The structure of doubly non-commuting isometries

TL;DR

This paper develops a classification and decomposition theory for n-tuples of doubly non-commuting isometries, generalizing known structures like non-commutative tori, and establishes dilation results for these operators.

Contribution

It introduces a simultaneous Wold decomposition and classification scheme for doubly non-commuting isometries, extending the theory of non-commutative tori.

Findings

Established a Wold decomposition for the isometries.

Classified n-tuples up to unitary equivalence using non-commutative tori invariants.

Proved a dilation theorem extending isometries to unitaries.

Abstract

Suppose that $n\geq 1$ and that, for all $i$ and $j$ with $1\leq i,j\leq n$ and $i\neq j$, $z_{ij}\in{\mathbb T}$ are given such that $z_{ji}=\overline{z}_{ij}$ for all $i\neq j$. If $V_1,\dotsc, V_n$ are isometries on a Hilbert space such that $V_i^\ast V_j^{\phantom{\ast}}\!=\overline{z}_{ij} V_j^{\phantom{\ast}}\!V_i^\ast$ for all $i\neq j$, then $(V_1,\dotsc,V_n)$ is called an $n$-tuple of doubly non-commuting isometries. The generators of non-commutative tori are well-known examples. In this paper, we establish a simultaneous Wold decomposition for $(V_1,\dotsc,V_n)$. This decomposition enables us to classify such $n$-tuples up to unitary equivalence. We show that the joint listing of a unitary equivalence class of a representation of each of the $2^n$ non-commutative tori that are naturally associated with the structure constants is a classifying invariant. A dilation theorem is also established, showing that an $n$-tuple of doubly non-commuting isometries can be extended to an $n$-tuple of doubly non-commuting unitary operators on an enveloping Hilbert space.