Journal of Operator Theory

Volume 43, Issue 1, Winter 2000 pp. 97-143.

Pure states on $\scriptstyle {\cal O}_{d}$

Authors: O. Bratteli (1), P.E.T. Jorgensen (2), A. Kishimoto (3), and R.F. Werner (4)
Author institution: (1) Department of Mathematics, University of OsloPB 1053, Blindern, N--0316 Oslo, Norway
(2) Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242--1419, USA
(3) Department of Mathematics, University of Hokkaido, Sapporo 060, Japan
(4) Mathematische Physik, Universitaet Braunschweig, Mendelssohnstr.\ 3, D-38016 Braunschweig, Germany

Summary: We study representations of the Cuntz algebras ${\cal O}_{d}$ and their associated decompositions. In the case that these representations are irreducible, their restrictions to the gauge-invariant subalgebra ${\rm UHF}_{d}$ have an interesting cyclic structure. If $S_{i}$, $1\leq i\leq d$, are representatives of the Cuntz relations on a Hilbert space ${\cal H}$, special attention is given to the subspaces which are invariant under $S_{i}^{\ast}$. The applications include wavelet multiresolutions corresponding to wavelets of compact support (to appear in the later paper [8]), and finitely correlated states on one-dimensional quantum spin chains.

Keywords: Cuntz algebra, representations of $C^*$-algebras, Hilbert space, endomorphism, completely positive maps, dilation, commutant, von Neumann algebras

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