An infinite series of Kronecker conjugate polynomials

Let be a field of characteristic , a transcendental over , and be the absolute Galois group of . Then two non-constant polynomials are said to be Kronecker conjugate if an element of fixes a root of if and only if it fixes a root of . If is a number field, and where is the ring of integers of , then and are Kronecker conjugate if and only if the value set equals modulo all but finitely many non-zero prime ideals of . In 1968 H. Davenport suggested the study of this latter arithmetic property. The main progress is due to M. Fried, who showed that under certain assumptions the polynomials and differ by a linear substitution. Further, he found non-trivial examples where Kronecker conjugacy holds. Until now there were only finitely many known such examples. This paper provides the first infinite series. The main part of the construction is group theoretic.

     

Formulae and continuity for the index of subfactors

Let be an inclusion of -factors, the trace state of , and , the set of projections in and , respectively. We prove that the Jones index for the inclusion is

This formula is exploited to obtain continuity results for the index. In particular, we obtain a formula for the index which expresses in terms of the positions of and , in , when and are finite-dimensional -subalgebras with dense union in and , respectively.

     

Invariance of spectrum for representations of -algebras on Banach spaces

Let be a Banach space, a unital -algebra, and an injective, unital homomorphism. Suppose that there exists a function such that, for all , and all ,

(a) ,

(b) ,

(c) .
Then for all , the spectrum of in equals the spectrum of as a bounded linear operator on . If satisfies an additional requirement and is a -algebra, then the Taylor spectrum of a commuting -tuple of elements of equals the Taylor spectrum of the -tuple in the algebra of bounded operators on . Special cases of these results are (i) if is a closed subspace of a unital -algebra which contains as a unital -subalgebra such that , and only if , then for each , the spectrum of in is the same as the spectrum of left multiplication by on ; (ii) if is a unital -algebra and is an essential closed left ideal in , then an element of is invertible if and only if left multiplication by on is bijective; and (iii) if is a -algebra, is a Hilbert -module, and is an adjointable module map on , then the spectrum of in the -algebra of adjointable operators on is the same as the spectrum of as a bounded operator on . If the algebra of adjointable operators on is a -algebra, then the Taylor spectrum of a commuting -tuple of adjointable operators on is the same relative to the algebra of adjointable operators and relative to the algebra of all bounded operators on .

     

On weighted inequalities for singular integrals

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in then the one-sided condition, , is a sufficient condition for the singular integral to be bounded in , , or from into weak- if . This one-sided condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in . The two-sided version of this result is also obtained: Muckenhoupts condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in or 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 .

     

Semi-free actions of zero-dimensional compact groups on Menger compacta

Let be the -dimensional universal Menger compactum, a -set in and a metrizable zero-dimensional compact group with the unit. It is proved that there exists a semi-free -action on such that is the fixed point set of every . As a corollary, it follows that each compactum with can be embedded in as the fixed point set of some semi-free -action on .

     

Inequalities based on a generalization of concavity

The concept of concavity is generalized to functions, , satisfying order differential inequalities, , and homogeneous two-point boundary conditions, , for some . A piecewise polynomial, which bounds the function, , below, is constructed, and then is employed to obtain that , where max and denotes the supremum norm. An analogous inequality for a related Green's function is also obtained. These inequalities are useful in applications of certain cone theoretic fixed point theorems.

     

Collapsing successors of singulars

Let be a singular cardinal in , and let be a model such that for some -cardinal with . We apply Shelah's pcf theory to study this situation, and prove the following results. 1) is not a -c.c generic extension of . 2) There is no ``good scale for ' in , so in particular weak forms of square must fail at . 3) If then and also . 4) If then .

     

When do two rational functions have the same Julia set?

It is proved that non-exceptional rational functions and on the Riemann sphere have the same measure of maximal entropy iff there exist iterates of and of and natural numbers such that

If one assumes only that have the same Julia set and no singular or parabolic domains of normality for the iterates, one also proves .

     

Superrigid subgroups of solvable Lie groups

Let be a discrete subgroup of a simply connected, solvable Lie group , such that has the same Zariski closure as . If is any finite-dimensional representation of , we show that virtually extends to a continuous representation of . Furthermore, the image of is contained in the Zariski closure of the image of . When is not discrete, the same conclusions are true if we make the additional assumption that the closure of is a finite-index subgroup of (and is closed and is continuous).

     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

Invariants of Skew Derivations

If is an automorphism and is a -derivation of a ring , then the subring of invariants is the set The main result of this paper is Theorem. Let be a -derivation of an algebra over a commutative ring such that

for all , where and .

(i)

If , then .

(ii)

If is a -stable left ideal of such that , then .

This theorem generalizes results on the invariants of automorphisms and derivations.

     

Open covers and the square bracket partition relation

An open cover of an infinite separable metric space is an -cover of if and for every finite subset of there is a such that . Let be the collection of -covers of . We show that the partition relation holds if, and only if, the partition relation holds.

     

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 .

     

On -summable sequences in the range of a vector measure

Let . Among other results, we prove that a Banach space has the property that every sequence lies inside the range of an -valued measure if and only if, for all sequences in satisfying that the operator is 1-summing, the operator is nuclear, being the conjugate number for . We also prove that, if is an infinite-dimensional -space for , then can't have the above property for any .

     

A commutativity theorem for semibounded operators in hilbert space

Let and be semibounded (bounded from below) operators in a Hilbert space and a dense linear manifold contained in the domains of , , , and , and such that for all in . It is shown that if the restriction of to is essentially self-adjoint, then and are essentially self-adjoint and and commute, i.e. their spectral projections permute.

     

Point spectrum and mixed spectral types for rank one perturbations

We consider examples of rank one perturbations with a cyclic vector for . We prove that for any bounded measurable set , an interval, there exist so that
eigenvalue agrees with up to sets of Lebesgue measure zero. We also show that there exist examples where has a.c. spectrum for all , and for sets of 's of positive Lebesgue measure, also has point spectrum in , and for a set of 's of positive Lebesgue measure, also has singular continuous spectrum in .

     

On the lengths of closed geodesics on a two-sphere

Let be an isolated closed geodesic of length on a compact Riemannian manifold which is homologically visible in the dimension of its index, and for which the index of the iterates has the maximal possible growth rate. We show that has a sequence , , of prime closed geodesics of length where and . The hypotheses hold in particular when is a two-sphere and the ``shortest' Lusternik-Schnirelmann closed geodesic is isolated and ``nonrotating'.

     

Exact topological analogs to orthoposets

An arbitrary orthoposet is shown to be isomorphic to , being a subbasis of a Hausdorff topological space satisfying 1) , 2) , and 3) every covering of by elements of possesses an at most 2-element subcovering. The couple turns out to be unique.