Local derivations of reflexive algebras

Let be a reflexive algebra in Banach space such that both and in , the invariant subspace lattice of , then every derivation of into itself is spatial. Furthermore, if is additionally reflexive, then the set of all inner derivations of into itself is topologically algebraically reflexive.

     

Fixed points of the mapping class group in the moduli spaces

Let be a compact oriented surface with or without boundary components. In this note we prove that if then there exist infinitely many integers such that there is a point in the moduli space of irreducible flat connections on which is fixed by any orientation preserving diffeomorphism of . Secondly we prove that for each orientation preserving diffeomorphism of and each there is some such that has a fixed point in the moduli space of irreducible flat connections on . Thirdly we prove that for all there exists an integer such that the 'th power of any diffeomorphism fixes a certain point in the moduli space of irreducible flat connections on .

     

Less Saturated Ideals

We prove the following:

(1)

If is weakly inaccessible then is not -saturated.

(2)

If is weakly inaccessible and is regular then is not -saturated.

(3)

If is singular then is not -saturated.

Combining this with previous results of Shelah, one obtains the following:

(A)

If then is not -saturated.

(B)

If then is not -saturated.

     

Weakly coupled bound states in quantum waveguides

We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a
bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain obtained by adding an arbitrarily small ``bump' to the tube (i.e., , open and connected, outside a bounded region) produces at least one positive eigenvalue below the essential spectrum of the Dirichlet Laplacian . For sufficiently small ( abbreviating Lebesgue measure), we prove uniqueness of the ground state of and derive the ``weak coupling' result using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube with Dirichlet boundary conditions at , replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment , , of . If denotes the resulting Laplace operator in , then has a discrete eigenvalue in no matter how small is.

     

A convolution estimate for a measure on a curve in

Let and fix an interval . If is the operator on defined by , then maps into .

     

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 .

     

A note on the fundamental groups of manifolds with almost nonnegative curvature

We show that given and , there exists a positive number such that if a closed -manifold satisfies and , then is almost abelian.

     

A note on the relative class number in function fields

Let be a finite field, , and . Let be the field extension of obtained by adjoining the -torsion on the Carlitz module. The class number of can be written as a product . The number is called the relative class number. In this paper a formula for is derived which is the analogue of the Maillet determinant formula for the relative class number of the cyclotomic field of -th roots of unity. Some consequences of this formula are also derived.

     

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.