Homogeneity of the pure state space of the Cuntz algebra

We prove that the automorphism group of a Cuntz algebra of finite order acts transitively on the set of pure states which are invariant under some gauge actions (which may depend on the states). The question of whether any pure state is invariant under some gauge action is left open, but for the senigroups of unital endomorphisms stronger transitivity properties can be established witout knowing the answer of this question.Comment: 11 pages, latex. Correction in the new version: In Corollary 1 and the preceding remarks one must assume that d is a power of a prim

Endomorphisms of B(H)

The unital endomorphisms of B(H) of (Powers) index n are classified by certain U(n)-orbits in the set of non-degenerate representations of the Cuntz algebra O_n on H. Using this, the corre- sponding conjugacy classes are identified, and a set of labels is given. This set is given as P modulo a certain non-smooth equivalence, where P is a set of pure state on the UHF algebra of Glimm type n^infinity. Several subsets of P, giving concrete examples of non- conjugate shifts, are worked out in detail, including sets of product states, and a set of nearest neighbor states.Comment: 46 pages, amste

Iterated function systems and permutation representations of the Cuntz algebra

We study a class of representations of the Cuntz algebras O_N, N=2,3,..., acting on L^2(T) where T=R/2\pi Z. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the O_N-irreducibles decompose when restricted to the subalgebra UHF_N\subset O_N of gauge-invariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of O_N on Hilbert space H such that the generators S_i as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L^2(T) to L^2(T^d), d>1 ; even to L^2(\mathbf{T}) where \mathbf{T} is some fractal version of the torus which carries more of the algebraic information encoded in our representations.Comment: 84 pages, 11 figures, AMS-LaTeX v1.2b, full-resolution figures available at ftp://ftp.math.uiowa.edu/pub/jorgen/PermRepCuntzAlg in eps files with the same names as the low-resolution figures included her

Wavelet filters and infinite-dimensional unitary groups

In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C^*-algebra O_N. A main tool in our analysis is the infinite-dimensional group of all maps T -> U(N) (where U(N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.Comment: AMS-LaTeX; 30 pages, 2 tables, 1 picture. Invited lecture by Jorgensen at International Conference on Wavelet Analysis and Its Applications, Zhongshan University, Guangzhou, China, in November 1999. Changes: Some references have been added and some technical points in several proofs have been clarified in this new revised versio

A connection between multiresolution wavelet theory of scale N and representations of the Cuntz algebra O_N

In this paper we give a short survey of a connection between the theory of wavelets in L^2(R) and certain representations of the Cuntz algebra on L^2(T).Comment: 13 pages, AMS-TeX version 2.1, uses LaTeX circle font lcircle10. To appear in J. Roberts, ed., Proceedings of the Rome Conference on Operator Algebras and Quantum Field Theory. Survey article; for complete proofs see funct-an/9612002 and funct-an/9612003 by the same author

Convergence of the cascade algorithm at irregular scaling functions

The spectral properties of the Ruelle transfer operator which arises from a given polynomial wavelet filter are related to the convergence question for the cascade algorithm for approximation of the corresponding wavelet scaling function.Comment: AMS-LaTeX; 38 pages, 10 figures comprising 42 EPS diagrams; some diagrams are bitmapped at 75 dots per inch; for full-resolution bitmaps see ftp://ftp.math.uiowa.edu/pub/jorgen/convcasc

Approximately inner derivations

Let $\alpha$ be an approximately inner flow on a $C^*$ algebra $A$ with generator $\delta$ and let $\delta_n$ denote the bounded generators of the approximating flows $\alpha^{(n)}$. We analyze the structure of the set \cd=\{x\in D(\delta): \lim_{n\to\infty}\delta_n(x)=\delta(x)\} of pointwise convergence of the generators. In particular we examine the relationship of \cd and various cores related to spectral subspaces.Comment: 17 page

Abundance of invariant and almost invariant pure states of C*-dynamical systems

We show that invariant states of C*-dynamical systems can be approximated in the weak*-topology by invariant pure states, or almost invariant pure states, under various circumstances.Comment: LaTeX2e, 19 page

Compactly supported wavelets and representations of the Cuntz relations, II

We show that compactly supported wavelets in L^2(R) of scale N may be effectively parameterized with a finite set of spin vectors in C^N, and conversely that every set of spin vectors corresponds to a wavelet. The characterization is given in terms of irreducible representations of orthogonality relations defined from multiresolution wavelet filters.Comment: 10 or 11 pages, SPIE Technical Conference, Wavelet Applications in Signal and Image Processing VII