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 .

     

Densities with the mean value property for harmonic functions in a Lipschitz domain

Let be a bounded domain in , , and let . We consider positive functions on such that for all bounded harmonic functions on . We determine Lipschitz domains having such with .

     

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.

     

Infinite loop spaces and Neisendorfer localization

There is a localization functor with the property that is the -completion of whenever is a finite dimensional complex. This same functor is shown to have the property that is contractible whenever is a connected infinite loop space with a torsion fundamental group. One consequence of this is that many finite dimensional complexes are uniquely determined, up to -completion, by the homotopy fiber of any map from into the classifying space .

     

On sums and products of integers

Erdos and Szemerédi conjectured that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of . Erdos and Szemerédi proved that this number must be at least for some and . In this paper it is proved that the result holds for .

     

Radial limit of lacunary Fourier series with coefficients in non-commutative symmetric spaces

Let be a rearrangement invariant space, an arbitrary set and a von Neumann algebra with a semifinite normal faithful trace. It is proved that the associated symmetric space of measurable operators has -RNP if and only if has -RNP extending in this way some previous results by Q. Xu.

     

Prime ideals in polynomial rings in several indeterminates

If is a prime ideal of a polynomial ring , where is a field, then is determined by an irreducible polynomial in . The purpose of this paper is to show that any prime ideal of a polynomial ring in -indeterminates over a not necessarily commutative ring is determined by its intersection with plus polynomials.

     

A formula with nonnegative terms for the degree of the dual variety of a homogeneous space

Let be a reductive group and a parabolic subgroup. For every -regular dominant weight let denote the variety embedded in the projective space by the embedding corresponding to the ample line bundle . Writing , we prove that the degree of the dual variety to is a polynomial with nonnegative coefficients in . In the case of homogeneous spaces we find an expression for the constant term of this polynomial.

     

Primitive characters of subgroups of -groups

Let be an -group, let be a subnormal subgroup of , and let be a Hall subgroup of . If the character is primitive, then is a power of 2. Furthermore, if is odd, then .

     

Intersection of sets with -connected unions

We show that if sets in a topological space are given so that all the sets are closed or all are open, and for each every of the sets have a -connected union, then the sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every or fewer members of a finite family of closed sets in have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.

     

Covering by complements of subspaces, II

Let be an -dimensional vector space over an algebraically closed field . Define to be the least positive integer for which there exists a family of -dimensional subspaces of such that every -dimensional subspace of has at least one complement among the 's. Using algebraic geometry we prove that .

     

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).

     

Factorization of an integrally closed ideal in two-dimensional regular local rings

Let be a two-dimensional regular local ring with algebraically closed residue field and be an -primary integrally closed ideal in . Let be the set of Rees valuations of and be the residue field of the valuation ring associated with . Assume that is any minimal reduction of . We show that if is the product of the distinct simple -primary integrally closed ideals in , then is generated by the image of over for all , and the converse of this is also true.

     

The optimality of James's distortion theorems

A renorming of , explored here in detail, shows that the copies of produced in the proof of the Kadec-Pelczynski theorem inside nonreflexive subspaces of cannot be produced inside general nonreflexive spaces that contain copies of . Put differently, James's distortion theorem producing one-plus-epsilon-isomorphic copies of inside any isomorphic copy of is, in a certain sense, optimal. A similar renorming of shows that James's distortion theorem for is likewise optimal.

     

Factorisation in nest algebras

We give a necessary and sufficient condition on an operator for the existence of an operator in the nest algebra of a continuous nest satisfying (resp. . We also characterise the operators in which have the following property: For every continuous nest there exists an operator in satisfying (resp. .

     

On invertibility in non-selfadjoint operator algebras

Let be a complete commutative subspace lattice on a Hilbert space. When is purely atomic, we give a necessary and sufficient condition for for every in , where and denote the spectrum of in and respectively. In addition, we discuss the properties of the spectra and the invertibility conditions for operators in .

     

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.

     

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 a class of subalgebras of and the intersection of their free maximal ideals

Let be a Tychonoff space and a subalgebra of containing . Suppose that is the set of all functions in with compact support. Kohls has shown that is precisely the intersection of all the free ideals in or in . In this paper we have proved the validity of this result for the algebra . Gillman and Jerison have proved that for a realcompact space , is the intersection of all the free maximal ideals in . In this paper we have proved that this result does not hold for the algebra , in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in . The paper terminates by showing that for any realcompact space , there exists in some sense a minimal algebra for which becomes -compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

     

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 .