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.

     

An Engel condition with derivation for left ideals

We generalize a number of results in the literature by proving the following theorem: Let be a semiprime ring, a nonzero derivation of , a nonzero left ideal of , and let . If for some positive integers , and all , the identity holds, then either or else the ideal of generated by and is in the center of . In particular, when is a prime ring, is commutative.

     

Hyperspaces and open monotone maps of hereditarily indecomposable continua

We prove the following theorems:

Theorem 1. Let be an -dimensional hereditarily indecomposable continuum. Then there exist -dimensional hereditarily indecomposable continua and monotone maps such that is an embedding and the space of all subcontinua of is embeddable in by .

Theorem 2. For every open monotone map with non-trivial sufficiently small fibers on a finite dimensional hereditarily indecomposable continuum with there exists a -dimensional subcontinuum such that and the restriction of to is also monotone and open.

The connection between these theorems and other results in Hyperspace theory is studied.

     

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.

     

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.

     

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 .

     

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.

     

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 .

     

On the curves of contact on surfaces in a projective space. III

Suppose a smooth curve is a set-theoretic complete intersection of two surfaces and with the multiplicity of along less than or equal to the multiplicity of along . One obtains a relation between the degrees of , and , the genus of , and the multiplicity of along in case has only ordinary singularities. One obtains (in the characteristic zero case) that a nonsingular rational curve of degree 4 in is not set-theoretically an intersection of 2 surfaces, provided one of them has at most ordinary singularities. The same result holds for a general nonsingular rational curve of degree .

     

Grothendieck operators on tensor products

We prove that for Banach spaces and operators , the tensor product is a Grothendieck operator, provided is a Grothendieck operator and is compact.

     

Groups with many normal-by-finite subgroups

A subgroup of a group is said to be normal-by-finite if the core of in has finite index in . In this article groups satisfying the minimal condition on subgroups which are not normal-by-finite and groups with finitely many conjugacy classes of subgroups which are not normal-by-finite are characterized.

     

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.

     

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 genera of smooth curves in higher dimensional varieties

We prove that for any smooth projective variety of dimension , there exists an integer , such that for any integer , there exists a smooth curve in with .

     

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 .

     

On asymptotic estimates for arithmetic cost functions

Let (the cost of ) be the minimum number of arithmetic operations needed to obtain starting from 1. We prove that for almost all , and, given , for all sufficiently large. We prove analogous results for costs of polynomials with integer coefficients.

     

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.

     

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 .

     

Cellular filtration of K-theory and determinants of -algebras

In this note, we will disprove the following conjecture raised by Exel-Loring: Let be a -algebra with trace and let be a determinant associated to . If is a continuous family of homomorphisms and is the canonical matrix valued function on which represents the Bott element in , then . It should be noticed that the conjecture has been proved by Exel-Loring for the case that is a smooth family of homomorphisms.

     

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 .