The structure of doubly non-commuting isometries

Abstract

Suppose that n ≥ 1 and that, for all i and j with 1 ≤ i, j ≤ n and i = j, z ij ∈ T are given such that z ji = z ij for all i = j. If V 1 , . . . , Vn are isometries on a Hilbert space such thatThe generators of non-commutative tori are well-known examples. In this paper, we establish a simultaneous Wold decomposition for (V 1 , . . . , Vn). 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 included.