The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators

In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of finitely many time-frequency shifts of one function. We also estimate the coefficient decay of the inverse of finite linear combinations of time-frequency shifts.

     

On injective or dense-range operators leaving a given chain of subspaces invariant

In this paper we prove the existence of dense-range or one-to-one compact operators on a separable Banach space leaving a given finite chain of subspaces invariant. We use this result to prove that a semigroup of bounded operators is reducible if and only if there exists an appropriate one-to-one compact operator such that the collection of compact operators is reducible.

     

On the commutant and orbits of conjugation

In this note, we analyse the relationship between the commutant of a bounded linear operator and the algebra of similarity that was introduced in the late 70s as a characterization of nest algebras. Necessary and sufficient conditions are also obtained for an operator to commute with real scalar generalized operators in the sense of Colojoara-Foias in Banach spaces. In the second part, we analyse the relationship between the generalized inverse, the generalized commutant and the orbits of conjugation.

     

Sequential Fourier-Feynman transform, convolution and first variation

Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebra of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous. Also we study the relationships involving this transform and the first variation.

     

An elementary proof for a characterization of *-isomorphisms

We give an elementary proof of a result which characterizes onto *-isomorphisms of the algebra of all the bounded linear operators on a Hilbert space . A known proof of this result (Arveson, 1976) relies on the theory of irreducible representations of -algebras, whereas the proof given by us is based on elementary properties of operators on a Hilbert space which can be found in any introductory text on Functional Analysis.

     

Sheaf cohomology and free resolutions over exterior algebras

We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part' of a resolution over the exterior algebra.

We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree 1$"> in the complex yields a new complex which is eventually exact.

As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.

     

Bundles of -correspondences over directed graphs and a theorem of Ionescu

We give a short proof of a recent theorem of Ionescu which shows that the Cuntz-Pimsner -algebra of a certain correspondence associated to a Mauldin-Williams graph is isomorphic to the graph algebra.

     

Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space . For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone , we construct a unique ``symbol valued trace', which extends the -trace on operators of small order. This construction is in the spirit of a trace due to Kontsevich and Vishik in the nonparametric case. Our trace allows us to construct various trace functionals in a systematic way. Furthermore, we study the higher-dimensional eta-invariants on algebras with parameter space . Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over . The eta-invariant of this family coincides with the spectral eta-invariant of the operator.

     

Composition operators on spaces of entire functions

In this paper we study composition operators on spaces of entire functions. We determine which entire functions induce bounded composition operators on the Paley-Wiener space, , and on the spaces. In addition, we characterize compact composition operators on these spaces. We also study the cyclic properties of composition operators acting on .

     

Exemples séparant certaines classes d'algèbres topologiques

Esterle (1979) and Zelazko (1996 and 1990) gave an example of an algebra which cannot be topologized as a topological algebra (with jointly continuous multiplication), and Müller (in 1991) showed that there exists a topological algebra, which is indeed a locally bounded algebra, which cannot be topologized as a locally convex algebra.

To complete this study on the separation between the classes of algebras, we construct for every a locally bounded algebra which is -normed for every which cannot be topologized as a locally -convex algebra, and we deduce an example of a locally pseudoconvex algebra which cannot be topologized as a locally -convex algebra for every . We also show the existence of a topological algebra which cannot be topologized as a locally pseudoconvex algebra.

     

Minimal projective resolutions

In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the -algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop' conjecture.

     

Compact Schur multipliers

Compact Schur multipliers on the algebra of all bounded linear operators on an infinite-dimensional separable complex Hilbert space will be identified as the elements of the Haagerup tensor product (the completion of in the Haagerup norm). Other ideals of Schur multipliers related to compact operators will also be characterized.

     

On the length of the spectral sequence of a Lie algebra extension

The length of the spectral sequence of a Lie algebra extension is at most the dimension of the quotient algebra. We show that this bound can be attained for arbitrarily large quotient algebras even when the algebra is nilpotent and the extension splits.

     

Smoothed projections in finite element exterior calculus

The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, has proved to be an effective tool in finite element exterior calculus. The advantage of these operators is that they are bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non-quasi-uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space-dependent smoothing operator that commutes with the exterior derivative.

     

The arithmetic and combinatorics of buildings for

In this paper, we investigate both arithmetic and combinatorial aspects of buildings and associated Hecke operators for with a local field. We characterize the action of the affine Weyl group in terms of a symplectic basis for an apartment, characterize the special vertices as those which are self-dual with respect to the induced inner product, and establish a one-to-one correspondence between the special vertices in an apartment and the elements of the quotient .

We then give a natural representation of the local Hecke algebra over acting on the special vertices of the Bruhat-Tits building for . Finally, we give an application of the Hecke operators defined on the building by characterizing minimal walks on the building for .

     

An affine PI Hopf algebra not finite over a normal commutative Hopf subalgebra

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra , provides a noetherian prime counterexample.

     

Positivity preserving finite element approximation

We consider finite element operators defined on ``rough' functions in a bounded polyhedron in . Insisting on preserving positivity in the approximations, we discover an intriguing and basic difference between approximating functions which vanish on the boundary of and approximating general functions which do not. We give impossibility results for approximation of general functions to more than first order accuracy at extreme points of . We also give impossibility results about invariance of positive operators on finite element functions. This is in striking contrast to the well-studied case without positivity.

     

The identity is isolated among composition operators

Let be the Banach algebra of bounded holomorphic functions on the open unit ball of a Banach space. We show that the identity operator is an isolated point in the space of composition operators on . This answers a conjecture of Aron, Galindo and Lindström.

     

Quiver varieties and finite dimensional representations of quantum affine algebras

We study finite dimensional representations of the quantum affine algebra using geometry of quiver varieties introduced by the author.

As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.