Dilations for operator-valued quantum measures

Abstract

This paper concerns the dilations of Banach space operator-valued quantum measures. While the recently developed general dilation theory leads to a projection (idempotent) valued dilation for any quantum measure over the projection lattice of a von Neumann algebra that does not contain a Type I2 direct summand, such a dilation does not necessarily guarantee the preservation of countable additivity of the quantum measure. So it remains an open question whether every countably additive B(X)-valued quantum measure can be dilated to a countably additive projection-valued measure. The main purpose of this paper is to prove that such a dilation exists if one of the following two conditions is satisfied: (i) either the underlying Banach space X is ℓp for some 1<p<2 or X has the Schur property, (ii) the quantum measure has bounded p-variation (i.e., the total p-variation is finite) for some 1≤p<∞. These are achieved by establishing a non-commutative version of a minimal dilation theory on the elementary dilation space equipped with an appropriate dilation norm. In particular, we introduce a p-variation norm on the elementary dilation space and prove that every operator-valued quantum measure with bounded p-variation has a projection-valued quantum measure dilation that preserves the boundedness of the p-variation.