Differences of vector-valued functions on topological groups

Let be a locally compact group equipped with right Haar measure. The right differences of functions on are defined by for . Let and suppose for some and all . We prove that is a right uniformly continuous function of . If is abelian and the Beurling spectrum does not contain the unit of the dual group , then we show . These results have analogues for functions , where is a separable or reflexive Banach space. Finally, we apply our methods to vector-valued right uniformly continuous differences and to absolutely continuous elements of left Banach -modules.

     

Banach spaces in which every -weakly summable sequence lies in the range of a vector measure

Let be a Banach space. For we prove that the identity map is -summing if and only if the operator is nuclear for every unconditionally summable sequence in , where is the conjugate number for . Using this result we find a characterization of Banach spaces in which every -weakly summable sequence lies inside the range of an -valued measure (equivalently, every -weakly summable sequence in , satisfying that the operator is compact, lies in the range of an -valued measure) with bounded variation. They are those Banach spaces such that the identity operator is -summing.

     

Vanishing of the leading term in Harish-Chandra's local character expansion

Harish-Chandra's formula for the character of an irreducible smooth representation of a reductive -adic group expresses near as a linear combination of the Fourier transforms of nilpotent -orbits in the Lie algebra of . In this note, we prove that if is tempered but not in the discrete series, then the coefficient attached to the zero nilpotent orbit vanishes.

     

sets of unbounded Loeb measure

For every and non-Borel subset of an internal set in a saturated nonstandard universe there exists an internal, unbounded, non-atomic measure so that is not finite for any Borel set in

     

-Fréchet-Urysohn property of Franklin compact spaces

Franklin compact spaces defined by maximal almost disjoint families of subsets of are considered from the view of its -sequentiality and -Fréchet-Urysohn-property for ultrafilters . Our principal results are the following: CH implies that for every -point there are a Franklin compact -Fréchet-Urysohn space and a Franklin compact space which is not -Fréchet-Urysohn; and, assuming CH, for every Franklin compact space there is a -point such that it is not -Fréchet-Urysohn. Some new problems are raised.

     

On rigidity of affine surfaces

Rigidity of nondegenerate Blaschke surfaces in is studied. The rigidity criteria are given in terms of , where is the curvature of the Blaschke connection . If the rank of is 2, then the surface is rigid. If , it is nonrigid. In the case where the rank of is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.

     

On the size of lemniscates of polynomials in one and several variables

In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set and , for a polynomial of degree . Usually, is taken to be monic, and either Cartan's Lemma or potential theory is used to estimate the size of , in terms of Hausdorff contents, planar Lebesgue measure , or logarithmic capacity cap. Here we normalize and show that cap and are the sharp estimates for the size of . Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on or product capacity and Favarov's capacity. Several of our estimates are sharp with respect to order in and .

     

On the dimension of infinite covers

We prove the following theorem and some generalizations. . Let be a connected CW complex which satisfies Poincaré duality of dimension . For any subgroup of , let denote the cover of corresponding to . If has infinite index in , then is homotopy equivalent to an -dimensional CW complex.

     

Non-commutative disc algebras and their representations

It is shown that the smallest closed subalgebra

generated by any sequence of isometries on a Hilbert space such that is completely isometrically isomorphic to the non-commutative ``disc' algebra introduced in Math. Scand. 68 (1991), 292--304. We also prove that for the Banach algebras and are not isomorphic. In particular, we give an example of two non-isomorphic Banach algebras which are completely isometrically embedded in each other. The completely bounded (contractive) representations of the ``disc' algebras on a Hilbert space are characterized. In particular, we prove that a sequence of operators is simultaneously similar to a contractive sequence (i.e., ) if and only if it is completely polynomially bounded. The first cohomology group of with coefficients in is calculated, showing, in particular, that the disc algebras are not amenable. Similar results are proved for the non-commutative Hardy algebras introduced in Math. Scand. 68 (1991), 292--304. The right joint spectrum of the left creation operators on the full Fock space is also determined.

     

Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any separable unital -subalgebra of ,

In this note, we exhibit a non-separable unital -subalgebra of for which (VDCT) fails.

     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

On contravariant finiteness of subcategories of modules of projective dimension

Let be an artin algebra. This paper presents a sufficient condition for the subcategory of to be contravariantly finite in , where is the subcategory of consisting of --modules of projective dimension less than or equal to . As an application of this condition it is shown that is contravariantly finite in for each when is stably equivalent to a hereditary algebra.

     

On a measure-theoretic problem of Arveson

A probability measure on a product space is said to be bistochastic with respect to measures on and on if the marginals and are exactly and . A solution is presented to a problem of Arveson about sets which are of measure zero for all such .

     

A rigidity theorem for the Clifford tori in

Let be the unit hypersphere in the 4-dimensional Euclidean space defined by . For each with , we denote by the Clifford torus in given by the equations and . The Clifford torus is a flat Riemannian manifold equipped with the metric induced by the inclusion map . In this note we prove the following rigidity theorem: If is an isometric embedding, then there exists an isometry of such that . We also show no flat torus with the intrinsic diameter is embeddable in except for a Clifford torus.

     

-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 .

     

Bounds for the operator norms of some Nörlund matrices

Suppose is a non-increasing sequence of non-negative numbers with , , , and is the lower triangular matrix defined by , , and , . We show that the operator norm of as a linear operator on is no greater than , for ; this generalizes, yet again, Hardy's inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the tend to a positive limit, the operator norm of on is exactly . We also give some cases when the operator norm of on is less than .

     

A topological characterization of linearity for quasi-traces

Let be a -algebra, and let be a (local) quasi-trace on . Then is linear if, and only if, the restriction of to the closed unit ball of is uniformly weakly continuous.