Polynomials in

A positive semidefinite polynomial is said to be if is a sum of squares in , but no fewer, and is a sum of squares in , but no fewer. If is not a sum of polynomial squares, then we set .

It is known that if , then . The Motzkin polynomial is known to be . We present a family of polynomials and a family of polynomials. Thus, a positive semidefinite polynomial in may be a sum of three rational squares, but not a sum of polynomial squares. This resolves a problem posed by Choi, Lam, Reznick, and Rosenberg.

     

Sufficient conditions for a linear functional to be multiplicative

A commutative Banach algebra is said to have the property if the following holds: Let be a closed subspace of finite codimension such that, for every , the Gelfand transform has at least distinct zeros in , the maximal ideal space of . Then there exists a subset of of cardinality such that vanishes on , the set of common zeros of . In this paper we show that if is compact and nowhere dense, then , the uniform closure of the space of rational functions with poles off , has the property for all . We also investigate the property for the algebra of real continuous functions on a compact Hausdorff space.

     

Between the Lindelöf property and countable tightness

We consider a class of compact spaces for which the space of probability Radon measures on has countable tightness in the topology. We show that that class contains those compact zero-dimensional spaces for which is weakly Lindelöf, and, under MA + CH, all compact spaces with having property (C) of Corson.

     

Derived subgroups and centers of capable groups

A group is said to be capable if it is isomorphic to the central factor group for some group . It is shown in this paper that if is finite and capable, then the index of the center in is bounded above by some function of the order of the derived subgroup . If is cyclic and its elements of order are central, then, in fact, .

     

A remark on generalised Putnam-Fuglede theorems

Given , the algebra of operators on a Hilbert space , define and by and . Let and be two classes of operators strictly larger than the class of normal operators. Define (resp., if (resp., for all and . This note shows that the equivalence holds for a number of the commonly considered classes of operators.

     

Finite groups and the fixed points of coprime automorphisms

Let be a prime, and let be a finite -group acted on by an elementary abelian -group . The following results are proved:

1. If and is nilpotent of class at most for any , then the group is nilpotent of -bounded class.

2. If and is nilpotent of class at most for any , then the derived group is nilpotent of -bounded class.

     

Acyclicity criteria for complexes associated with an alternating map

When is a Gorenstein ideal of grade in a local ring , results of Boffi and Sánchez, and of Kustin and Ulrich show that for each one can construct in a canonical way a finite free complex that is ``approximately" a resolution for the ideal . Kustin and Ulrich also provide a sufficient condition that is acyclic, and a sufficient condition that is a resolution of . We complete these two acyclicity criteria by showing that the corresponding sufficient conditions are also necessary.

     

Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Let be a -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ). Let be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset of and let be arbitrary. Then the iteration sequence defined by , converges strongly to a fixed point of , provided that and have certain properties. If is a Hilbert space, then converges strongly to the unique fixed point of closest to .

     

Asymptotic behavior of Fourier transforms of self-similar measures

Let be a self-similar probability measure on satisfying where 0$"> and Let be the Fourier transform of A necessary and sufficient condition for to approach zero at infinity is given. In particular, if and for then 0$"> if and only if is a PV-number and is not a factor of . This generalizes the corresponding theorem of Erdös and Salem for the case

     

Nonstandard topologies with bases that consist only of standard sets

Let be an infinite set, a set of pseudo-metrics on and If is limited (finite) for every and every then, for each we can define a pseudo-metric on by writing st We investigate the conditions under which the topology induced on by has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem.

     

On stability of -semigroups

We prove that if is a -semigroup on a Hilbert space , then (a) if and only if , for all , and (b) is exponentially stable if and only if , for all . Analogous, but weaker, statements also hold for semigroups on Banach spaces.

     

A remark on a theorem of J. Tits

Let be a rank two Chevalley group and be the corresponding Moufang polygon. J. Tits proved that is the universal completion of the amalgam formed by three subgroups of : the stabilizer of a point of , the stabilizer of a line incident with , and the stabilizer of an apartment passing through and . We prove a slightly stronger result, in which the exact structure of is not required. Our result can be used in conjunction with the ``weak -pair" theorem of Delgado and Stellmacher in order to identify subgroups of finite groups generated by minimal parabolics.

     

A matrix-valued Choquet-Deny theorem

Let be a positive matrix-valued measure on a locally compact abelian group such that is the identity matrix. We give a necessary and sufficient condition on for the absence of a bounded non-constant matrix-valued function on satisfying the convolution equation . This extends Choquet and Deny's theorem for real-valued functions on .

     

On the Berezin-Toeplitz calculus

We consider the problem of composing Berezin-Toeplitz operators on the Hilbert space of Gaussian square-integrable entire functions on complex -space, . For several interesting algebras of functions on , we have for all in the algebra, where is the Berezin-Toeplitz operator associated with and is a ``twisted' associative product on the algebra of functions. On the other hand, there is a function for which is bounded but for any .

     

Low-dimensional unitary representations of

We characterize all simple unitarizable representations of the braid group on complex vector spaces of dimension . In particular, we prove that if and denote the two generating twists of , then a simple representation (for ) is unitarizable if and only if the eigenvalues of are distinct, satisfy and 0$"> for , where the are functions of the eigenvalues, explicitly described in this paper.

     

A splitting theorem for degrees

We prove that, for any , and with _{T}A\oplus U$"> and r.e., in , there are pairs and such that ; ; and, for any and from and any set , if and , then . We then deduce that for any degrees , , and such that and are recursive in , , and is into , can be split over avoiding . This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.

     

Some results related to the Corach-Porta-Recht inequality

Let be the algebra of all bounded operators on a complex Hilbert space and let be an invertible self-adjoint (or skew-symmetric) operator of . Corach-Porta-Recht proved that

The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht Inequality; (ii) a necessary condition (resp. necessary and sufficient condition, when for the invertible positive operators to satisfy the operator-norm inequality for all in ; (iii) a necessary and sufficient condition for the invertible operator in to satisfy

     

A short proof of ergodicity of Babillot-Ledrappier measures

Let be a compact manifold, and let be a transitive homologically full Anosov flow on . Let be a -cover for , and let be the lift of to . Babillot and Ledrappier exhibited a family of measures on , which are invariant and ergodic with respect to the strong stable foliation of . We provide a new short proof of ergodicity.

     

The Fefferman-Stein type inequality for the Kakeya maximal operator

Let , , be the Kakeya maximal operator defined as the supremum of averages over tubes of the eccentricity . We shall prove the so-called Fefferman-Stein type inequality for ,

in the range , , with some constants and independent of and the weight .