On the validity of the Cauchy–Schwarz inequality for the bracket map

Abstract

Due to its properties, the bracket map associated with a dual integrable unitary representation of a locally compact group can be viewed as a certain operator-valued inner product; however, in the non-commutative setting, the Cauchy–Schwarz property for bracket is no longer present in its full strength. In this paper, we show that fulfillment of the property, even in weaker forms, has strong consequences on the underlying group \(G\) and the corresponding von Neumann algebra \(\mbox{VN}(G)\). In particular, we show that for unimodular group \(G\), positive elements of the \(L^1\) space over \(\mbox{VN}(G)\) which are affiliated with the commutant of \(\mbox{VN}(G)\) are precisely those for which the weaker variant of the inequality is fulfilled, and that the validity of the Cauchy–Schwarz property for the appropriate set of elements indicates the existence of a closed abelian subgroup or abelian von Neumann subalgebra of \(\mbox{VN}(G)\).