The cocycle identity holds under stopping

TL;DR

This paper generalizes the cocycle identity to include stopping with finite quantum stop times, extending previous results on stopped CCR flows and their cocycles.

Contribution

It introduces a generalized cocycle identity applicable when both parameters are finite quantum stop times, broadening the theoretical framework.

Findings

The cocycle identity holds under finite quantum stopping times.

The generalization applies to CCR flows and their isometric cocycles.

The result extends previous deterministic stopping time results.

Abstract

In recent work of the authors, it was shown how to use any finite quantum stop time to stop the CCR flow and its strongly continuous isometric cocycles (Q. J. Math. 65:1145-1164, 2014). The stopped cocycle was shown to satisfy a stopped form of the cocycle identity, valid for deterministic increments of the time used for stopping. Here, a generalisation of this identity is obtained, where both cocycle parameters are replaced with finite quantum stop times.