Pure states on O_d

TL;DR

This paper investigates the structure of pure states on the Cuntz algebra O_d, focusing on irreducible representations, their restrictions to gauge-invariant subalgebras, and applications to wavelet theory and quantum spin chains.

Contribution

It provides a detailed analysis of irreducible representations of O_d and explores their cyclic structures and invariant subspaces, with applications to wavelets and quantum states.

Findings

Irreducible representations have cyclic structures on gauge-invariant subalgebras.

Invariant subspaces under S_i^* are characterized.

Applications to wavelet multiresolutions and quantum spin chains.

Abstract

We study representations of the Cuntz algebras O_d and their associated decompositions. In the case that these representations are irreducible, their restrictions to the gauge-invariant subalgebra UHF_d have an interesting cyclic structure. If S_i, 1 \leq i \leq d, are representatives of the Cuntz relations on a Hilbert space H, special attention is given to the subspaces which are invariant under S_i^*. The applications include wavelet multiresolutions corresponding to wavelets of compact support (to appear in the later paper \cite{BEJ97}), and finitely correlated states on one-dimensional quantum spin chains.