Neumann Algebra to a Weight

Abstract

The aim of this paper is to extend the result of the paper [2] (on the possibility of the extension of a Hilbert-space-valued unbounded orthogonal vector measure on all projections on a Hilbert space to a vector weight) to the case of an arbitrary semifinite von Neumann algebra.Throughout the paper, let M be a von Neumann algebra which acts on a Hilbert space, endowed with an inner product < .,. >. We will denote by X pr and X + the sets of all orthoprojections and positive operators in X(C M), respectively. We will examine measures on projections with values in a Hilbert space K complemented with an improper element oo. The following assumptions will be needed in this case: / + oo = oo (/ € K), 00 + 00 = 00, A • 00 = oo(A >0), 0 • 00 = 0 (here 6 denotes the zero vector in K). We first give a definition of a scalar unbounded measure on projections (see Definition 2.2 [6]