Noncommutative spaces

Let be a von Neumann algebra with a faithful, finite, normal tracial state , and let be a finite, maximal subdiagonal algebra of . Let be the closure of in the noncommutative Lebesgue space . Then possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an factorization theorem, Nehari's Theorem, and harmonic conjugates which are bounded.

     

Finite and -resolvability

A topological space is -resolvable if has disjoint dense subsets. In this paper, we prove that if is -resolvable for each positive integer , then is -resolvable.

     

The relationship between fragmentable spaces and class spaces

In this note, we show that each fragmentable space introduced by Jayne and Rogers in 1985 is of class which was introduced by Kenderov in 1984. Our example shows that a space which is of class may not be a fragmentable space.

     

The classical Banach spaces

In this paper we study some structural and geometric properties of the quotient Banach spaces , where is an arbitrary set, is an Orlicz function, is the corresponding Orlicz space on and , being the ideal of elements with finite support. The results we obtain here extend and complete the ones obtained by Leonard and Whitfield (Rocky Mountain J. Math. 13 (1983), 531-539). We show that is not a dual space, that , if for every , that has no smooth points, that it cannot be renormed equivalently with a strictly convex or smooth norm, that is a Grothendieck space, etc.

     

On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

Products of images

Let be the \u{C}ech-Stone remainder . We show that there exists a large class of images of such that whenever is a subset of of cardinality at most the continuum, then is again an image of . The class contains all separable compact spaces, all compact spaces of weight at most and all perfectly normal compact spaces.

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

Immersed

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an -manifold into , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed -manifold in . We also study generic projections of an embedded -manifold in into and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in . The problem of lifting a map into to an embedding into is also studied.

     

Enriched -Partitions

An (ordinary) -partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard
tableaux associated with Schur's -functions. In this paper, we introduce and develop a theory of enriched -partitions; like ordinary -partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched -partitions given here are the tableaux associated with Schur's -functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.

     

On van Mill's example of a normed

In 1987 van Mill constructed an infinite-dimensional normed space which is not homeomorphic with the product . We give a short proof of this property of van Mill's example.

     

Stable orders of stunted lens spaces mod

Let be the stunted lens space mod and its stable order. If , then was determined by H. Toda (1963). In this paper, we determine the number for .

     

Support functionals and smooth points in abstract spaces

By presenting some properties of support functionals in abstract spaces, we get some sufficient and necessary conditions for smooth points in abstract (function) spaces. Moreover, the notion of the smallest support semi-norm is introduced and an explicit form for this functional in abstract function spaces is also given.

     

Criteria for -continuity

Bernoullicity is the strongest mixing property that a measure-theoretic dynamical system can have. This is known to be intimately connected to the so-called metric on processes, introduced by Ornstein. In this paper, we consider families of measures arising in a number of contexts and give conditions under which the measures depend -continuously on the parameters. At points where there is -continuity, it is often straightforward to establish that the measures have the Bernoulli property.

     

-regular Witt rings

We improve Kula's bounds on the size of possible -regular Witt rings.

     

Oscillatory singular integrals on and Hardy spaces

We consider boundedness properties of oscillatory singular integrals on and Hardy spaces. By constructing a phase function, we prove that boundedness may fail while boundedness holds for all . This shows that the theory and theory for such operators are fundamentally different.

     

Affine and homeomorphic embeddings into

It is shown that

(1)

a locally compact convex subset of a topological vector space that admits a sequence of continuous affine functionals separating points of affinely embeds into a Hilbert space;

(2)

an infinite-dimensional locally compact convex subset of a metric linear space has a central point;

(3)

every -compact locally convex metric linear space topologically embeds onto a pre-Hilbert space.

     

A uniform

A uniform estimate of Bessel functions is obtained, which is used to get a characterization of the measures on the unit sphere of in terms of the mixed norm of the Fourier transform of the measures.