Direct Integral Decomposition Research Articles

DYNAMICAL MCDUFF-TYPE PROPERTIES FOR GROUP ACTIONS ON VON NEUMANN ALGEBRAS

AbstractWe consider the notion of strong self-absorption for continuous actions of locally compact groups on the hyperfinite II $_1$ factor and characterize when such an action is tensorially absorbed by another given action on any separably acting von Neumann algebra. This extends the well-known McDuff property for von Neumann algebras and is analogous to the core theorems around strongly self-absorbing C $^*$ -dynamics. Given a countable discrete group G and an amenable action $G\curvearrowright M$ on any separably acting semifinite von Neumann algebra, we establish a type of measurable local-to-global principle: If a given strongly self-absorbing G-action is suitably absorbed at the level of each fibre in the direct integral decomposition of M, then it is tensorially absorbed by the action on M. As a direct application of Ocneanu’s theorem, we deduce that if M has the McDuff property, then every amenable G-action on M has the equivariant McDuff property, regardless whether M is assumed to be injective or not. By employing Tomita–Takesaki theory, we can extend the latter result to the general case, where M is not assumed to be semifinite.

The Plancherel formula of Fourier integral operators: The case of Sp(1,1)

In this paper, we consider the Plancherel formula for Fourier integral operator on the space [Formula: see text], where [Formula: see text] is the quaternion skew field. By using the metaplectic representations of a noncompact classical dual pair [Formula: see text] and combining with the Plancherel formula of Poisson transform, we obtain the direct integral decomposition of all irreducible unitary representations of Fourier integral operator.

On the Equivalence of Causal Propagators of the Dirac Equation in Vacuum-Destabilising External Fields

In QED, an external electromagnetic field can be accounted for non-perturbatively by replacing the causal propagators used in Feynman diagram calculations with Green’s functions for the Dirac equation under the external field. If the external field destabilises the vacuum, then it is a difficult problem to determine which Green’s function is appropriate, and multiple approaches have been developed in the literature whose equivalence, in many cases, is not clear. In this paper, we demonstrate for a broad class of external fields that includes all that act for a finite time, that four Green’s functions used in the literature are equivalent: Schwinger’s “proper-time” propagator; the “causal propagator” used in the “Bogoliubov transformation” method based on the canonical quantization of the field operator; and two defined using analytic continuation from complexified parameters. To do so, we formulate Schwinger’s “proper-time quantum mechanics" as a Schrödinger wave mechanics, and use this to re-derive Schwinger’s expression for the propagator as a statement relating solutions of the inhomogeneous Dirac equation to those of the inhomogeneous “proper-time Dirac equation”. This is done by constructing direct integral spectral decompositions of the Hamiltonians of both equations, and deriving a form for solutions of the inhomogeneous Dirac equation in terms of these decompositions. We then show that all four propagators return solutions of the inhomogeneous Dirac equation that satisfy the same boundary condition, which under a physically reasonable assumption is sufficient to specify the solution uniquely.

Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

<p style='text-indent:20px;'>Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the almost periodic homogenization result by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.</p>

Decomposing the Wavelet Representation for Shifts by Wallpaper Groups

The wavelet group and wavelet representation associated with shifts coming from a two dimensional crystal symmetry group \(\Gamma \) and dilations by powers of 3, are defined and studied. The main result is an explicit decomposition of this \(3\Gamma \)-wavelet representation into irreducible representations of the wavelet group. Because we prove that the \(3\Gamma \)-wavelet representation is multiplicity free, this direct integral decomposition is essentially unique.

Bloch wave homogenisation of quasiperiodic media

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

Smoothing operators and C∗-algebras for infinite dimensional Lie groups

A smoothing operator for a unitary representation [Formula: see text] of a (possibly infinite dimensional) Lie group [Formula: see text] is a bounded operator [Formula: see text] whose range is contained in the space [Formula: see text] of smooth vectors of [Formula: see text]. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map [Formula: see text] is smooth. For unitary representations [Formula: see text] which are semibounded, i.e. there exists an element [Formula: see text] such that all operators [Formula: see text] from the derived representation, for [Formula: see text] in a neighborhood of [Formula: see text], are uniformly bounded from above, we show that [Formula: see text] coincides with the space of smooth vectors for the one-parameter group [Formula: see text]. As the main application of our results on smoothing operators, we present a new approach to host [Formula: see text]-algebras for infinite dimensional Lie groups, i.e. [Formula: see text]-algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of [Formula: see text]. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.

On reduction theory and Brown measure for closed unbounded operators

The theory of direct integral decompositions of both bounded and unbounded operators is further developed; in particular, results about spectral projections, functional calculus and affiliation to von Neumann algebras are proved. For operators belonging to or affiliated to a tracial von Neumann algebra that is a direct integral von Neumann algebra, the Brown measure is shown to be given by the corresponding integral of Brown measures.

Complex Semigroups for Oscillator Groups

An oscillator group $G$ is a semidirect product of a Heisenberg group with a one-parameter group. In this article we construct Olshanski semigroups for infinite-dimensional oscillator groups. These are complex involutive semigroups which have a polar decomposition. The main application will be for representations $\pi$ of $G$ which are semibounded, i.e., there exists a non-empty open subset $U$ of the Lie algebra $\mathfrak{g}$ such that the operators $id\pi(x)$ from the derived representation are uniformly bounded from above for $x\in U$. More precisely we show that every semibounded representation of an oscillator group $G$ extends to a non-degenerate holomorphic representation of such a semigroup and conversely each non-degenerate holomorphic representation of such a semigroup gives rise to a semibounded representation of $G$. The main application of this result is a classification of representations of the canonical commutation relations with a positive Hamiltonian, which will be obtained in a subsequent paper. Moreover it yields direct integral decomposition into irreducible ones and implies the existence of a dense subspace of analytic vectors for semibounded representations of $G$.

Crossed product algebras and direct integral decomposition for Lie supergroups

For every finite dimensional Lie supergroup $(G,\mathfrak g)$, we define a $C^*$-algebra $\mathcal A:=\mathcal A(G,\mathfrak g)$, and show that there exists a canonical bijective correspondence between unitary representations of $(G,\mathfrak g)$ and nondegenerate $*$-representations of $\mathcal A$. The proof of existence of such a correspondence relies on a subtle characterization of smoothing operators of unitary representations. For a broad class of Lie supergroups, which includes nilpotent as well as classical simple ones, we prove that the associated $C^*$-algebra is CCR. In particular, we obtain the uniqueness of direct integral decomposition for unitary representations of these Lie supergroups.

Wave front sets of reductive Lie group representations

If G is a Lie group, H⊂G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ∈ig∗ to be in the wave front set of IndHGτ. In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a real, reductive algebraic group and π is a unitary representation of G that is weakly contained in the regular representation, then the authors give a geometric description of WF(π) in terms of the direct integral decomposition of π into irreducibles. Special cases of this result were previously obtained by Kashiwara–Vergne, Howe, and Rossmann. The authors give applications to harmonic analysis problems and branching problems.

Regular representations of time‐frequency groups

In this paper, we study the Plancherel measure of a class of non‐connected nilpotent groups which is of special interest in Gabor theory. Let G be a time‐frequency group. That is where , are translations and modulations operators acting in and B is a non‐singular matrix. We compute the Plancherel measure of the left regular representation of G which is denoted by L. The action of G on induces a representation which we call a Gabor representation. Motivated by the admissibility of this representation, we compute the decomposition of L into direct integral of irreducible representations by providing a precise description of the unitary dual and its Plancherel measure. As a result, we generalize Hartmut Führ's results which are only obtained for the restricted case where , and Even in the case where G is not type I, we are able to obtain a decomposition of the left regular representation of G into a direct integral decomposition of irreducible representations when . Some interesting applications to Gabor theory are given as well. For example, when B is an integral matrix, we are able to obtain a direct integral decomposition of the Gabor representation of G.

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

We study two-dimensional magnetic Schrodinger operators with a magnetic field that is equal to b>0 for x > 0 and (-b) for x < 0. This magnetic Schrodinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schrodinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schrodinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. We study the perturbation by slowly decaying negative potentials and establish the asymptotic behavior of the eigenvalue counting function for the infinitely-many eigenvalues below the bottom of the essential spectrum.

THE CONTINUOUS SPECTRUM IN DISCRETE SERIES BRANCHING LAWS

If G is a reductive Lie group of Harish-Chandra class, H is a symmetric subgroup, and π is a discrete series representation of G, the authors give a condition on the pair (G, H) which guarantees that the direct integral decomposition of π|H contains each irreducible representation of H with finite multiplicity. In addition, if G is a reductive Lie group of Harish-Chandra class, and H ⊂ G is a closed, reductive subgroup of Harish-Chandra class, the authors show that the multiplicity function in the direct integral decomposition of π|H is constant along "continuous parameters". In obtaining these results, the authors develop a new technique for studying multiplicities in the restriction π|H via convolution with Harish-Chandra characters. This technique has the advantage of being useful for studying the continuous spectrum as well as the discrete spectrum.

A universal envelope for Gaussian processes and their kernels

The central theme in our paper deals with mathematical issues involved in the answer to the following question: How can we generate stochastic processes from their correlation data? Since Gaussian processes are determined by moment information up to order two, we focus on the Gaussian case. Two functional analytic tools are used here, in more than one variant. They are: operator factorization; and direct integral decompositions in the form of Karhunen-Loeve expansions. We define and study a new interplay between the theory of positive definition functions, and their reproducing kernels, on the one hand, and Gaussian stochastic processes, on the other. The three classes of processes we study are as follows: Processes associated with: (a) arbitrarily given sigma finite regular measures on a fixed Borel measure space; (b) with Hilbert spaces of sigma-functions; and (c) with systems of self-similar measures arising in the theory of iterated function systems. Starting with a non-degenerate positive definite function K on some fixed set S, we show that there is a choice of a universal sample space Ω, which can be realized as a “boundary” of (S,K). Its boundary-theoretic properties are analyzed, and we point out their relevance to the study of electrical networks on countable infinite graphs.

Admissibility for quasiregular representations of exponential solvable Lie groups

Let $N$ be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra $\mathfrak{n}$ of dimension $n.$ Let $H$ be a subgroup of the automorphism group of $N.$ Assume that $H$ is a commutative, simply connected, connected Lie group with Lie algebra $\mathfrak{h}.$ Furthermore, let us assume that the linear adjoint action of $\mathfrak{h}$ on $\mathfrak{n}$ is diagonalizable with non-purely imaginary eigenvalues. Let $\tau=\mathrm{Ind}%_{H}^{N\rtimes H} 1$. We obtain an explicit direct integral decomposition for $\tau$, including a description of the spectrum as a sub-manifold of $(\mathfrak{n}+\mathfrak{h})^{\ast}$, a formula for the multiplicity function of the unitary irreducible representations occurring in the direct integral, and a precise intertwining operator. Finally, we completely settle the admissibility question of $\tau$. In fact, we show that if $G=N\rtimes H$ is unimodular, then $\tau$ is never admissible, and if $G$ is nonunimodular, $\tau$ is admissible if and only if the intersection of $H$ and the center of $G$ is equal to the identity of the group.

On the relation between rigging inner product and master constraint direct integral decomposition

Canonical quantization of constrained systems with first-class constraints via Dirac’s operator constraint method proceeds by the theory of Rigged Hilbert spaces, sometimes also called refined algebraic quantization. This method can work when the constraints form a Lie algebra. When the constraints only close with nontrivial structure functions, the Rigging map can no longer be defined. To overcome this obstacle, the master constraint method has been proposed which replaces the individual constraints by a weighted sum of absolute squares of the constraints. Now the direct integral decomposition (DID) methods, which are closely related to Rigged Hilbert spaces, become available and have been successfully tested in various situations. It is relatively straightforward to relate the rigging inner product to the path integral that one obtains via reduced phase space methods. However, for the master constraint, this is not at all obvious. In this paper we find sufficient conditions under which such a relation can be established. Key to our analysis is the possibility to pass to equivalent, Abelian constraints, at least locally in phase space. Then the master constraint DID for those Abelian constraints can be directly related to the rigging map and therefore has a path integral formulation.

FULL REGULARITY FOR A C*-ALGEBRA OF THE CANONICAL COMMUTATION RELATIONS

The Weyl algebra — the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect, in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space (S, B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalized group algebra, explained below) for the σ-representation theory of the Abelian group S where σ(·,·) ≔ eiB(·,·)/2. As an easy application, it then follows that for every regular representation of [Formula: see text] on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result).

Up–down representations and ergodic theory in nilpotent Lie groups

Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation \({({\rm Ind}_\Gamma^G 1)\vert_A}\) . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G.

Direct integral decomposition for periodic function spaces and application to Bloch waves

In this paper, we study a direct integral decompositionfor the spaces $L^2(O)$ and $H^1(O)$ based on $(\xi,Y^*)-$periodicfunctions. Using this decomposition we can write the Green'soperator (associated to the classical Stokes system in fluid mechanics)in terms of a family of self-adjoint compact operators which depend on theparameter $\xi$. As a consequence, we obtain theso-called Bloch waves associated to the Stokes system in the caseof a periodic perforated domain.