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arXiv2004

No More Than Mechanics. I

One can introduce so-called {\em Plain Mechanics} having an {\bf operator realization}. Then the set of one-dimension representations of this operator realization may be identified with the Classical Mechanics. Different irreducible infinite-dimension representations may be recognized as Quantum Mechanics for different $\hbar$ (the Planck constant). It can be done in the such manner that the following diagram will be commutative. Plain Mechanics / \ / \ / \ Quantum Mechanics --> Classical Mechanics h->0 Here the horizontal arrow is well known correspondence between Quantum and Classical Mechanics if Planck constant tensing to zero. A {\em realization} of this scheme for a particle in $n$-dimensional space by two-sided convolutions on the Heisenberg group is constructed. We also introduce the {\em motion equations} for observables in this realization. The left arrow of the given diagram carries this equation to the Heisenberg one and the right arrow maps it to the Hamilton equation.

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arXiv1994

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.

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arXiv1994

Completely Positive Maps on Coxeter Groups, Deformed Commutation Relations, and Operator Spaces

In this article we prove that quasi-multiplicative (with respect to the usual length function) mappings on the permutation group $\SSn$ (or, more generally, on arbitrary amenable Coxeter groups), determined by self-adjoint contractions fulfilling the braid or Yang-Baxter relations, are completely positive. We point out the connection of this result with the construction of a Fock representation of the deformed commutation relations $d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=δ_{ij}\id$, where the matrix $t_{js}^{ir}$ is given by a self-adjoint contraction fulfilling the braid relation. Such deformed commutation relations give examples for operator spaces as considered by Effros, Ruan and Pisier. The corresponding von Neumann algebras, generated by $G_i=d_i+d_i^*$, are typically not injective.

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arXiv2010

Hilbert C*-modules over monotone complete C*-algebras

The aim of the present paper is to describe self-duality and C*- reflexivity of Hilbert {\bf A}-modules $\cal M$ over monotone complete C*-algebras {\bf A} by the completeness of the unit ball of $\cal M$ with respect to two types of convergence being defined, and by a structural criterion. The derived results generalize earlier results of {\sc H.~Widom} [Duke Math.~J.~23, 309-324, MR 17 \# 1228] and {\sc W.~L.~Paschke} [Trans. Amer.~Math.~Soc.~182, 443-468, MR 50 \# 8087, Canadian J.~Math.~26, 1272-1280, MR 57 \# 10433]. For Hilbert C*-modules over commutative AW*-algebras the equivalence of the self-duality property and of the Kaplansky-Hilbert property is reproved, (cf. {\sc M.~Ozawa} [J.~Math.~Soc.~Japan 36, 589-609, MR 85m:46068] ). Especially, one derives that for a C*-algebra {\bf A} the {\bf A}-valued inner pro\-duct of every Hilbert {\bf A}-module $\cal M$ can be continued to an {\bf A}-valued inner product on it's {\bf A}-dual Banach {\bf A}-module $\cal M$' turning $\cal M$' to a self-dual Hilbert {\bf A}-module if and only if {\bf A} is monotone complete (or, equivalently, additively complete) generalizing a result of {\sc M.~Hamana} [Internat.~J.~Math.~3(1992), 185-204]. A classification of countably generated self-dual Hilbert {\bf A}-modules over monotone complete C*-algebras {\bf A} is established. The set of all bounded module operators ${\bf End}_A (\cal M)$ on self-dual Hilbert {\bf A}-modules $\cal M$ over monotone complete C*-algebras {\bf A} is proved again to be a monotone complete

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arXiv1994

Approximate unitary equivalence of homomorphisms from O_{\infty}

We prove that if two homomorphisms from O_{\infty} to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows that O_{\infty} is classifiable in the sense of Rordam. In particular, Rordam's classification theorem for direct limits of matrix algebras over even Cuntz algebras extends to direct limits involving both matrix algebras over even Cuntz algebras and corners of O_{\infty}, for which the K_0-group can be an arbitrary countable abelian group with no even torsion.

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arXiv1994

Crossed products whose primitive ideal spaces are generalized trivial $\widehat G$-bundles

We characterize when the primitive ideal space of a crossed product $\acg$ of a \cs-algebra $A$ by a locally compact abelian group $G$ is a $σ$-trivial $\ghat G$-space for the dual $\ghat G$-action. Specifically, we show that $\Prim(\acg)$ is $σ$-trivial if and only if the quasi-orbit space is Hausdorff, the map which assigns to each quasi-orbit $\w$ a certain subgroup $\ttg(α^\w)$ of the Connes spectrum of the system $(A_\w,G,α^\w)$ is continuous, and there is a generalized Green twisting map for $(A,G,α)$. Our proof requires a substantial generalization of a theorem of Olesen and Pedersen in which we show that there is a generalized Green twisting map for $(A,G,α)$ if and only if $\acg$ is isomorphic to a generalized induced algebra.

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arXiv1994

Landau Hamiltonians with Random Potentials: Localization and the Density of States

We prove the existence of localized states at the edges of the bands for the two-dimensional Landau Hamiltonian with a random potential, of arbitrary disorder, provided that the magnetic field is sufficiently large. The corresponding eigenfunctions decay exponentially with the magnetic field and distance. We also prove that the integrated density of states is Lipschitz continuous away from the Landau energies. The proof relies on a Wegner estimate for the finite-area magnetic Hamiltonians with random potentials and exponential decay estimates for the finite-area Green's functions. The proof of the decay estimates for the Green's functions uses fundamental results from two-dimensional bond percolation theory.

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arXiv1994

An existence result for non-coercive non-convex problems

We consider a class of one--dimensional non--convex non--coercive problems in the Calculus of Variations. We prove an existence result for this class of problems using a Liapunov type theorem on the range of non--atomic measures.

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arXiv1994

Time-Optimal Control Problem for the Swing and the Ski

This paper is concerned with a class of time-optimal control problems for the swing and the ski. We first consider the motion of a man standing on a swing. For simplicity, we neglect friction and air resistance and assume that the mass of the swinger is concentrated in his baricenter $B$. For the ski, we use the model considered by Aldo Bressan (1991), with a special approximation of the skier. Our main concern is the existence and the structure of time-optimal controls. As first step, we introduce an auxiliary control system with control entering linearly and show the relationships between this and the original system. We then perform a detailed study of the auxiliary system, using geometric techniques. In turn this provides accurate information on the time-optimal controls for the original system.

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arXiv1994

Berezin's quantization on flag manifolds and spherical modules

It is shown that the theory of spherical Harish-Chandra modules naturally provides the algebras of covariant, contravariant and mixed symbols on generalized flag manifolds. The general proof of the correspondence principle for all these symbol algebras is given.

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arXiv1994

On dense subspaces in a class of Fréchet function spaces on R^n

When dealing with concrete problems in a function space on R^n, it is sometimes helpful to have a dense subspace consisting of functions of a particular type, adapted to the problem under consideration. We give a theorem that allows one to write down many of such subspaces in commonly occurring Fréchet function spaces. These subspaces are all of the form $\{pf_0 | p\in{\cal P}\}$ where $f_0$ is a fixed function and ${\cal P}$ is an algebra of functions. Classical results like the Stone-Weierstrass theorem for polynomials and the completeness of the Hermite functions are related by this theorem.

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arXiv1995

Connection between Different Function Theories in Clifford Analysis

We describe an explicit connection between solutions to equations $Df=0$ (the Generalized Cauchy-Riemann equation) and $(D+M)f=0$, where operators $D$ and $M$ commute. The described connection allows to construct a ``function theory'' (the Cauchy theorem, the Cauchy integral, the Taylor and Laurent series etc.) for solutions of the second equation from the known function theory for solution of the first (generalized Cauchy-Riemann) equation. As well known, many physical equations related to the orthogonal group of rotations or the Lorentz group (the Dirac equation, the Maxwell equation etc.) can be naturally formulated in terms of the Clifford algebra. For them our approach gives an explicit connection between solutions with zero and non-zero mass (or external fields) and provides with a family of formulas for calculations. \keywords{Dirac equation with mass, Clifford analysis.} \AMSMSC{30G35}{34L40, 81Q05}

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arXiv1995

Lusin's C-property is not valid for functional Hilbert modules

We show that elements of Hilbert $A$-module obtained by completion of the space of square-integrable functions on a space with measure $X$ taking values in a $C^*$-algebra $A$ cannot be viewed as $A$-valued functions on $X$ defined almost everywhere

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arXiv1995

Unconditional Integrability for Dual Actions

The dual action of a locally compact abelian group, in the context of C*-algebraic bundles, is shown to satisfy an integrability property, similar to Rieffel's proper actions. The tools developed include a generalization of Bochner's integral as well as a Fourier inversion formula for operator valued maps.

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arXiv1995

C*-Algebras with the Approximate Positive Factorization Property

We say that a unital C*-algrebra A has the approximate positive factorization property (APFP) if every element of A is a norm limit of products of positive elements of A. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of C*-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial K_1, and stable rank 1. In the unital case, the K_0 group separates the tracial states. The APFP passes to matrix algebras. and if I is an ideal in A such that I and A/I have the APFP, then so does A. We also give some new examples of C*-algebras with the APFP, including type II_1 factors and infinite-dimensional simple unital direct limits with slow dimension growth, real rank zero, and trivial K_1 group. An infinite- dimensional simple unital direct limit with slow dimension growth and with the APFP must have real rank zero, but we also give examples of unital algebras with the APFP which do not have real rank zero. Our analysis also leads to the introduction of a new concept of rank for a C*-algebra that may be of interest in the future.

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