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.

     

Unbounded quasi-integrals

Let be a locally compact Hausdorff space. We define a quasi-measure in , a quasi-integral on , and a quasi-integral on . We show that all quasi-integrals on are bounded, continuity properties of the quasi-integral on , representation of quasi-integrals on in terms of quasi-measures, and unique extension of quasi-integrals on to .

     

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.

     

Real groups transitive on complex flag manifolds

Let be a complex flag manifold. The compact real form of is transitive on . If is a noncompact real form, such transitivity is rare but occasionally happens. Here we work out a complete list of Lie subgroups of transitive on and pick out the cases that are noncompact real forms of .

     

Le degré de Lindelöf est -invariant

Two Tychonoff spaces and are said to be -equivalent if and are linearly homeomorphic. It is shown that if and are -equivalent, then the Lindelöf numbers of and are the same. The proof given is a strengthening of the one given by N.V. Velichko to show that the Lindelöf property is -invariant.

     

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, .

     

Spectral synthesis for

Let be a locally compact group, the Fourier algebra of and the von Neumann algebra generated by the left regular representation of . We introduce the notion of -spectral set and -Ditkin set when is an -invariant linear subspace of , thus providing a unified approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of -spectral sets and -Ditkin sets, and an injection theorem for -spectral sets.

     

Hahn-Banach operators

We consider real spaces only.

Definition. An operator between Banach spaces and is called a Hahn-Banach operator if for every isometric embedding of the space into a Banach space there exists a norm-preserving extension of to .

A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to characterize pairs of finite-dimensional normed spaces such that there exists a Hahn-Banach operator of rank . The latter result is a generalization of a recent result due to B. L. Chalmers and B. Shekhtman.

     

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.

     

Approximating discrete valuation rings by regular local rings

Let be a field of characteristic zero and let be a discrete rank-one valuation domain containing with . Assume that the fraction field of has finite transcendence degree over . For every positive integer , we prove that can be realized as a directed union of regular local -subalgebras of of dimension .

     

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.

     

On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .

We prove that the following are equivalent:

1. is Lipschitz homeomorphic to .

2. is Lipschitz homeomorphic to .

3. is Lipschitz homeomorphic to .

This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.

     

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 .

     

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 .

     

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.

     

Lattice properties of subspace families in an inner product space

Let be a separable inner product space over the field of real numbers. Let (resp., denote the orthomodular poset of all splitting subspaces (resp., complete-cocomplete subspaces) of . We ask whether (resp., can be a lattice without being complete (i.e. without being Hilbert). This question is relevant to the recent study of the algebraic properties of splitting subspaces and to the search for ``nonstandard' orthomodular spaces as motivated by quantum theories. We first exhibit such a space that is not a lattice and is a (modular) lattice. We then go on showing that the orthomodular poset may not be a lattice even if . Finally, we construct a noncomplete space such that with being a (modular) lattice. (Thus, the lattice properties of (resp. do not seem to have an explicit relation to the completeness of though the Ammemia-Araki theorem may suggest the opposite.) As a by-product of our construction we find that there is a noncomplete such that all states on are restrictions of the states on for being the completion of (this provides a solution to a recently formulated problem).

     

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.

     

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.

     

On an asymptotic behavior of elements of order in irreducible representations of the classical algebraic groups with large enough highest weights

The behavior of the images of a fixed element of order in irreducible representations of a classical algebraic group in characteristic with highest weights large enough with respect to and this element is investigated. More precisely, let be a classical algebraic group of rank over an algebraically closed field of characteristic 2$">. Assume that an element of order is conjugate to that of an algebraic group of the same type and rank naturally embedded into . Next, an integer function on the set of dominant weights of and a constant that depend only upon , and a polynomial of degree one are defined. It is proved that the image of in the irreducible representation of with highest weight contains more than Jordan blocks of size if and are not too small and .