Functional Analysis

We provide additional results in connection with Krein's formula, which describes the resolvent difference of two self-adjoint extensions A_1 and A_2 of a densely defined closed symmetric linear operator A with (possibly infinite) equal deficiency indices. In particular, we explicitly derive the linear fractional transformation relating the operator-valued Weyl-Titchmarsh M-functions M_1(z) and M_2(z) corresponding to A_1 and A_2.

The adiabatic approximation in quantum mechanics is considered in the case where the self-adjoint hamiltonian H 0 (t) , satisfying the usual spectral gap assumption in this context, is perturbed by a term of the form ϵ H 1 (t) . Here ϵ→0 is the adiabaticity parameter and H 1 (t) is a self-adjoint operator defined on a smaller domain than the domain of H 0 (t) . Thus the total hamiltonian H 0 (t)+ϵ H 1 (t) does not necessarily satisfy the gap assumption, ∀ϵ>0 . It is shown that an adiabatic theorem can be proven in this situation under reasonnable hypotheses. The problem considered can also be viewed as the study of a time-dependent system coupled to a time-dependent perturbation, in the limit of large coupling constant.

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann algebras of local observables associated with wedge regions act geometrically as pure Lorentz transformations. Such a property, satisfied by the local algebras generated by Wightman fields because of the Bisognano-Wichmann theorem, is regarded as a natural primitive assumption.

Suppose that (G,T) is a second countable locally compact transformation group given by a homomorphism $\ell:G\to\Homeo(T)$, and that A is a separable continuous-trace \cs-algebra with spectrum T . An action $\alpha:G\to\Aut(A)$ is said to cover ℓ if the induced action of G on T coincides with the original one. We prove that the set $\brgt$ of Morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product: $[A,\alpha][B,\beta] = [A\Ttensor B,\alpha\tensor\beta]$, and we refer to $\brgt$ as the Equivariant Brauer Group. We give a detailed analysis of the structure of $\brgt$ in terms of the Moore cohomology of the group G and the integral cohomology of the space T . Using this, we can characterize the stable continuous-trace \cs-algebras with spectrum T which admit actions covering ℓ . In particular, we prove that if $G=\R$, then every stable continuous-trace \cs-algebra admits an (essentially unique) action covering~ ℓ , thereby substantially improving results of Raeburn and Rosenberg. Versions of this paper in *.dvi and *.ps form are available via World wide web servers at this http URL

These lecture notes contain an exposition of basic ideas of K-theory and cyclic cohomology. I begin with a list of examples of various situations in which the K-functor of Grothendieck appears naturally, including the rudiments of the topological and algebraic K-theory, K-theory of C^*-algebras, and K-homology. I then discuss elementary properties of cyclic cohomology using the Cuntz-Quillen version of the calculus of noncommutative differential forms on an algebra. As an example of the relation between the two theories we describe the Chern homomorphism and various index-theorem type statements. The remainder of the notes contains some more detailed calculations in cyclic and reduced cyclic cohomology. A key tool in this part is Goodwillie's theorem on the cyclic complex of a semi-direct product algebra. The final chapter gives an exposition of the entire cyclic cohomology of Banach algebras from the point of view of supertraces on the Cuntz algebra. The results discussed here include the simplicial normalization of the entire cyclic cohomology, homotopy invariance and the action of derivations.

This is a contribution to the program of featuring even geometry as a ``collective effect in infinite-dimensional odd geometry,'' as suggested by Manin. We show that the (Gel'fand) spectrum of the locally convex nonstandard hull (in the sense of Luxemburg) of a grassmannian algebra with infinitely many odd generators contains a nontrivial analytic part.

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.

For a wide class of pairs of unbounded selfadjoint operators with bounded commutator we construct a K-theoretical integer invariant which is continuous, is equal to zero for commuting operators and is equal to one for the pair (x, i d/dx).

We consider two independent q-Gaussian random variables X and Y and a function f chosen in such a way that f(X) and X have the same distribution. For 0 < q < 1 we find that at least the fourth moments of X + Y and f(X) + Y are different. We conclude that no q-deformed convolution product can exist for functions of independent q-Gaussian random variables.

We explore a function theory connected with the principal series representation of SL(2,R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. Keywords: Complex analysis, Cauchy integral formula, Hardy space, Taylor expansion, Cauchy-Riemann equations, Dirac operator, group representations, SL(2,R), discrete series, principal series, wavelet transform, coherent states.