Functional Analysis

The known correspondence between the Kronig-Penney model and certain Jacobi matrices is extended to a wide class of Schroedinger operators on graphs. Examples include rectangular lattices with and without a magnetic field, or comb-shaped graphs leading to a Maryland-type model.

A new approach to the Lance-Blecher theorem is presented resting on the interpretation of elements of Hilbert C*-module theory in terms of multiplier theory of operator C*-algebras: The Hilbert norm on a Hilbert C*-module allows to recover the values of the inducing C*-valued inner product in a unique way, and two Hilbert C*-modules {M_1, <.,.>_1}, {M_2, <.,.>_2} are isometrically isomorphic as Banach C*-modules if and only if there exists a bijective C*-linear map S: M_1 --> M_2 such that the identity <.,.>_1 \equiv <S(.),S(.)>_2 is valid. In particular, the values of a C*-valued inner product on a Hilbert C*-module are completely determined by the Hilbert norm induced from it. In addition, we obtain that two C*-valued inner products on a Banach C*-module inducing equivalent norms to the given one give rise to isometrically isomorphic Hilbert C*-modules if and only if the derived C*-algebras of ''compact'' module operators are *-isomorphic. The involution and the C*-norm of the C*-algebra of ''compact'' module operators on a Hilbert C*-module allow to recover its original C*-valued inner product up to the following equivalence relation: <.,.>_1 \sim <.,.>_2 if and only if there exists an invertible, positive element a of the center of the multiplier C*-algebra M(A) of A such that the identity <.,.>_1 \equiv a \cdot <.,.>_2 holds.

In this paper, we describe a natural method to extend left invariant weights on C*-algebraic quantum groups. This method is then used to improve the left invariance property of a left invariant weight. We also prove some kind of uniqueness result for left Haar weights on C*-algebraic quantum groups arising from algebraic ones.

The core of this article is a general theorem with a large number of specializations. Given a manifold N and a finite number of one-parameter groups of point transformations on N with generators Y, X (1) ,⋯, X (d) , we obtain, via functional integration over spaces of pointed paths on N (paths with one fixed point), a one-parameter group of functional operators acting on tensor or spinor fields on N . The generator of this group is a quadratic form in the Lie derivatives $\La_{X_{(\a)}}$ in the $X_{(\a)}$-direction plus a term linear in $\La_Y$. The basic functional integral is over L 2,1 paths $x: {\bf T} \ra N$ (continuous paths with square integrable first derivative). Although the integrator is invariant under time translation, the integral is powerful enough to be used for systems which are not time translation invariant. We give seven non trivial applications of the basic formula, and we compute its semiclassical expansion. The methods of proof are rigorous and combine Albeverio H\oegh-Krohn oscillatory integrals with Elworthy's parametrization of paths in a curved space. Unlike other approaches we solve Schrödinger type equations directly, rather than solving first diffusion equations and then using analytic continuation.

In this paper we discuss trace properties of d + -summable K -cycles considered by A.Connes in [\rfr(Conn4)]. More precisely we give a proof of a trace theorem on the algebra $\A$ of a K --cycle stated in [\rfr(Conn4)], namely we show that a natural functional on $\A$ is a trace functional. Then we discuss whether this functional gives a trace on the whole universal graded differential algebra $\Q(\A)$. On the one hand we prove that the regularity conditions on K -cycles considered in [\rfr(Conn4)] imply the trace property on $\Q(\A)$. On the other hand, by constructing an explicit counterexample, we remark that the sole K -cycle assumption is not sufficient for such a property to hold.

Conditions for a function (number sequence) to be a multiplier on the space of integrable functions on R ( T ) are given. This generalizes recent results of Giang and Moricz.

We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach-Lie groups and some monothetic subgroups of the unitary group of a separable Hilbert space.

We show that invariant states of C*-dynamical systems can be approximated in the weak*-topology by invariant pure states, or almost invariant pure states, under various circumstances.

This is a revised and corrected version of a preprint circulated in 1990 in which various non-self-adjoint limit algebras are classified. The principal invariant is the scaled K 0 group together with the algebraic order on the scale induced by partial isometries in the algebra.

Given a Fell bundle $\B$, over a discrete group Γ , we construct its reduced cross sectional algebra $C^*_r(\B)$, in analogy with the reduced crossed products defined for C*-dynamical systems. When the reduced and full cross sectional algebras of $\B$ are isomorphic, we say that the bundle is amenable. We then formulate an approximation property which we prove to be a sufficient condition for amenability. A theory of Γ -graded C*-algebras possessing a conditional expectation is developed, with an eye on the Fell bundle that one naturally associates to the grading. We show, for instance, that all such algebras are isomorphic to $C^*_r(\B)$, when the bundle is amenable. We also study induced ideals in graded C*-algebras and obtain a generalization of results of Stratila and Voiculescu on AF-algebras, and of Nica on quasi-lattice ordered groups. A brief comment is made on the relevance, to our theory, of a certain open problem in the theory of exact C*-algebras. An application is given to the case of an F n -grading of the Cuntz-Krieger algebras O A , recently discovered by Quigg and Raeburn. Specifically, we show that the Cuntz-Krieger bundle satisfies the approximation property, and hence is amenable, for all matrices A with entries in {0,1}, even if A does not satisfy the well known property (I) studied by Cuntz and Krieger in their paper.