A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

It is established that if and are connected locally symmetric spaces of noncompact type where has finite volume, and is a totally geodesic immersion, then the closure of in is an immersed ``algebraic' submanifold. It is also shown that if in addition, the real ranks of and are equal, then the the closure of in is a totally geodesic submanifold of The proof is a straightforward application of Ratner's Theorem combined with the structure theory of symmetric spaces.

     

Relative modular theory for a weight

We consider the balanced weight of a semi-finite weight and a (not necessarily faithful) normal positive functional on a von Neumann algebra , and discuss how the modular operator and the modular conjugation are described under the identification of the standard Hilbert space with , where is the support projection of and .

     

A mean value theorem on bounded symmetric domains

Let be a Cartan domain of rank and genus and , , the Berezin transform on ; the number can be interpreted as a certain invariant-mean-value of a function around . We show that a Lebesgue integrable function satisfying , , must be -harmonic. In a sense, this result is reminiscent of Delsarte's two-radius mean-value theorem for ordinary harmonic functions on the complex -space , but with the role of radius played by the quantity .

     

Polynomially convex hulls of graphs on the sphere

Let be the graph of a continuous map of the unit sphere of into , and the polynomially convex hull of . Several examples of for are given, which have different properties from the known ones for .

     

On the semisimplicity of pure sheaves

We obtain a criteria for a pure sheaf to be semisimple. As a corollary, we prove the following: Let and be two schemes over a finite field , and let be a proper smooth morphism. Assume is normal and geometrically connected, and assume there exists a closed point in such that the Frobenius automorphism acts semisimply on , where is the geometric fiber of at (this last assumption is unnecessary if the semisimplicity conjecture is true). Then is a semisimple sheaf on . This verifies a conjecture of Grothendieck and Serre provided the semisimplicity conjecture holds. As an application, we prove that the galois representations of function fields associated to the -adic cohomologies of surfaces are semisimple. We also get a theorem of Zarhin about the semisimplicity of the Galois representations of function fields arising from abelian varieties. The proof relies heavily on Deligne's work on Weil conjectures.

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .

     

On the dimension of almost -dimensional spaces

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are -dimensional by proving that almost -dimensional spaces are at most -dimensional. These homeomorphism groups are almost -dimensional and at least -dimensional by classical results of Brechner and Bestvina. In this note we prove that almost -dimensional spaces for are -dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is -dimensional.

     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Global bifurcation in generic systems of nonlinear Sturm-Liouville problems

We consider the system of coupled nonlinear Sturm-Liouville boundary value problems where , are real spectral parameters. It will be shown that if the functions and are `generic' then for all integers , there are smooth 2-dimensional manifolds , , of `semi-trivial' solutions of the system which bifurcate from the eigenvalues , , of , , respectively. Furthermore, there are smooth curves , , along which secondary bifurcations take place, giving rise to smooth, 2-dimensional manifolds of `non-trivial' solutions. It is shown that there is a single such manifold, , which `links' the curves , . Nodal properties of solutions on and global properties of are also discussed.

     

Semilinear transformations

In previous papers, nice trinomial equations were given for unramified coverings of the once punctured affine line in nonzero characteristic with the projective general group and the general linear group as Galois groups where is any integer and is any power of . These Galois groups were calculated over an algebraically closed ground field. Here we show that, when calculated over the prime field, as Galois groups we get the projective general semilinear group and the general semilinear group . We also obtain the semilinear versions of the local coverings considered in previous papers.

     

The maximal ideal space of with respect to the Hadamard product

It is shown that the space of all regular maximal ideals in the Banach algebra with respect to the Hadamard product is isomorphic to The multiplicative functionals are exactly the evaluations at the -th Taylor coefficient. It is a consequence that for a given function in and for a function holomorphic in a neighborhood of with and for all the function is in

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

Finite rank perturbations and distribution theory

Perturbations of a selfadjoint operator by symmetric finite rank operators from to are studied. The finite dimensional family of selfadjoint extensions determined by is given explicitly.

     

Gaussian estimates and regularized groups

We show that if a bounded analytic semigroup on satisfies a Gaussian estimate of order and is the generator of its consistent semigroup on , then generates a -regularized group on where . We obtain the estimate of () and the -independence of , and give applications to Schrödinger operators and elliptic operators of higher order.

     

Poncelet's theorem in space

A plane polygon inscribed in a conic and circumscribed to a conic can be continuously `rotated', as it were. One of the many proofs consists in viewing each side of as translation by a torsion point of an elliptic curve. In the -space version, involving torsion points of hyperelliptic Jacobians, there is a -dimensional family of rotations, where of the hyperelliptic curve; the polygon is now inscribed in one and circumscribed to quadrics.

     

The connected stable rank of the purely infinite simple -algebras

Suppose that is a unital purely infinite simple -algebra. If the class [1] of the unit 1 in has torsion, then ; if [1] is torsion-free in , then . If is a non-unital purely infinite simple -algebra, then .

     

A class of differentiable toral maps which are topologically mixing

We show that on the 2-torus there exists a open set of regular maps such that every map belonging to is topologically mixing but is not Anosov. It was shown by Mañé that this property fails for the class of toral diffeomorphisms, but that the property does hold for the class of diffeomorphisms on the 3-torus . Recently Bonatti and Diaz proved that the second result of Mañé is also true for the class of diffeomorphisms on the -torus ().

     

Some corollaries of Frobenius' normal -complement theorem

For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

     

A finiteness theorem for a class of exponential congruences

For given elements and belonging to the ring of integers of a number field we consider the set of all tuples in for which divides for any and prove under some reasonable assumptions that the set of solutions is finite.