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

UHF flows and the flip automorphism

A UHF flow is an infinite tensor product type action of the reals on a UHF algebra $A$ and the flip automorphism is an automorphism of $A\otimes A$ sending $x\otimes y$ into $y\otimes x$. If $\alpha$ is an inner perturbation of a UHF flow on $A$, there is a sequence $(u_n)$ of unitaries in $A\otimes A$ such that $\alpha_t\otimes \alpha_t(u_n)-u_n$ converges to zero and the flip is the limit of \Ad u_n. We consider here whether the converse holds or not and solve it with an additional assumption: If $A\otimes A\cong A$ and $\alpha$ absorbs any UHF flow $\beta$ (i.e., $\alpha\otimes\beta$ is cocycle conjugate to $\alpha$), then the converse holds; in this case $\alpha$ is what we call a universal UHF flow.Comment: 18 page

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

Cauchy Problem and Green's Functions for First Order Differential Operators and Algebraic Quantization

Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.Comment: revised version: typos; reordering of sec 2; results unchange

Black Hole Horizons and Thermodynamics: A Quantum Approach

We focus on quantization of the metric of a black hole restricted to the Killing horizon with universal radius $r_0$. After imposing spherical symmetry and after restriction to the Killing horizon, the metric is quantized employing the chiral currents formalism. Two ``components of the metric'' are indeed quantized: The former behaves as an affine scalar field under changes of coordinates, the latter is instead a proper scalar field. The action of the symplectic group on both fields is realized in terms of certain horizon diffeomorphisms. Depending on the choice of the vacuum state, such a representation is unitary. If the reference state of the scalar field is a coherent state rather than a vacuum, spontaneous breaking of conformal symmetry arises and the state contains a Bose-Einstein condensate. In this case the order parameter fixes the actual size of the black hole with respect to $r_0$. Both the constructed state together with the one associated with the affine scalar are thermal states (KMS) with respect to Schwarzschild Killing time when restricted to half horizon. The value of the order parameter fixes the temperature at the Hawking value as well. As a result, it is found that the quantum energy and entropy densities coincide with the black hole mass and entropy, provided the universal parameter $r_0$ is suitably chosen, not depending on the size of the actual black hole in particular.Comment: 21 pages, revised and published version, title change

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

Classification of the direct limits of involution simple associative algebras and the corresponding dimension groups

A classification of (countable) direct limits of finite dimensional involution simple associative algebras over an algebraically closed field of arbitrary characteristic is obtained. This also classifies the corresponding dimension groups. The set of invariants consists of two supernatural numbers and two real parameters