Problem of Integration with Respect to Unbounded Measures on the Set of Projections

Abstract

We note if j is a normal weight on M, then is a measure on projections and if a measure on projections can be extended to a normal weight, then the problem of constructing an integral with respect to this measure reduces to the problem of constructing an integral with respect to the weight. We therefore present several methods of constructing noncommutative integration which gives a survey of the contemporary state of the theory in the von Neumann algebra (M) with respect to weightj. For every aÎ [0,1], the Banach space is isometrically isomorphic to the space Lp(t) and the space is, by definition, the Banach space completion of  in the norm .We construct the scale of L_p(j) spaces  with respect to a faithful normal semifinite (f.n.s.) weight j on a von Neumann algebra M. These spaces are realized by operators. This is achieved by extending the original algebra M, and the Hilbert space where M originally acted is altered, as well. In the construction of the scale, the concept of an operator-valued weight is used. We discuss the problem of integration with respect to measures on projections which remains open for unbounded measures (m(1) = +¥) and their structure has been studied only for the algebra ?(?).

Keywords: Von Neumann algebra, Faithful normal semifinite trace (f.n.s.)t, weight, isometrically isomorphic, projections, Banach spaces and L_p-spaces