Functional Analysis

The paper contains examples of Fredholm n-tuples of operators that are of index 0 but cannot be perturbed by compact operators to n-tuples with exact Koszul complex.

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 $d_id_j^*-\sum_{r,s} t_{js}^{ir} d_r^*d_s=\delta_{ij}\id$, where the matrix t ir js 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.

We consider the (real) nonlinear wave equation □u+ m 2 u+λ u 3 =0,m>0,λ>0, on four-\-dimensional Minkowski space. We introduce the complex structure and show that the (nonlinear) operator of dynamics, the wave and scattering operators define complex analytic maps on the space of initial Cauchy data with finite energy. In other words, let R(φ,π)=φ+i μ −1 π be the map of initial data on the positive frequency part of the solution of the free Klein-\-Gordon equation with these initial data. The operators RU(t) R −1 , RW R −1 , and RS R −1 are defined correctly and are complex analytic on the complex Hilbert space $H^1({\R}^3,\C).$

This paper studies direct limits of full upper triangular matrix algebras with embeddings which are not *-extendible. A representation of the limit algebra is found so that the generated C*-algebra is the C*-envelope. Some examples are described.

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}

In a previous paper [ B,V-1 ] , an algebra of holomorphic ``perikernels'' on a complexified hyperboloid X (c) d−1 (in C d ) has been introduced; each perikernel K can be seen as the analytic continuation of a kernel K on the unit sphere S d−1 in an appropriate ``cut-domain'' , while the jump of K across the corresponding ``cut'' defines a Volterra kernel K (in the sense of J. Faraut [ Fa-1 ] ) on the one-sheeted hyperboloid X d−1 (in R d ). \par In the present paper, we obtain results of harmonic analysis for classes of perikernels which are invariant under the group SO(d,C) and of moderate growth at infinity. For each perikernel K in such a class, the Fourier-Legendre coefficients of the corresponding kernel K on S d−1 admit a carlsonian analytic interpolation F ~ (λ) in a half-plane, which is the ``spherical Laplace transform''\ of the associated Volterra kernel K on X d−1 . Moreover, the composition law K= K 1 ∗ (c) K 2 for perikernels (interpreted in terms of convolutions for the

We consider a quantum mechanical particle living on a graph and discuss the behaviour of its wavefunction at graph vertices. In addition to the standard (or delta type) boundary conditions with continuous wavefunctions, we investigate two types of a singular coupling which are analogous to the delta' interaction and its symmetrized version for particle on a line. We show that these couplings can be used to model graph superlattices in which point junctions are replaced by complicated geometric scatterers. We also discuss the band spectra for rectangular lattices with the mentioned couplings. We show that they roughly correspond to their Kronig-Penney analogues: the delta' lattices have bands whose widths are asymptotically bounded and do not approach zero, while the delta lattice gap widths are bounded. However, if the lattice-spacing ratio is an irrational number badly approximable by rationals, and the delta coupling constant is small enough, the delta lattice has no ggaps above the threshold of the spectrum. On the other hand, infinitely many gaps emerge above a critical value of the coupling constant; for almost all ratios this value is zero.

We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its ``purification''. As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein-Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its ``purification'' induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III_1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.

Given a measurable twisted action of a second-countable, locally compact group G on a separable C*-algebra A, we prove the existence of a topology on AxG making it a continuous bundle, whose cross sectional C*-algebra is isomorphic to the Busby--Smith--Packer--Raeburn crossed product.

In this paper we analyze the convergence of the splitting method for shallow water equations. In particular, we give an analytical estimation of the time step which is necessary for the convergence and then we study the behaviour of the motion of the shallow water in the Venice lagoon by using the splitting method with a finite element space discretization. The numerical calculations show that the splitting method is convergent if the time step of the first part is sufficiently small and that it gives a good agreement with the experimental data.