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Palle E. T. Jorgensen

Publications and source records attributed to Palle E. T. Jorgensen.

154 records · Page 9Linked to original sources
arXiv1999

Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams

We first study situations where the stable AF-algebras defined by two square primitive nonsingular incidence matrices with nonnegative integer matrix elements are isomorphic even though no powers of the associated automorphisms of the corresponding dimension groups are isomorphic. More generally we consider neccessary and sufficient conditions for two such matrices to determine isomorphic dimension groups. We give several examples.

math.OA↗
arXiv1998

Ruelle operators: Functions which are harmonic with respect to a transfer operator

Let $ N \in \mathbb{N} $, $ N \geq 2 $, be given. Motivated by wavelet analysis, we consider a class of normal representations of the $ C^* $-algebra $ \mathfrak{A}_{N} $ on two unitary generators $ U $, $ V $ subject to the relation \[ UVU^{-1}=V^{N}. \] The representations are in one-to-one correspondence with solutions $ h \in L^{1}(\mathbb{T}) $, $ h \geq 0 $, to $ R(h)=h $ where $ R $ is a certain transfer operator (positivity-preserving) which was studied previously by D. Ruelle. The representations of $ \mathfrak{A}_{N} $ may also be viewed as representations of a certain (discrete) $ N $-adic $ ax+b $ group which was considered recently by J.-B. Bost and A. Connes.

math.FA↗
arXiv1997

Pure states on O_d

We study representations of the Cuntz algebras O_d and their associated decompositions. In the case that these representations are irreducible, their restrictions to the gauge-invariant subalgebra 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 H, special attention is given to the subspaces which are invariant under S_i^*. The applications include wavelet multiresolutions corresponding to wavelets of compact support (to appear in the later paper \cite{BEJ97}), and finitely correlated states on one-dimensional quantum spin chains.

funct-an↗
arXiv1997

Dense analytic subspaces in fractal $L^{2}$-spaces

We consider self-similar measures $μ$ with support in the interval $0\leq x\leq 1$ which have the analytic functions $\left\{e^{i2πnx}:n=0,1,2,... \right\} $ span a dense subspace in $L^{2}(μ) $. Depending on the fractal dimension of $μ$, we identify subsets $P\subset \mathbb{N}_{0}=\{0,1,2,... \} $ such that the functions $\{e_{n}:n\in P\} $ form an orthonormal basis for $L^{2}(μ) $. We also give a higher-dimensional affine construction leading to self-similar measures $μ$ with support in $\mathbb{R}^ν$. It is obtained from a given expansive $ν$-by-$ν$ matrix and a finite set of translation vectors, and we show that the corresponding $L^{2}(μ) $ has an orthonormal basis of exponentials $e^{i2πλ\cdot x}$, indexed by vectors $λ$ in $\mathbb{R}^ν$, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.

funct-an↗
arXiv1997

Spectral asymptotics of periodic elliptic operators

We demonstrate that the structure of complex second-order strongly elliptic operators $H$ on ${\bf R}^d$ with coefficients invariant under translation by ${\bf Z}^d$ can be analyzed through decomposition in terms of versions $H_z$, $z\in{\bf T}^d$, of $H$ with $z$-periodic boundary conditions acting on $L_2({\bf I}^d)$ where ${\bf I}=[0,1>$. If the semigroup $S$ generated by $H$ has a Hölder continuous integral kernel satisfying Gaussian bounds then the semigroups $S^z$ generated by the $H_z$ have kernels with similar properties and $z\mapsto S^z$ extends to a function on ${\bf C}^d\setminus\{0\}$ which is analytic with respect to the trace norm. The sequence of semigroups $S^{(m),z}$ obtained by rescaling the coefficients of $H_z$ by $c(x)\to c(mx)$ converges in trace norm to the semigroup $\hat{S}^z$ generated by the homogenization $\hat{H}_z$ of $H_z$. These convergence properties allow asymptotic analysis of the spectrum of $H$.

funct-an↗
arXiv1997

Unitary Representations of Lie Groups with Reflection Symmetry

We consider the following class of unitary representations $π$ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $τ$, and that the Hilbert space $\mathbf{H} (π)$ carries a unitary operator $J$ such that $Jπ=(π\circ τ)J$ (i.e., selfsimilarity). (ii) An added symmetry is implied if $\mathbf{H} (π)$ further contains a closed subspace $\mathbf{K}_0 $ having a certain order-covariance property, and satisfying the $\mathbf{K}_0 $-restricted positivity: $< v \mid Jv > \ge 0$, $\forall v\in \mathbf{K}_0 $, where $< \cdot \mid \cdot >$ is the inner product in $\mathbf{H} (π)$. From (i)--(ii), we get an induced dual representation of an associated dual group $G^c$. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context when $G$ is semisimple and hermitean; but when $G$ is the $(ax+b)$-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinite-dimensional irreducible representations. We describe a class of $G$, containing the latter two, which admits a classification of the possible spaces $\mathbf{K}_0 \subset \mathbf{H} (π)$ satisfying the axioms of selfsimilarity and order-covariance.

funct-an↗
arXiv1996

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/2πZ. 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.

funct-an↗
arXiv1996

Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N

In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras O_N, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L^2(T) by (S_iξ)(z)=m_i(z)ξ(z^N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over L^2(T). This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations of O_N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.

funct-an↗
arXiv1993

Harmonic analysis of fractal measures induced by representations of a certain C$^*$-algebra

We describe a class of measurable subsets $Ω$ in $\br^d$ such that $L^2(Ω)$ has an orthogonal basis of frequencies $e_λ(x)=e^{i2πλ\cdot x}(x\inΩ)$ indexed by $λ\inΛ\subset\br^d$. We show that such spectral pairs $(Ω,Λ)$ have a self-similarity which may be used to generate associated fractal measures $μ$ with Cantor set support. The Hilbert space $L^2(μ)$ does not have a total set of orthogonal frequencies, but a harmonic analysis of $μ$ may be built instead from a natural representation of the Cuntz C$^*$- algebra which is constructed from a pair of lattices supporting the given spectral pair $(Ω,Λ)$. We show conversely that such a pair may be reconstructed from a certain Cuntz-representation given to act on $L^2(μ)$.

math.OA↗