Measurable bundles of Banach lattices

The aim of this paper is to outline a portion of the theory of liftable bundles of Banach lattices and to give some applications to representation of dominated operators.     

On holomorphic Banach vector bundles over Banach spaces

Let X be a Banach space with a countable unconditional basis (e.g., X = ℓ ₂ Hilbert space), Ω ⊂ X pseudoconvex open, E → Ω a locally trivial holomorphic vector bundle with a Banach space Z for fiber type, the sheaf of germs of holomorphic sections of E → Ω, and Z ₁ the Banach space . Then and Ω × Z ₁ are holomorphically isomorphic, is acyclic and E is so to speak stably trivial over Ω in a generalized sense. We also show that if E is continuously trivial over Ω, then E is holomorphically trivial over Ω. In particular, if Z = ℓ ₂ or Ω is contractible, then E is holomorphically trivial over Ω. Some applications are also given. To my dear little daughter, Sári Mangala, on her third birthday. Supported in part by a Research Initiation Grant from Georgia State University.     

The Socle and finite dimensionality of some Banach algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact groupG, G is compact if there exists a measure μ in Soc(L ¹(G)) such that μ(G) ≠ 0. We also prove thatG is finite if Soc(M(G)) is closed and every nonzero left ideal inM(G) contains a minimal left ideal.     

Finite separability in varieties of associative rings

A subset M of a universal algebra A is called finitely separated in A if, for any element x∈A/M, there exists an homomorphism ϕ of A into a finite algebra, for which ϕ(x)∋ϕ(M). A ring is said to be S-separable (R-separable) if its subrings (resp., right ideals) are all finitely separated in it. We give equational (in the language of identities) and indicator (in the language of “prohibited” rings) characterizations of varieties consisting of S-separable (R-separable) rings. Moreover, varieties are described in which, not all, but finitely generated rings only share the properties mentioned. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 201–227, march–April, 1999.     

A reflexivity criterion for stalks of a Banach bundle

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 36, No. 4, pp. 851–856, July–August, 1995.     

Characterization of finite dimensionality for semi-simple Banach algebras

Some conditions which are equivalent to finite dimensionality for a semi-simple Banach algebra A are given, and simplification of the proofs of previously known results are included. These conditions include local finiteness, spectral finiteness and the condition that socle A exists and is equal to A.     

Distribution of finite-dimensional and separable stalks of an ample Banach bundle

The topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle are finite-dimensional or separable. A connection is established between the property of the stalks of a bundle to be finite-dimensional or separable with the analogous property of the stalks of the ample hull of the bundle. A new criterion is obtained for existence of the dual bundle.     

A splitting theorem for holomorphic Banach bundles

This paper is motivated by Grothendieck’s splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold X and a holomorphic Banach bundle E → X that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that E splits into the sum of a finite rank bundle and a trivial bundle, provided .     

Lindelöf type of generalization of separability in Banach spaces

We will introduce the countable separation property (CSP) of Banach spaces X, which is defined as follows: X has CSP if each family E of closed linear subspaces of X whose intersection is the zero space contains a countable subfamily E₀ with the same intersection. All separable Banach spaces have CSP and plenty of examples of non-separable CSP spaces are provided. Connections of CSP with Marku?evi?-bases, Corson property and related geometric issues are discussed.     

Finite representation of Banach spaces-smoothness

We discuss the finite representability of a Banach space E in another Banach space F, assuming that F satisfies certain smoothness conditions. We apply these results to develop a classification of superreflexive spaces into isomorphic classes, called k-superreflexive spaces (k = 1, 2,…, ∞); and we derive a strong converse to Dvoretsky's near-sphericity theorem. Further, our main theorem complements an important recent result of Krivine.     

Finite quotients of rings and applications to subgroup separability of linear groups

In this paper we apply results from algebraic number theory to subgroup separability of linear groups. We then state applications to subgroup separability of free products with amalgamation of hyperbolic -manifold groups.

     

Finite spectra and quasinilpotent equivalence in Banach algebras

This paper further investigates the implications of quasinilpotent equivalence for (pairs of) elements belonging to the socle of a semisimple Banach algebra. Specifically, not only does quasinilpotent equivalence of two socle elements imply spectral equality, but also the trace, determinant and spectral multiplicities of the elements must agree. It is hence shown that quasinilpotent equivalence is established by a weaker formula (than that of the spectral semidistance). More generally, in the second part, we show that two elements possessing finite spectra are quasinilpotent equivalent if and only if they share the same set of Riesz projections. This is then used to obtain further characterizations in a number of general, as well as more specific situations. Thirdly, we show that the ideas in the preceding sections turn out to be useful in the case of C*-algebras, but now for elements with infinite spectra; we give two results which may indicate a direction for further research.     

Finite termination of the proximal point algorithm in Banach spaces

In this paper, we show that the convex optimization problem can be solved by the proximal point algorithm in a finite number of steps under the assumption that the solution set is a set of weak sharp minima.     

Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices

The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted ℓ ^p -spaces. In particular, the stability of the finite section method on ℓ ² implies its stability on weighted ℓ ^p -spaces. Our approach uses recent results from the theory of Banach algebras of matrices with off-diagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of unstructured non-hermitian matrices as well as to least squares problems.     

Cyclic conjugacy separability and conjugacy separability of certain HNN extensions

Abstract

In this note, we give a criterion for certain HNN extensions of cyclic conjugacy separable (respectively conjugacy separable) groups with infinite cyclic associated subgroups to be again cyclic conjugacy separable (respectively conjugacy separable).

Communicated by Alexander Olshanskii     

Finite dimensionality and compactness of attractors for von Karman equations with nonlinear dissipation

Asymptotic behaviour of dynamics governed by PDE system describing nonlinear vibrations of a shell immersed in a supersonic gas is considered. The undelying dynamics is modeled by a nonlinear hyperbolil-like shell equation with a nonlinear dissipation. It is shown that all finite energy ("weak") solutions converge to a global, compact attractor which is also finite-dimensional.