Renorming James tree space

We show that the James tree space can be renormed to be Lipschitz separated. This negatively answers the question of J. Borwein, J. Giles and J. Vanderwerff as to whether every Lipschitz separated Banach space is an Asplund space.

     

Weakly compact operators into sequence spaces: A counterexample

In a recent paper Gutiérrez and Villanueva have used, without giving a detailed proof, an analogue of a well-known result of Ryan characterizing the weakly compact operators from a Banach space into the space of null sequences in a Banach space . In this note a counterexample is given showing that in the statement of Gutiérrez and Villanueva an additional condition is needed.

     

Baire spaces and Vietoris hyperspaces

We prove that if the Vietoris hyperspace of all nonempty closed subsets of a space is Baire, then all finite powers of must be Baire spaces. In particular, there exists a metrizable Baire space whose Vietoris hyperspace is not Baire. This settles an open problem of R. A. McCoy stated in 1975.

     

Transfinite sequences of continuous and Baire class 1 functions

The set of continuous or Baire class 1 functions defined on a metric space is endowed with the natural pointwise partial order. We investigate how the possible lengths of well-ordered monotone sequences (with respect to this order) depend on the space .

     

The complete isomorphism class of an operator space

Suppose is an infinite-dimensional operator space and is a positive integer. We prove that for every there exists an operator space such that the formal identity map is a complete isomorphism, is an isometry, and . This provides a non-commutative counterpart to a recent result of W. Johnson and E. Odell.

     

Algebraic structure in the loop space homology Bockstein spectral sequence

Let be a finite, -dimensional, -connected CW complex. We prove the following theorem:

If is an odd prime, then the loop space homology Bockstein spectral sequence modulo is a spectral sequence of universal enveloping algebras over differential graded Lie algebras.

     

Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities

Let be a p. i. algebra with 1 in characteristic zero, satisfying a Capelli identity. Then the cocharacter sequence is asymptotic to a function of the form , where and .

     

On Bernstein type theorems in Finsler spaces with the volume form induced from the projective sphere bundle

By using the volume form induced from the projective sphere bundle of the Finsler manifold, we study the Finsler minimal submanifolds. It is proved that such a volume form for the Randers metric in a Randers space is just that for the Riemannian metric , and therefore the Bernstein type theorem in the special Randers space of dimension is true. Moreover, a Bernstein type theorem in the -dimensional Minkowski space is established by considering the volume form induced from the projective sphere bundle.

     

Analytic norms in Orlicz spaces

It is shown that an Orlicz sequence space admits an equivalent analytic renorming if and only if it is either isomorphic to or isomorphically polyhedral. As a consequence, we show that there exists a separable Banach space admitting an equivalent -Fréchet norm, but no equivalent analytic norm.

     

The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms

A uqr mapping of an -manifold is a mapping which is rational with respect to a bounded measurable conformal structure on . Remarkably, the only closed manifolds on which locally (but not globally) injective uqr mappings act are Euclidean space forms. We further characterize space forms admitting uniformly quasiregular self mappings and we show that the space forms admitting branched uqr maps are precisely the spherical space forms. We further show that every non-injective uqr map of a Euclidean space form is a quasiconformal conjugate of a conformal map. This is not true if the non-injective hypothesis is removed.

     

Homogeneity of powers of spaces and the character

A space is said to be power-homogeneous if some power of it is homogeneous. We prove that if a Hausdorff space of point-countable type is power-homogeneous, then, for every infinite cardinal , the set of points at which has a base of cardinality not greater than , is closed in . Every power-homogeneous linearly ordered topological space also has this property. Further, if a linearly ordered space of point-countable type is power-homogeneous, then is first countable.

     

On condensations of -spaces onto compacta

A condensation is a one-to-one onto mapping. It is established that, for each -compact metrizable space , the space of real-valued continuous functions on in the topology of pointwise convergence condenses onto a metrizable compactum. Note that not every Tychonoff space condenses onto a compactum.

     

Loop spaces and the compression theorem

For a smooth, finite-dimensional manifold with a submanifold we study the topology of the straight loop space , the space of loops whose intersections with are subject to a certain transversality condition. Our main tool is Rourke and Sanderson's compression theorem. We prove that the homotopy type of the straight loop space of a link in depends only on the number of link components.

     

Realizing alternating groups as monodromy groups of genus one covers

We prove that if , a generic Riemann surface of genus 1 admits a meromorphic function (i.e., an analytic branched cover of ) of degree such that every branch point has multiplicity and the monodromy group is the alternating group . To prove this theorem, we construct a Hurwitz space and show that it maps (generically) onto the genus one moduli space.

     

Filling analytic sets by the derivatives of -smooth bumps

If is an infinite-dimensional Banach space, with separable dual, and is an analytic set such that any point can be reached from  by a continuous path contained (except for the point ) in the interior of , then is the range of the derivative of a -smooth function on  with bounded nonempty support.

     

Strictly singular non-compact operators on hereditarily indecomposable Banach spaces

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic Banach space. The construction of this operator relies on the existence of transfinite -spreading models in the dual of the space.

     

Complemented invariant subspaces and interpolation sequences

It is shown that the invariant subspace of the Bergman space of the unit disc, generated by a finite union of Hardy interpolation sequences, is complemented in .

     

Embeddings of some classical Banach spaces into modulation spaces

We give sufficient conditions for a tempered distribution to belong to certain modulation spaces by showing embeddings of some Besov-Triebel-Lizorkin spaces into modulation spaces. As a consequence we have a new proof that the Hölder-Lipschitz space embeds into the modulation space when d$">. This embedding plays an important role in interpreting recent modulation space approaches to pseudodifferential operators.

     

Identification of some real interpolation spaces

We interrelate the real interpolation spaces associated with the couples , and , proving among others the identities

for all .

     

Universal spaces for almost -dimensionality

We find universal functions for the class of lower semi-continuous (LSC) functions with at most -dimensional domain. In an earlier paper we proved that a space is almost -dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most -dimensional domain. We conclude that the class of almost -dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdos space.