The structure of doubly non-commuting isometries

Suppose that $n\ge 1$ and that, for all $i$ and $j$ with $1\le i,j\le n$ and $i\mathrm{\ne }j$, $z_{ij}\in \mathbb{T}$ are given such that $z_{ji}={\bar{z}}_{ij}$ for all $i\mathrm{\ne }j$. If $V_1,\dots ,V_n$ 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 $i\mathrm{\ne }j$, then $(V_1,\dots ,V_n)$ 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 $(V_1,\dots ,V_n)$. 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.