The structure of doubly non-commuting isometries

Abstract

Suppose that $n\ge 1$ and that, for all $i$ and $j$ with $1\le i,j\le n$ and $i\ne j$, $z_{ij}\in \mathbb{T}$ are given such that $z_{ji}={\overline{z}}_{ij}$ for all $i\ne j$. If $V_1,\dots ,V_n$ are isometries on a Hilbert space such that $V_i^{\ast }{V}_j^{}={\overline{z}}_{ij}{V}_j^{}{V}_i^{\ast }$ for all $i\ne j$, 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.

A remark on the relation between the dilation theorem in this paper and other dilation theorems for multiple operators in the literature has been added. Otherwise, there are only a few minor editorial changes compared to the first version; some typos have also been corrected. Final version, to appear in Advances in Mathematics

References in corpus (1)

• On C*-algebras generated by pairs of q-commuting isometries