Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

Characterization of Amalgamated Free Blocks of a Graph von Neumann Algebra

We identify two noncommutative structures naturally associated with countable directed graphs. They are formulated in the language of operators on Hilbert spaces. If G is a countable directed graphs with its vertex set V(G) and its edge set E(G), then we associate partial isometries to the edges in E(G) and projections to the vertices in V(G). We construct a corresponding von Neumann algebra as a groupoid crossed product algebra of an arbitrary fixed von Neumann algebra M and the graph groupoid induced by G, via a graph-representation (or a groupoid action) α. Graph groupoids are well-determined (categorial) groupoids. The graph groupoid of G has its binary operation, called admissibility. This has concrete local parts , for all e ∈ E(G). We characterize of , induced by the local parts of , for all e ∈ E(G). We then characterize all amalgamated free blocks of . They are chracterized by well-known von Neumann algebras: the classical group crossed product algebras , and certain subalgebras (M) of operator-valued matricial algebra . This shows that graph von Neumann algebras identify the key properties of graph groupoids. Received: December 20, 2006. Revised: March 07, 2007. Accepted: March 13, 2007.     

Passive Systems with a Normal Main Operator and Quasi-selfadjoint Systems

Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass and the Q-function of T is given. Received: December 16, 2007., Accepted: March 4, 2008.     

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras of Bergman Type Operators with Piecewise Continuous Coefficients

The C ^*-algebra generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L ²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to simpler C ^*-algebras associated with points and pairs . Applying a symbol calculus for the abstract unital C ^*-algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras at points being the discontinuity points of coefficients. A symbol calculus for the C ^*-algebra is constructed and a Fredholm criterion for the operators is obtained.     

Tolokonnikov’s Lemma for Real H ∞ and the Real Disc Algebra

We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space , the space of bounded holomorphic (possibly operator-valued) functions on the unit disc all of whose matrix-entries (with respect to fixed orthonormal bases) are functions having real Fourier coefficients, or equivalently, each matrix entry f satisfies for all z ∈ . Tolokonnikov’s Lemma for means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in , such that F = [ f f _c ] for some f _c in . In control theory, Tolokonnikov’s Lemma implies that if a function has a right coprime factorization over , then it has a doubly coprime factorization in . We prove the lemma for the real disc algebra as well. In particular, and are Hermite rings. The work of the first author was supported by Magnus Ehrnrooth Foundation. Received: December 5, 2006. Revised: February 4, 2007.     

Minimal faithful representations of reductive Lie algebras

Let G and H be Lie groups with Lie algebras and . Let G be connected. We prove that a Lie algebra homomorphism is exact if and only if it is completely positive. The main resource is a corresponding theorem about representations on Hilbert spaces. This article summarizes the main results of [1]. Received: 6 December 2005     

On H. Weyl and J. Steiner Polynomials

The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set . A polynomial of this class describes the volume of the set V + tB ⁿ as a function of t, where t is a positive number and B ⁿ denotes the unit ball in . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold , where is isometrically embedded with positive codimension in . A Weyl polynomial describes the volume of a tubular neighborhood of its associated as a function of the tube’s radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of into for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial’s roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.

Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole day. The experiment yielded no results.

Submitted: March 5, 2007., Revised: February 1, 2008., Accepted: February 2, 2008.     

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

Let E be a non empty set, let P : = E × E, := {x × E|x ∈ E}, := {E × x|x ∈ E}, and := {C ∈ 2 ^P |∀X ∈ : |C ∩ X| = 1} and let . Then the quadruple resp. is called chain structure resp. maximal chain structure. We consider the maximal chain structure as an envelope of the chain structure . Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes. Here we study in detail the groups of automorphisms , , , related to a maximal chain structure . The set of all chains can be turned in a group such that the subgroup of generated by the left-, by the right-translations and by ι the inverse map of is isomorphic to (cf. (2.14)).     

An Operator Corona Theorem for Some Subspaces of <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

Let E, E_* be separable Hilbert spaces. If S is an open subset of , then denotes the space of all functions that are holomorphic in , and bounded and continuous on . In this article we prove the following results:

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

The C*-algebra generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C*-algebras associated with points and pairs We construct a symbol calculus for unital C*-algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈∂Π being the discontinuity points of coefficients. A symbol calculus for the C*- algebra and a Fredholm criterion for the operators are obtained. Finally, a C*-algebra isomorphism between the quotient algebra where is the ideal of compact operators, and its analogue for the unit disk is constructed.     

A note on weak convergence in L^{1}_{\rm loc}({\mathbb{R}})

Let be a sequence of Borel measurable functions satisfying, for a function the inequalities

C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

We establish a symbol calculus for the C*-subalgebra of generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators where is the Cauchy singular integral operator and The C*-algebra is invariant under the transformations

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

The Inverse Problem for Perturbed Harmonic Oscillator on the Half-Line with a Dirichlet Boundary Condition

We consider the perturbed harmonic oscillator in , where is a real-valued potential. We prove that the mapping spectral data = {eigenvalues of T _D } {norming constants} is one-to-one and onto. The complete characterization of the set of spectral data which corresponds to is given. Dedicated to Vladimir Buslaev on the occasion of his 70th birthday Submitted: September 27, 2006. Accepted: January 9, 2007.     

On the Matlis duals of local cohomology modules

Let ( ) be a commutative Noetherian local ring with non-zero identity, an ideal of R and M a finitely generated R-module with . Let D(–) := Hom _R (–, E) be the Matlis dual functor, where is the injective hull of the residue field . We show that, for a positive integer n, if there exists a regular sequence and the i-th local cohomology module H ⁱ _a (M) of M with respect to is zero for all i with i > n then The author was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 85130023). Received: 9 August 2006     

One-sided Grothendieck quotients

We introduce one-sided thick subcategories of an arbitrary preadditive category and define a quotient category . When is abelian, this concept specializes to Grothendieck’s quotient for two-sided thick . We determine the left noetherian rings for which the injective modules form a left thick subcategory. We exhibit a class of one-sided thick subcategories in categories of coherent functors which are ubiquitous in representation theory. Received: 14 November 2006 Revised: 12 March 2007     

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.