On isometric dilations of product systems of C*-correspondences and applications to families of contractions associated to higher-rank graphs

TL;DR

This paper investigates conditions for isometric dilations of product systems of C*-correspondences over N^r, focusing on higher-rank graphs, and characterizes when minimal dilations are *-regular or doubly commuting.

Contribution

It provides new sufficient conditions for the existence of isometric dilations and clarifies the relationship between regularity and double commutativity in this context.

Findings

A minimal isometric dilation is *-regular iff it is doubly commuting.

Conditions for non-regular isometric dilations are established.

Detailed analysis of product systems associated with higher-rank graphs.

Abstract

Let E be a product system of C*-correspondences over N^r. Some sufficient conditions for the existence of a not necessarily regular isometric dilation of a completely contractive representation of E are established and difference between regular and *-regular dilations discussed. It is in particular shown that a minimal isometric dilation is *-regular if and only if it is doubly commuting. The case of product systems associated with higher-rank graphs is analysed in detail.