100 research outputs found
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 be an approximately inner flow on a algebra with
generator and let denote the bounded generators of the
approximating flows . 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
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