On invariants dual to the Bass numbers

Let be a commutative Noetherian ring, and let be an -module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers were defined for all primes and all integers by use of the minimal injective resolution of . It is well known that . On the other hand, if is finitely generated, the Betti numbers are defined by the minimal free resolution of over the local ring . In an earlier paper of the second author (1995), using the flat covers of modules, the invariants were defined by the minimal flat resolution of over Gorenstein rings. The invariants were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that

for any cotorsion module . Comparing this with the computation of the Bass numbers, we see that is replaced by and the localization is replaced by (which was called the colocalization of at the prime ideal by Melkersson and Schenzel).

     

Sequential type Korovkin theorem on -test functions

Let be a sequence of bounded linear operators on such that and for every . It is proved that for every .

     

Rudin's orthogonality problem and the Nevanlinna counting function

Let be a holomorphic function taking the open unit disk into itself. We show that the set of nonnegative powers of is orthogonal in if and only if the Nevanlinna counting function of , , is essentially radial. As a corollary, we obtain that the orthogonality of for a univalent implies for some constant . We also show that if is orthogonal, then the closure of must be a disk.

     

Continuous functions on compact groups

We show that every scalar valued continuous function on a compact group may be written as for all , where are vectors in a separable Hilbert space , and is a strongly continuous unitary valued function on which is a product of unitary representations and antirepresentations of on . This product is countable, but always converges uniformly on . Moreover the supremum norm of is matched by . This may be viewed as a `Fourier product representation' for , and complements a result of Eymard for the Fourier algebra. For `Fourier polynomials' we show that the Hilbert space may be taken to be finite dimensional, and the product finite, which is more or less obvious except in that we are able to match the correct norm. The main ingredients of the proof are the Peter-Weyl theory, Tannaka's duality theorem, and a method developed with Paulsen using a characterization of operator algebras due to the author, Ruan and Sinclair. We also give the analogues of these formulae for compact quantum groups.

     

On the von Neumann-Jordan constant for Banach spaces

Let be the von Neumann-Jordan constant for a Banach space . It is known that for any Banach space ; and is a Hilbert space if and only if . We show that: (i) If is uniformly convex, is less than two; and conversely the condition implies that admits an equivalent uniformly convex norm. Hence, denoting by the infimum of all von Neumann-Jordan constants for equivalent norms of , is super-reflexive if and only if . (ii) If , (the same value as that of -space), is of Rademacher type and cotype for any with , where ; the converse holds if is a Banach lattice and is finitely representable in or .

     

A Strict Version of the Non-commutative Urysohn Lemma

Given a pair , of -commuting, hereditary -subalgebras of a unital -algebra , such that is -unital and , there is an element in , with , such that is strictly positive in and is strictly positive in in . Moreover, is strictly positive in in .

     

Inner derivations on ultraprime normed algebras

We show that, for every ultraprime Banach algebra , there exists a positive number satisfying for all in , where denotes the centre of and denotes the inner derivation on induced by . Moreover, the number depends only on the ``constant of ultraprimeness' of .

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

Combinatorial aspects of F-sets

We discuss F filters and show that the minimum size of a filter base generating an undiagonalizable filter included in some F filter is the better known bounded evasion number . An application to -sets from trigonometric series is given by showing that if is an -set and has size less than , then is again an -set.

     

Commutative group algebras of -summable abelian groups

In this note we study the commutative modular and semisimple group rings of -summable abelian -groups, which group class was introduced by R. Linton and Ch. Megibben. It is proved that is -summable if and only if is -summable, provided is an abelian group and is a commutative ring with 1 of prime characteristic , having a trivial nilradical. If is a -summable -group and the group algebras and over a field of characteristic are -isomorphic, then is a -summable -group, too. In particular provided is totally projective of a countable length.

Moreover, when is a first kind field with respect to and is -torsion, is -summable if and only if is a direct sum of cyclic groups.

     

Hypercomplex structures on four-dimensional Lie groups

The purpose of this paper is to classify invariant hypercomplex structures on a -dimensional real Lie group . It is shown that the -dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group of the quaternions, the multiplicative group of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyperbolic spaces, and , respectively, and the semidirect product . We show that the spaces and possess an of (inequivalent) invariant hypercomplex structures while the remaining groups have only one, up to equivalence. Finally, the corresponding hyperhermitian -manifolds are determined.

     

Hypersurfaces in a sphere with constant mean curvature

Let be a closed hypersurface of constant mean curvature immersed in the unit sphere . Denote by the square of the length of its second fundamental form. If , is a small hypersphere in . We also characterize all with .

     

Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

A classical result of W. Bade states that if is any complete Boolean algebra of projections in an arbitrary Banach space then, for every there exists an element (called a Bade functional for with respect to in the dual space , with the following two properties: (i) is non-negative on and, (ii) whenever satisfies It is shown that a Fréchet space has this property if and only if it does not contain an isomorphic copy of the sequence space

     

Beals-Cordes-type characterizations of pseudodifferential operators

We show that, if is the representation of on given by (2.11), and is a bounded operator on , then belongs to if and only if

is a function on with values in the Banach space .

     

Hyperspaces and open monotone maps of hereditarily indecomposable continua

We prove the following theorems:

Theorem 1. Let be an -dimensional hereditarily indecomposable continuum. Then there exist -dimensional hereditarily indecomposable continua and monotone maps such that is an embedding and the space of all subcontinua of is embeddable in by .

Theorem 2. For every open monotone map with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum with there exists a -dimensional subcontinuum such that and the restriction of to is also monotone and open.

The connection between these theorems and other results in Hyperspace theory is studied.

     

Primitive characters of subgroups of -groups

Let be an -group, let be a subnormal subgroup of , and let be a Hall subgroup of . If the character is primitive, then is a power of 2. Furthermore, if is odd, then .

     

An invariance of domain result for multi-valued maximal monotone operators whose domains do not necessarily contain any open sets

Let be a real, reflexive, locally uniformly convex Banach space with locally uniformly convex. Let be a maximal monotone operator and open and bounded. Assume that is pathwise connected and such that and Then If, moreover, is of type () on then may be replaced above by The significance of this result lies in the fact that it holds for multi-valued mappings which do not have to satisfy It has also been used in this paper in order to establish a general ``invariance of domain' result for maximal monotone operators, and may be applied to a greater variety of problems involving partial differential equations. No degree theory has been used. In addition to the above, necessary and sufficient conditions are given for the existence of a zero (in an open and bounded set ) of a completely continuous perturbation of a maximal monotone operator such that is locally monotone on

     

The structure of hypersurfaces with some curvature conditions

Let be a hypersurface in , and let denote the mean curvature and the scalar curvature of respectively. We show that if is compact and , then is diffeomorphic to . Also we prove that if is complete, is constant and , then is or or .

     

On functions arising as potentials on spaces of homogeneous type

On a space of homogeneous type we consider functions in , , which are potentials of order of functions. We show that these functions belong to the class of smooth functions of Calderón-Scott. This result has applications to tangential convergence.