Remarks on commuting exponentials in Banach algebras, II

Suppose that and are elements of a complex unital Banach algebra such that the spectrum of is -congruence-free and . We show that then is the sum of nilpotent elements. If denotes the spectral radius of , then we show that the additional assumption implies that

     

Integral representation of linear functionals on spaces of unbounded functions

Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.

     

A note on Pontrjagin forms

Let be a principal bundle over a manifold of dimension , and let be its -dimensional Pontrjagin class. In this paper, we aim at answering the following question: Which representatives of the class can be realised as the Pontrjagin form of some connection on ?

     

Relative boundedness and relative compactness for linear operators in Banach spaces

If and are linear operators acting between Banach spaces, we show that compactness of relative to does not in general imply that has -bound zero. We do, however, give conditions under which the above implication is valid.

     

A co-Frobenius Hopf algebra with a separable Galois extension is finite

If is a co-Frobenius Hopf algebra over a field, having a Galois -object which is separable over , its ring of coinvariants, then is finite dimensional.

     

Convergence of paths for pseudo-contractive mappings in Banach spaces

Let be a real Banach space, let be a closed convex subset of , and let , from into , be a pseudo-contractive mapping (i.e. for all and 1)$">. Suppose the space has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path , , defined by the equation is continuous and strongly converges to a fixed point of as , provided that satisfies the weakly inward condition.

     

Semigroups and weights for group representations

Let be a finite group. Consider a pair of linear characters of subgroups of with and agreeing on . Naturally associated with is a finite monoid . Semigroup representation theory then yields a representation of . If is irreducible, we say that is a weight for . When the underlying field is the field of complex numbers, we obtain a formula for the character of in terms of and . We go on to construct weights for some familiar group representations.

     

Pyramidal vectors and smooth functions on Banach spaces

We prove that if , are Banach spaces such that has nontrivial cotype and has trivial cotype, then smooth functions from into have a kind of ``harmonic" behaviour. More precisely, we show that if is a bounded open subset of and is -smooth with uniformly continuous Fréchet derivative, then is dense in . We also give a short proof of a recent result of P. Hájek.

     

Betti numbers of modules of essentially monomial type

Let be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of -modules of essentially monomial type. These modules are defined with respect to various -regular sequences. For example, finite length modules of monomial type over regular local rings of dimension are modules of essentially monomial type with respect to -regular sequences of length . If a module is of essentially monomial type with respect to an -regular sequence of length , then the rank of its -th syzygy is at least and its -th Betti number is at least .

     

Four-genus and four-dimensional clasp number of a knot

For a knot in the -sphere, by using the linking form on the first homology group of the double branched cover of the -sphere, we investigate some numerical invariants, -genus , nonorientable -genus and -dimensional clasp number , defined from the four-dimensional viewpoint. T. Shibuya gave an inequality , and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.

     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

     

The Mardesic factorization theorem for extension theory and c-separation

We shall prove a type of Mardesic factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces.

A corollary to our result will be that for any weight and any finitely homotopy dominated CW-complex , there exists a Hausdorff compactum with weight and which is universal for the property and weight . The condition means that for every closed subset of and every map , there exists a map which is an extension of . The universality means that for every compact Hausdorff space whose weight is and for which is true, there is an embedding of into .

We shall show, on the other hand, that there exists a CW-complex which is not finitely homotopy dominated but which has the property that for each weight , there exists a Hausdorff compactum which is universal for the property and weight .

     

Asymptotically isometric copies of czynski

We introduce the notion of a Banach space containing an asymptotically isometric copy of . A well known result of Bessaga and Peczynski states a Banach space contains a complemented isomorphic copy of if and only if contains an isomorphic copy of if and only if contains an isomorphic copy of . We prove an asymptotically isometric analogue of this result.

     

A ``universal' dynamical system generated by a continuous map of the interval

In this paper we show that there is a continuous map of the interval such that any -limit set of any continuous map can be transformed by a homeomorphism to an -limit set of . Consequently, any nowhere-dense compact set and any finite union of compact intervals is a homeomorphic copy of an -limit set of .

     

Fixed point and selection theorems in hyperconvex spaces

It is shown that a set-valued mapping of a hyperconvex metric space which takes values in the space of nonempty externally hyperconvex subsets of always has a lipschitzian single valued selection which satisfies for all . (Here denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded -lipschitzian self-mappings of is itself hyperconvex. Several related results are also obtained.

     

Strongly asymptotically stable Frobenius-Perron operators

Let be a -finite measure space and let be a Frobenius-Perron operator.

In 1997 Bartoszek and Brown proved that if overlaps supports and if there exists , 0$"> on , such that , then is (strongly) asymptotically stable.

In the note we prove that instead of assuming that 0$"> on , it is enough to assume that and . More precisely, we prove that is asymptotically stable if and only if overlaps supports and there exists , , , such that .

     

On Burgess's theorem and related problems

Let be a graph. We determine all graphs which are -like. We also prove that if are graphs, then in order that each -like continuum be -indecomposable for some it is necessary and sufficient that if is a graph, then is not -like for some integer with . This generalizes a well known theorem of Burgess.

     

Incompressible surfaces in handlebodies and closed 3-manifolds of Heegaard genus 2

In this paper, we shall prove that for any integer 0$">, 1) a handlebody of genus 2 contains a separating incompressible surface of genus , 2) there exists a closed 3-manifold of Heegaard genus which contains a separating incompressible surface of genus .

     

On singular critical points of positive operators in Krein spaces

We give an example of a positive operator in a Krein space with the following properties: the nonzero spectrum of consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of are uniformly bounded and the point is a singular critical point of