The Weyl transform and the Stone-Von Neumann theorem

We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of “generalized weighted composition” operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint.

We investigate the geometric properties for Hilbert C*-modular frames. We show that any dual frame pair in a Hilbert C*-module is an orthogonal compression of a Riesz basis and its canonical dual for some larger Hilbert C*-module. This generalizes the Hilbert space dual frame pair dilation theory due to Casazza, Han and Larson to dual Hilbert C*-modular frame pairs. Additionally, for any Hilbert C*-modular dual frame pair induced by a group of unitary operators, we show that there is a dilated dual pair which inherits the same group structure.

We construct a representation of the Weyl group in the p-adic Hilbert space of functions which are square integrable with respect to a p-adic valued Gaussian distribution. The operators corresponding to position and momentum are determined by groups of unitary operators with parameters restricted to some balls in the field of p-adic numbers. A surprising fact is that the radii of these balls are connected by `an uncertainty relation' which can be considered as a p-adic analogue of the Heisenberg uncertainty relations. The p-adic Hilbert space representation of the Weyl group is the basis for a calculus of pseudo-differential operators and for an operator quantization over p-adic numbers.

We show that a well-known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group. 1. Uncertainty principles related to Lie group representations Let G be a Lie group with Lie algebra g, and let (n, H) be a unitary representation of G. Then each element Xe g generates a closed, skew-adjoint operator n(X) on H by n(exptX)x-x n(X)x = hm '-o t with domain D(n(X)) consisting of all xe H for which the limit exists. The uncertainty principle related to n says that for operators generated by X, Y and [X, Y] the following holds ''n(X)x''''n(Y)x''>^'(n([X,Y])x,x)' for all xeD(n(X))nD(n(Y))nD(n([X, Y])). We would like to advocate this as a natural way to achieve uncertainty principles. It was first proposed in Kraus [4] for Lie groups with three dimensions or fewer, and the dimension constraint was recently removed by Christensen in [2]. For example, the classical Heisenberg uncertainty principle for functions on Un is easily derived in this way from the Schrodinger representation of the Heisenberg group (see [3, p. 212]). It is the purpose of this note to point out that the uncertainty principle for the circle, which was motivated by Breitenberger [1] and further discussed in [5; 6; 7; 8; 9; 10] is obtained similarly from the principal series representation of the Euclidean motion group of U2. * Corresponding author; e-mail: vepjan@math.ku.dk Mathematical Proceedings of the Royal Irish Academy, 104A (2), 249-252 (2004) © Royal Irish Academy This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms 250 Mathematical Proceedings of the Royal Irish Academy 2. The Euclidean motion group and a unitary representation Let G be the Euclidean motion group G=j(r,z)=(^or ^|rER,zecj Its Lie algebra is fl={(j j)|reR,zec} Let H be the Hubert space H = L2(J) of square integrable functions on the circle T = {s e C| 's' = 1}, with inner product (/, g) = JT f(t)g(t)dt. As in [1 1, chapter V] the following defines a unitary representation of G on ^i: 7Tfl(r, z)/(?) = ^Re(z^/(^-^), (j e T) where a e C. For simplicity we assume in the following that a = l, which is sufficient for our purpose. The representation n' will be denoted n. 3. Operators generated from the representation We now generate three operators from elements of the Lie algebra g. Let X,YU r2egbe *-G s), .-(s i) ^(o ?) then exp(ijr)=^ J), exParl)=^ i) and QxV(tY2)=(^ ^ and we then get 7t(X)f(s) = lim i-+0 / i->0 ? where /'(a) = jtf{eus). This operator has domain {/eL2(T)|?h-»/V0 absolutely continuous with /7 e L2(T)} (3.1) Also rv^^ v hm 1*0, t)f(s) f(s) hm e^s)fis) fis) . ,_, . 7t( rv^^ Fj)/^) = v hm i-o t t-+o t and /VUM r hm 40, iY)/(j) /(j) hm r e^fis) fis) . . . _. _ . . ni /VUM Y2)fis) = hm r t^o i /->o t when s=e10 . Both of these operators are defined on H. This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms Christensen and Schlichtkrull Euclidean motion group 251 4. The uncertainty principle Since [X,YX] = Y21 [X,Y2]=-Yl the uncertainty principle gives 1(7*7, )/,/>| = '(n([X, Y2])fJ)' The first author would like to thank the people at NUI Galway for the help and friendliness he received while writing his thesis.

We show that all groups of a distinguished class of «large» topological groups, that of Roelcke precompact Polish groups, have Kazhdan’s Property (T). This answers a question of Tsankov and generalizes previous results by Bekka (for the infinite-dimensional unitary group) and by Evans and Tsankov (for oligomorphic groups). Further examples include the group \({{\,\mathrm{Aut}\,}}(\mu )\) of measure-preserving transformations of the unit interval and the group \({{\,\mathrm{Aut}\,}}^*(\mu )\) of non-singular transformations of the unit interval.More precisely, we prove that the smallest cocompact normal subgroup \(G^\circ \) of any given non-compact Roelcke precompact Polish group G has a free subgroup \(F\le G^\circ \) of rank two with the following property: every unitary representation of \(G^\circ \) without invariant unit vectors restricts to a multiple of the left-regular representation of F. The proof is model-theoretic and does not rely on results of classification of unitary representations. Its main ingredient is the construction, for any \(\aleph _0\)-categorical metric structure, of an action of a free group on a system of elementary substructures with suitable independence conditions.

1* Introduction. We present a generic C*-algebra in illustration of several peculiar phenomena that may occur in the theory of representations. Let F2 denote the free group on two generators. If π is the universal unitary representation of F2 on a Hubert space £ίf, then the full group C -algebra C*(F2) is the C*-subalgebra of ^ ( < a r ) generated by the set {π(g):geF2} (see [4, §13.9]). Alternatively, we can re-define C*(F2), in an operator-theoretical setting, as follows:

Let be a unitary operator on a Hubert space H. Then in particular, (i) is a contraction, i.e. || T 1; and (ii) The spectrum of is a subset of the unit circle, i.e. Sp (T)cC, where C denotes the set of complex numbers of absolute value one. Call an arbitrary operator a unimodular contraction if it satisfies conditions (i) and (ii) above. Then several questions immediately come to mind. Do there exist nonunitary unimodular contractions? If so, what is the nature of their spectra, e.g. what subsets of the unit circle arise as spectra of nonunitary unimodular contractions; when does the spectrum contain point, residual, or continuous spectrum? Under what conditions is a unimodular contraction unitary? What is the nature of operator algebras containing nonunitary unimodular contractions? In this paper examples are given of nonunitary unimodular contractions. It is shown (Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum. It is also shown (Theorem 1) that nonunitary unimodular contractions exist only in infinite von Neumann algebras. This result is applied to a mapping problem of operator algebras.

In this chapter we will introduce the Feynman path integral. We will derive the simplest path integral from conventional quantum mechanics. In doing so we may perhaps give the impression that path integration is merely a useful trick for solving a certain class of quantum mechanics problems. As familiarity with the technique grows, however, the gentle reader may come to follow Feynman in his belief that the path integral captures the fundamental physics, and that hamiltonians and Hubert space are merely useful mathematical methods for evaluating path integrals.KeywordsPath IntegralPerturbation SeriesGaussian IntegralEuclidean TimeClassical Statistical MechanicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.

One-dimensional polyacetylene is studied as a model of statistical mechanics. In a semiclassical approximation the system is equivalent to a quantumXY model interacting with unbounded classical spins in one-dimensional lattice spaceZ. By establishing uniform estimates, an infinite-volume-limit Hilbert space, a strongly continuous time evolution group of unitary operators, and an invariant vector are constructed. Moreover, it is proven that any infinite-limit state satisfies Gibbs conditions. Finally, a modification of Araki's relative entropy method is used to establish the uniqueness of Gibbs states.

Weakly wandering vectors for compactly generated groups of unitary operators

We determine precisely for which irreducible unitary highest weight representation of the group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the countable direct limit of the finite-dimensional unitary groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the corresponding 1-cohomology space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> does not vanish. This occurs in particular if a highest weight, viewed as an integer-valued function on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℕ</mml:mi> </mml:math> , is finitely supported. In a second step, we extend the finitely supported highest weight representations to norm-continuous unitary representations of the Banach-completions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of the direct limit <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo>(</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with respect to the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> th Schatten norm for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> . For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , the corresponding 1-cohomology spaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> do not vanish either, except in three cases. We conclude that these groups do not have Kazhdan’s Property (T). On the other hand, for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , the first cohomology spaces all vanish because <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has property (FH) as a bounded topological group.

For every Tikhonov space X the free abelian topological group A(X) and the free locally convex vector space L(X) admit a topologically faithful unitary representation. For compact spaces X this is due to Jorge Galindo.

The relation between the manifold topology, observables, and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated with positions and displacements on the manifold. The guiding, physically motivated, principles are (i) locality, i.e., the generating role of the algebras localized in small, topologically trivial, regions, (ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, and (iii) the exclusion of additional local degrees of freedom with respect to the Schrödinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, “topological,” subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering M̃ of M and on the decomposition of L2(M̃) according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the permutation group in particle statistics are discussed.

Motivated by the task to decide whether two quantum states are equally entangled we consider the orbits under the action of the group of all one-qubit operations. To investigate the orbit structure of this group of local unitary operations we propose to use methods from classical invariant theory as well as new results. Two approaches are presented. The first uses the orbit separation property of invariant rings to distinguish among nonequivalent quantum states. In this context we study the Molien series which describes the structure of the invariant ring as a graded ring. We give a closed formula for the Molien series of the group of one-qubit operations. Our second approach makes use of an equivalence relation, the so-called graph of the action, which relates two points iff they are on the same orbit. For finite groups which factor, are synchronous direct sums, or tensor products we analyze the structure of the graph of the action. This yields new algorithms for the computation of the graph of the action.