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.

     

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.

     

Compact operators and the geometric structure of -algebras

Given a -algebra and an element , we give necessary and sufficient geometric conditions equivalent to the existence of a representation of so that is a compact or a finite-rank operator. The implications of these criteria on the geometric structure of -algebras are also discussed.

     

-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.

     

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?

     

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 .

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

-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 .

     

The range of a ring homomorphism from a commutative -algebra

We prove that if a commutative semi-simple Banach algebra is the range of a ring homomorphism from a commutative -algebra, then is -equivalent, i.e. there are a commutative -algebra and a bicontinuous algebra isomorphism between and . In particular, it is shown that the group algebras , and the disc algebra are not ring homomorphic images of -algebras.

     

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

     

On purely inseparable extensions and their generators

Let be a field of characteristic and a polynomial algebra in two variables. By a -generator of we mean an element of for which there exist and such that . We also define a -line of to mean any element of whose coordinate ring is that of a -generator. Then we prove that if is such that is a -line of (where is an indeterminate over ), then is a -generator of . This is analogous to the well-known fact that if is such that is a line of , then is a variable of . We also prove that if is a -line of for which there exist and such that , then is in fact a -generator of .

     

and the associated line bundles

Let be a compact semi-simple Lie group, and let be a maximal unipotent subgroup of the complexified group . In this paper, we classify all the -invariant Kaehler structures on . For each Kaehler structure , let be the line bundle with connection whose curvature is . We then study the holomorphic sections of , which constitute a -representation space.

     

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 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 .

     

Integer-valued polynomials on Krull rings

If is a subring of a Krull ring such that is a valuation ring for every finite index , in Spec, we construct polynomials that map into the maximal possible (for a monic polynomial of fixed degree) power of , for all in Spec simultaneously. This gives a direct sum decomposition of Int, the -module of polynomials with coefficients in the quotient field of that map into , and a criterion when Int has a regular basis (one consisting of 1 polynomial of each non-negative degree).

     

Stability of semigroups commuting with a compact operator

It is proved that if are bounded -semigroups on Banach spaces and , resp., and , are bounded operators with dense ranges such that intertwines with and commutes with , then is strongly stable provided ---the generator of ---does not have eigenvalue on . An analogous result holds for power-bounded operators.

     

-regular maps on smooth manifolds

A continuous map is said to be -regular if whenever are distinct points of , then are linearly independent over . For smooth manifolds we obtain new lower bounds on the minimum for which a -regular map can exist in terms of the dual Stiefel-Whitney classes of .