Bilocal derivations of standard operator algebras

In this paper, we shall show the following two results: (1) Let be a standard operator algebra with , if is a linear mapping on which satisfies that maps into for all , then is of the form for some in . (2) Let be a Hilbert space, if is a norm-continuous linear mapping on which satisfies that maps into for all self-adjoint projection in , then is of the form for some in .

     

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 .

     

A note on GK dimension of skew polynomial extensions

Let be a finitely generated commutative domain over an algebraically closed field , an algebra endomorphism of , and a -derivation of . Then if and only if is locally algebraic in the sense that every finite dimensional subspace of is contained in a finite dimensional -stable subspace.

Similarly, if is a finitely generated field over , a -endomorphism of , and a -derivation of , then if and only if is an automorphism of finite order.

     

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.

     

Volumes of hyperbolic Haken manifolds, II

We show that if is a closed hyperbolic 3-manifold and if has a non-abelian free quotient, then the volume of is greater than . If, in addition, contains no genus- surface groups, then the volume of is greater than . Using these results we show that if there are infinitely many primitive homology classes in which are not represented by fibroids, then the volume of is greater than .

     

Product of distinct simple integrally closed ideals in 2-dimensional regular local rings

Let be a two-dimensional regular local ring and an -primary integrally closed ideal in . In this paper, we give equivalent conditions for to be a product of distinct simple -primary integrally closed ideals (i.e., , where are distinct simple -primary integrally closed ideals of ) in terms of the regularity of for all and in terms of how to choose a minimal generating set for over its minimal reductions.

     

On Bernstein-Sato polynomials

We show that for fixed and the set of Bernstein-Sato polynomials of all the polynomials in at most variables of degrees at most is finite. As a corollary, we show that there exists an integer depending only on and such that generates as a module over the ring of the -linear differential operators of , where is an arbitrary field of characteristic 0, is the ring of polynomials in variables over and is an arbitrary non-zero polynomial of degree at most .

     

On a theorem of Privalov and normal functions

A well known result of Privalov asserts that if is a function which is analytic in the unit disc , then has a continuous extension to the closed unit disc and its boundary function is absolutely continuous if and only if belongs to the Hardy space . In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, we prove that for any positive continuous function defined in with , as , there exists a function analytic in which is not a normal function and with the property that , for all sufficiently close to .

     

Noncommutative spaces

Let be a von Neumann algebra with a faithful, finite, normal tracial state , and let be a finite, maximal subdiagonal algebra of . Let be the closure of in the noncommutative Lebesgue space . Then possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an factorization theorem, Nehari's Theorem, and harmonic conjugates which are bounded.

     

Some Schrödinger operators with dense point spectrum

Given any sequence of positive energies and any monotone function on with , , we can find a potential on such that are eigenvalues of and .

     

There is no degree invariant half-jump

We prove that there is no degree invariant solution to Post's problem that always gives an intermediate degree. In fact, assuming definable determinacy, if is any definable operator on degrees such that on a cone then is low or high on a cone of degrees, i.e., there is a degree such that for every or for every .

     

Calibrated thin

Let be a compact metric space, and let be a calibrated thin -ideal. Then is . This solves an open problem, which was posed by Kechris, Louveau and Woodin. Using our result we obtain a new proof of Kaufman's theorem concerning -sets and -sets.

     

muni de l'arithmétique faible de Penzin est décidable

We prove the decidability of the additive ordered group equipped with a predicate for , the multiplication restricted to and the -adic valuation ranging in .

     

Hilbert Cmodules in which all closed submodules are complemented

Let be a Calgebra. If there exists a full Hilbert module such that for each closed submodule , then is *-isomorphic to a Calgebra of (not neccesarily all) compact operators on a Hilbert space.

     

Poincaré flows

We study flows on a compact metric space with the property that corresponding to every non-zero element of there is either a cross section associated with or one associated with . We obtain necessary and sufficient conditions for this to hold; on the -dimensional torus these conditions take a classical form.

     

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.

     

On normal solvability of the Riemann problem with singular coefficient

Suppose is a singular matrix function on a simple, closed, rectifiable contour . We present a necessary and sufficient condition for normal solvability of the Riemann problem with coefficient in the case where admits a spectral (or generalized Wiener-Hopf) factorization with essentially bounded. The boundedness of is not required when takes injective values a.e. on .

     

Orbifolds with lower Ricci curvature bounds

We show that the first betti number of a compact Riemannian orbifold with Ricci curvature and diameter is bounded above by a constant , depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the is bounded above by the dimension , and that if, in addition, , then is a flat torus .

     

Isotopy and identities in alternative algebras

In this paper we show how to construct an isomorphism between an alternative algebra over a field of characteristic and its isotope , where is an element of Zhevlakov's radical of . This leads to the equivalence of any polynomial identity in alternative algebras and the isotope identity .

     

On the Ganea conjecture for manifolds

Using a result of Singhof, we prove that provided is a connected closed PL manifold with and is the -sphere, .