Functional Analysis

For an infinitesimal symplectic action of a Lie algebra ${\goth g}$ on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and Lie algebra ${\goth g}$. We obtain its second crossed product in case ${\goth g}=R$ and show an infinitesimal version for a theorem type of Takesaki duality.

In view of the applications to the asymptotic analysis of a family of obstacle problems, we consider a class of convex local functionals F(u,A) , defined for all functions u in a suitable vector valued Sobolev space and for all open sets A in R n . Sufficient conditions are given in order to obtain an integral representation of the form F(u,A)= ∫ A f(x,u(x))dμ+ν(A) , where μ and ν are Borel measures and f is convex in the second variable.

The interpolated free group factors L(F_r), 1 < r <= \infty, are defined and proofs of their properties with respect to compression by projections and taking free products are proved. Hence it follows that all the free group factor are isomorphic to each other or none of them are. These factors were defined and these properties were proved independently by F. Radulescu, and those given in this paper are equivalent, but use different techniques. Specifically, we develop algebraic techniques that allow us to show that R*R = L(F_2), where R is the hyperfinite II_1 factor.

We present an interpolation between the bosonic and fermionic relations. This interpolation is given by an object which we call `generalized Brownian motion' and which is characterized by a generalization of the pairing rule for the calculation of the moments of bosonic and fermionic fields. We develop some basic theory for such generalized Brownian motions and consider more closely one example, which turns out to be intimately connected with Voiculescu's concept of `free product'.

The purpose of this paper is to describe some conjectures and results on the existence and uniqueness of invariant measures on formal completions of Kac-Moody groups and associated homogeneous spaces. Existence is rigorously established in all affine type cases.

In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras O_N, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L^2(T) by (S_i\xi)(z)=m_i(z)\xi(z^N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over L^2(T). This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations of O_N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.

We embed the quantum Heisenberg manifold in a crossed product algebra. This enables us to show that, in the irrational case, all tracial states on $\dc$ induce the same homomorphism on the K_0-group. We conclude that two irrational quantum Heisenberg manifolds $\dc$ and D c μ ′ ν ′ are isomorphic if and only if the parameters (μ,ν) and ( μ ′ , ν ′ ) belong to the same orbit under the usual action of $GL_2(\ZZ)$ on the torus.

Let M be a Banach C*-module over a C*-algebra A carrying two A -valued inner products <.,. > 1 , <.,. > 2 which induce equivalent to the given one norms on M . Then the appropriate unital C*-algebras of adjointable bounded A -linear operators on the Hilbert A -modules {M,<.,. > 1 } and {M,<.,. > 2 } are shown to be ∗ -isomorphic if and only if there exists a bounded A -linear isomorphism S of these two Hilbert A -modules satisfying the identity <.,. > 2 ≡<S(.),S(.) > 1 . This result extends other equivalent descriptions due to L.~G.~Brown, H.~Lin and E.~C.~Lance. An example of two non-isomorphic Hilbert C*-modules with ∗ -isomorphic C*-algebras of ''compact''/adjointable bounded module operators is indicated.

We study a class of representations of the Cuntz algebras O_N, N=2,3,..., acting on L^2(T) where T=R/2\pi Z. The representations arise in wavelet theory, but are of independent interest. We find and describe the decomposition into irreducibles, and show how the O_N-irreducibles decompose when restricted to the subalgebra UHF_N\subset O_N of gauge-invariant elements; and we show that the whole structure is accounted for by arithmetic and combinatorial properties of the integers Z. We have general results on a class of representations of O_N on Hilbert space H such that the generators S_i as operators permute the elements in some orthonormal basis for H. We then use this to extend our results from L^2(T) to L^2(T^d), d>1 ; even to L^2(\mathbf{T}) where \mathbf{T} is some fractal version of the torus which carries more of the algebraic information encoded in our representations.

C*-algebras generalizing Cuntz-Krieger algebras can be associated to hyperbolic homeomorphisms of compact metric spaces. They satisfy a non-commutative form of Spanier-Whitehead duality with respect to K-theory. We prove this for the case of subshifts of finite type. The special feature of the present situation is that the constructions are all done on the full Fock space and are very explicit, while the general theorem requires much more abstract machinery.