Orthogonal decompositions and twisted isometries

Abstract

Let [Formula: see text]. Let [Formula: see text] be [Formula: see text] commuting unitaries on some Hilbert space [Formula: see text], and suppose [Formula: see text], [Formula: see text]. An [Formula: see text]-tuple of isometries [Formula: see text] on [Formula: see text] is called [Formula: see text]-twisted isometry with respect to [Formula: see text] (or simply [Formula: see text]-twisted isometry if [Formula: see text] is clear from the context) if [Formula: see text]’s are in the commutator [Formula: see text], and [Formula: see text], [Formula: see text] We prove that each [Formula: see text]-twisted isometry admits a von Neumann–Wold type orthogonal decomposition, and prove that the universal [Formula: see text]-algebra generated by [Formula: see text]-twisted isometries is nuclear. We exhibit concrete analytic models of [Formula: see text]-twisted isometries, and establish connections between unitary equivalence classes of the irreducible representations of the [Formula: see text]-algebras generated by [Formula: see text]-twisted isometries and the unitary equivalence classes of the nonzero irreducible representations of twisted noncommutative tori. Our motivation of [Formula: see text]-twisted isometries stems from the classical rotation [Formula: see text]-algebras and Heisenberg group [Formula: see text]-algebras.