Functional Analysis

We prove that if two nonzero homomorphisms from the Cuntz algebra O_infinity 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_infinity 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_infinity for which the K_0 group can be an arbitrary countable abelian group with no even torsion.

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.

We prove that every AF-algebra is isomorphic to a crossed product of a commutative AF-algebra by a partial automorphism. The case of UHF-algebras is treated in detail.

Given an elliptic operator~ L on a bounded domain~ Ω⊆ R n , and a positive Radon measure~ μ on~ Ω , not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains~ Ω h ⊆Ω with the following property: for every~ f∈ H −1 (Ω) the sequence~ u h of the solutions of the Dirichlet problems~ L u h =f in~ Ω h , u h =0 on~ ∂ Ω h , extended to 0 in~ Ω∖ Ω h , converges to the solution of the \lq\lq relaxed Dirichlet problem\rq\rq\ Lu+μu=f in~ Ω , u=0 on~ ∂Ω .

Consider the Laplacian in a straight planar strip of width d , with the Neumann boundary condition at a segment of length 2a of one of the boundaries, and Dirichlet otherwise. For small enough a this operator has a single eigenvalue ϵ(a) ; we show that there are positive c 1 , c 2 such that − c 1 a 4 ≤ϵ(a)− (π/d) 2 ≤− c 2 a 4 . An analogous conclusion holds for a pair of Dirichlet strips, of generally different widths, with a window of length 2a in the common boundary.

We provide an overview of the connections between Bell's inequalities and algebraic structure.

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.

We consider the Dirichlet Laplacian for a strip in $\,\R^2$ with one straight boundary and a width a(1+λf(x)) , where f is a smooth function of a compact support with a length 2b . We show that in the critical case, ∫ b −b f(x)dx=0 , the operator has no bound states for small |λ| if b<( 3 – √ /4)a . On the other hand, a weakly bound state exists provided ∥ f ′ ∥<1.56 a −1 ∥f∥ ; in that case there are positive c 1 , c 2 such that the corresponding eigenvalue satisfies − c 1 λ 4 ≤ϵ(λ)−(π/a ) 2 ≤− c 2 λ 4 for all |λ| sufficiently small.

We consider the discrete spectrum of the Dirichlet Laplacian on a manifold consisting of two adjacent parallel strips or planar layers coupled by a finite number N of windows in the common boundary. If the windows are small enough, there is just one isolated eigenvalue. We find upper and lower asymptotic bounds on the gap between the eigenvalue and the essential spectrum in the planar case, as well as for N=1 in three dimensions. Based on these results, we formulate a conjecture on the weak-coupling asymptotic behaviour of such bound states.

We consider hilbert spaces of holomorphic functions in Cartan domains (in particular in ball and polydisk) and operator of restriction of holomorphic function to a submanifold in Shilov boundary. We discuss conditions when this operator exists. Using such 'trace theorems' it is possible to construct discrete increments to spectra of some unitary representation and to catch singular unitary representations in the spectra. We also discuss spectral problems related to Berezin type kernels on riemann symmetric spaces.