Kadison-Singer from mathematical physics: An introduction

We give an informal overview of the Kadison-Singer extension problem with emphasis on its initial connections to Dirac's formulation of quantum mechanics. Let H be an infinite dimensional separable Hilbert space, and B(H) the algebra of all bounded operators in H. In the language of operator algebras, the Kadison-Singer problem asks whether or not for a given MASA D in B(H), every pure state on D has a unique extension to a pure state on B(H). In other words, are these pure-state extensions unique? It was shown recently by Pete Casazza and co-workers that this problem is closely connected to central open problems in other parts of mathematics (harmonic analysis, combinatorics (via Anderson pavings), Banach space theory, frame theory), and applications (signal processing, internet coding, coding theory, and more).Comment: 12 pages, LaTeX2e "amsart" document class, grew out of a workshop at the AIM institute (with NSF support) in Palo Alto in September, 2006. v2: fine tuning. More details, clarifications, explanations, citations/ references have been added, most of the additions are motivated by suggestions coming in from KS IMA participant

Certain representations of the Cuntz relations, and a question on wavelets decompositions

We compute the Coifman-Meyer-Wickerhauser measure $\mu$ for certain families of quadrature mirror filters (QMFs), and we establish that for a subclass of QMFs, $\mu$ contains a fractal scale. In particular, these measures $\mu$ are not in the Lebesgue class.Comment: v.2 has a new title and additional material in the introduction. Prepared using the amsproc.cls document clas

Unitary matrix functions, wavelet algorithms, and structural properties of wavelets

Some connections between operator theory and wavelet analysis: Since the mid eighties, it has become clear that key tools in wavelet analysis rely crucially on operator theory. While isolated variations of wavelets, and wavelet constructions had previously been known, since Haar in 1910, it was the advent of multiresolutions, and subband filtering techniques which provided the tools for our ability to now easily create efficient algorithms, ready for a rich variety of applications to practical tasks. Part of the underpinning for this development in wavelet analysis is operator theory. This will be presented in the lectures, and we will also point to a number of developments in operator theory which in turn derive from wavelet problems, but which are of independent interest in mathematics. Some of the material will build on chapters in a new wavelet book, co-authored by the speaker and Ola Bratteli, see http://www.math.uiowa.edu/~jorgen/ .Comment: 63 pages, 10 figures/tables, LaTeX2e ("mrv9x6" document class), Contribution by Palle E. T. Jorgensen to the Tutorial Sessions, Program: ``Functional and harmonic analyses of wavelets and frames,'' 4-7 August 2004, Organizers: Judith Packer, Qiyu Sun, Wai Shing Tang. v2 adds Section 2.3.4, "Matrix completion" with reference

Representations of Cuntz algebras, loop groups and wavelets

A theorem of Glimm states that representation theory of an NGCR C*-algebra is always intractable, and the Cuntz algebra O_N is a case in point. The equivalence classes of irreducible representations under unitary equivalence cannot be captured with a Borel cross section. Nonetheless, we prove here that wavelet representations correspond to equivalence classes of irreducible representations of O_N, and they are effectively labeled by elements of the loop group, i.e., the group of measurable functions A:T-->U_N(C). These representations of O_N are constructed here from an orbit picture analysis of the infinite-dimensional loop group.Comment: 6 pages, LaTeX2e "amsproc" class; expanded version of an invited lecture given by the author at the International Congress on Mathematical Physics, July 2000 in Londo

Closed subspaces which are attractors for representations of the Cuntz algebras

We analyze the structure of co-invariant subspaces for representations of the Cuntz algebras O_N for N = 2,3,..., N < infinity, with special attention to the representations which are associated to orthonormal and tight-frame wavelets in L^2(R) corresponding to scale number N.Comment: 32 pages, LaTeX2e "birkart" document class; accepted for publication in the Proceedings of the 2002 IWOTA conference at Virginia Tech in Blacksburg, VA. v4 revision: changes and corrections to Theorem 4.4 and Corollary 7.1. Also Theorem 4.4 is relabeled "Proposition 4.4", and clarifying remarks are adde

Use of operator algebras in the analysis of measures from wavelets and iterated function systems

In this paper, we show how a class of operators used in the analysis of measures from wavelets and iterated function systems may be understood from a special family of representations of Cuntz algebras

A geometric approach to the cascade approximation operator for wavelets

This paper is devoted to an approximation problem for operators in Hilbert space, that appears when one tries to study geometrically the cascade algorithm in wavelet theory. Let $H$ be a Hilbert space, and let $\pi$ be a representation of $L^\infty(T)$ on $H$. Let $R$ be a positive operator in $L^\infty(T)$ such that $R(1)=1$, where $1$ denotes the constant function $1$. We study operators $M$ on $H$ (bounded, but non-contractive) such that $\pi(f)M=M\pi(f(z^2))$ and $M^* \pi(f)M=\pi(R^* f)$, $f \in L^\infty (T)$, where the $*$ refers to Hilbert space adjoint. We give a complete orthogonal expansion of $H$ which reduces $\pi$ such that $M$ acts as a shift on one part, and the residual part is $H^{(\infty)}=\bigcap_n[M^n H]$, where $[M^n H]$ is the closure of the range of $M^n$. The shift part is present, we show, if and only if $\ker(M^*) \neq \{0\}$. We apply the operator-theoretic results to the refinement operator (or cascade algorithm) from wavelet theory. Using the representation $\pi$, we show that, for this wavelet operator $M$, the components in the decomposition are unitarily, and canonically, equivalent to spaces $L^2(E_n) \subset L^2(R)$, where $E_n \subset R$, $n=0,1,2,...,\infty$, are measurable subsets which form a tiling of $R$; i.e., the union is $R$ up to zero measure, and pairwise intersections of different $E_n$'s have measure zero. We prove two results on the convergence of the cascade algorithm, and identify singular vectors for the starting point of the algorithm.Comment: AMS-LaTeX; 47 pages, 3 tables, 2 figures comprising 3 EPS diagram

Some recent trends from research mathematics and their connections to teaching: Case studies inspired by parallel developments in science and technology

We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed relatively constant over a long period of time, they must be varied in accordance with the needs and the demands of students, and they must constantly updated keeping an eye to current research and to modern international trends in technology. Our thoughts and suggestions on the use of these trends in teaching have been tried out by the author, and they are now in textbooks, some by the author.Comment: 9 pages, "article" document class, expanded version of an invited presentation "Teaching of mathematics at various levels in an international university-system, and connections to research and to current trends in technology" at the Symposium on Mathematics Education Reform (2006 Beijing, China) http://www.math.ohiou.edu/~shen/calculus/schedule.htm

Some second-order partial differential equations associated with Lie groups

In this note we survey results in recent research papers on the use of Lie groups in the study of partial differential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem natural in the context of Lie groups. The research is joint with D.W. Robinson, as well as other researchers who are listed in the references.Comment: 11 pages, 1 EPS graphic, LaTeX2e amsart document class, AMS Cyrillic font used in bibliography. This paper is an expanded version of a lecture given by the author at the National Research Symposium on Geometric Analysis and Applications at the the Centre for Mathematics and its Applications at The Australian National University in June of 200

A family of measures associated with iterated function systems

Let $(X,d)$ be a compact metric space, and let an iterated function system (IFS) be given on $X$, i.e., a finite set of continuous maps $\sigma_{i}$: $X\to X$, $i=0,1,..., N-1$. The maps $\sigma_{i}$ transform the measures $\mu$ on $X$ into new measures $\mu_{i}$. If the diameter of $\sigma_{i_{1}}\circ >... \circ \sigma_{i_{k}}(X)$ tends to zero as $k\to \infty$, and if $p_{i}>0$ satisfies $\sum_{i}p_{i}=1$, then it is known that there is a unique Borel probability measure $\mu$ on $X$ such that \mu =\sum_{i}p_{i} \mu_{i} \tag{*}. In this paper, we consider the case when the $p_{i}$s are replaced with a certain system of sequilinear functionals. This allows us to study the variable coefficient case of (*), and moreover to understand the analog of (*) which is needed in the theory of wavelets.Comment: 14 pages including references. Corrections made on pp.4 and 1