Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.     

Zero sums of idempotents in Banach algebras

The problem treated in this paper is the following.Let p ₁,...,p _k be idempotents in a Banach algebra B, and assume p ₁+...+p _k =0.Does it follow that p _j =0,j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four.     

Similarity implies homotopy of idempotents in Banach algebras of stable rank one

We establish the property stated in the title by means of an elementary construction: in a Banach algebra of stable rank one, any two similar idempotents can be connected by a piecewise affine path of idempotents consisting of at most 3 affine steps. Received: 1 February 2006     

Logarithmic residues in Banach algebras

Letf be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivativef′f ^?1 around a Cauchy domainD vanishes. Does it follow thatf takes invertible values on all ofD? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.     

An improvement of a recent closed graph theorem

We obtain a new closed graph theorem which is a substantial improvement of a recent result.     

Equivariant connected sums of compact self-dual manifolds

Partially supported by NSF grant #DMS-9306950 and UC Riverside grant #5-510000-07427-5     

Reproducing kernels for Hardy spaces on multiply connected domains

Associated with a boundedg-holed (g0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.     

On the approximate conformal mapping of multiply connected domains

Summary A method is presented for constructing approximations to the standard mappings for multiply connected regions given by Nehari [5]. The case of mapping onto a slit annulus is considered in detail, and computational results are presented for several examples.     

Embeddings of canonical modules and resolutions of connected sums

For an ideal $I_{m,n}$ generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of $R/I_{m,n}$ to $R/I_{m,n}$ itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by $I_{m,n}$. Using this embedding, we give a resolution of connected sums of several copies of certain Artin $\mathsf{k}$-algebras where $\mathsf{k}$ is a field.     

The K-Theory of C *-Algebras with Finite Dimensional Irreducible Representations

We study the K-theory of unital C^*-algebras A satisfying the condition that all irreducible representations are finite and of some bounded dimension. We construct computational tools, but show that K-theory is far from being able to distinguish between various interesting examples. For example, when the algebra A is n-homogeneous, i.e., all irreducible representations are exactly of dimension n, then K_*(A) is the topological K-theory of a related compact Hausdorff space, this generalises the classical Gelfand-Naimark theorem, but there are many inequivalent homogeneous algebras with the same related topological space. For general A we give a spectral sequence computing K_*(A) from a sequence of topological K-theories of related spaces. For A generated by two idempotents, this becomes a 6-term long exact sequence.     

On certain finite-dimensional algebras generated by two idempotents

This paper is concerned with algebras generated by two idempotents P and Q satisfying (PQ)^m=(QP)^m and (PQ)^m-1≠(QP)^m-1. The main result is the classification of all these algebras, implying that for each m?2 there exist exactly eight nonisomorphic copies. As an application, it is shown that if an element of such an algebra has a nondegenerate leading term, then it is group invertible, and a formula for the explicit computation of the group inverse is given.     

Interpolation of Hardy spaces on circular domains

In the paper, abstract interpolation of Hardy spaces on circular domains is studied. In particular, an extension of Peter Jones's result on the interpolation of Hardy spaces on the disc to multiply‐connected domains is presented. Moreover, some applications in approximation theory are given.     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market

Numerous researchers have applied the martingale approach for models driven by Lévy processes to study optimal investment problems. The aim of this paper is to apply the martingale approach to obtain a closed form solution for the optimal investment, consumption and insurance strategies of an individual in the presence of an insurable risk when the insurable risk and risky asset returns are described by Lévy processes and the utility is a constant absolute risk aversion (CARA). The model developed in this paper can potentially be applied to absorb large insurable losses in the absence of insurance protection and to examine the level of diminishing current utility and consumption.     

Positive Representations of L 1 of a Vector Measure

We characterize the vector measures n on a Banach lattice such that the map provides a quasi-norm which is equivalent to the canonical norm of the space L₁(n) of integrable functions as an specific type of transformations of positive vector measures that we call cone-open transformations. We also prove that a vector measure m on a Banach space X constructed as a cone-open transformation of a positive vector measure can be considered in some sense as a positive vector measure by defining a new order on X.     

Spaces of p-integrable Functions with Respect to a Vector Measure

We study several properties of the Banach lattices L^p (m) and L^p_w (m) of p-integrable scalar functions and weakly p-integrable scalar functions with respect to a countably additive vector measure m. The relation between these two spaces plays a fundamental role in our analysis. This research has been partially supported by La Consejería de Educatión y Ciencia de la Junta de Andalucía.