Functional Analysis

In this note we explain the relationship of the Wodzicki residue of (certain powers of) an elliptic differential operator P \ acting on sections of a complex vector bundle E \ over a closed compact manifold M \ and the asymptotic expansion of the trace of the corresponding heat operator e −tP . In the special case of a generalized laplacian △ \ and dimM>2 , we thereby obtain a simple proof of the fact already shown in [KW], that the Wodzicki residue res( △ − n 2 +1 ) \ is the integral of the second coefficient of the heat kernel expansion of △ \ up to a proportional factor.

A version of Paley-Wiener like theorem for connected, simply connected nilpotent Lie groups is proven.

The Riemann-Lebesque Theorem is commonly proved in a few strokes using the theory of Lebesque integration. Here, the upper bound 2π| c k (f)|≤ S k (f)− s k (f) for the Fourier coefficients c k is proved in terms of majoring and minoring Riemann sums S k (f) and s f (k) , respectively, for Riemann integrable functions f(x) . This proof has been used in a course on methods of applied mathematics.

We show a noncommutative Rohlin type theorem for automorphisms of a certain class of purely infinite simple C ∗ -algebras. This class consists of the purely infinite unital simple C ∗ -algebras which are in the bootstrap category N and have trivial K 1 -groups.

We give a short and very general proof of the fact that the property of a dense Fréchet subalgebra of a Banach algebra being local, or closed under the holomorphic functional calculus in the Banach algebra, is preserved by tensoring with the n×n matrix algebra of the complex numbers.

In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to the theory of subfactors. An important role is played by a notion of amenability inspired by the work of Popa.

We prove a Wiener energy estimate for relaxed Dirichlet problems Lu+μu=ν in Ω , with L an uniformly elliptic operator with bounded coefficients, μ a measure of M 0 (Ω) , ν a Kato measure and Ω a bounded open set of R N , N≥2 . Choosing a particular μ , we obtain an energy estimate also for classical variational Dirichlet problems.

We explore the use of bi-orthogonal basis for continuous wavelet transformations, thus relaxing the so-called admissibility condition on the analyzing wavelet. As an application, we determine the eigenvalues and corresponding radial eigenfunctions of the Hamiltonian of relativistic Hydrogen-like atoms.

For a von Neumann algebra M on a Hilbert space, A. Connes has constructed a module S and a derivation of M into S, such that M is approximately finite dimensional if and only if that derivation is inner. The paper contains a generalization of this result to the situation with a 2-cocycle instead. The cocycle is the obvious generalization, and the module is closely related to Connes, but isn't a dual module.

In this paper we give a short survey of a connection between the theory of wavelets in L^2(R) and certain representations of the Cuntz algebra on L^2(T).