Continuity of the support of a state

Let be a finite von Neumann algebra and a projection. It is well known that the map which assigns its support projection to a positive normal functional of is not continuous. In this note it is shown that if one restricts to the set of positive normal functionals with support equivalent to a fixed , endowed with the norm topology, and the set of projections of is considered with the strong operator topology, then the support map is continuous. Moreover, it is shown that the support map defines a homotopy equivalence between these spaces. This fact together with previous work implies that, for example, the set of projections of the hyperfinite II factor, in the strong operator topology, has trivial homotopy groups of all orders .

     

On classes of maps which preserve finitisticness

We shall prove the following: Let be a refinable map between paracompact spaces. Then is finitistic if and only if is finitistic. Let be a hereditary shape equivalence between metric spaces. Then if is finitistic, is finitistic.

     

Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of 0$">. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

     

On representable linearly compact modules

For a flat module we prove that is a functor from the category of linearly compact modules to itself and is exact. Moreover, is representable when is linearly compact and representable. This gives an affirmative answer to a question of L. Melkersson (1995) for linearly compact modules without the condition of finite Goldie dimension. The set of attached prime ideals of the co-localization of a linearly compact representable module with respect to a multiplicative set in is described.

     

Basic characters of the unitriangular group (for arbitrary primes)

Let denote the (upper) unitriangular group of degree over the finite field with elements. In this paper we consider the basic (complex) characters of and we prove that every irreducible (complex) character of is a constituent of a unique basic character. This result extends a previous result which was proved by the author under the assumption , where is the characteristic of the field .

     

Cohomological dimension of certain algebraic varieties

Let be an ideal of a commutative Noetherian ring . For finitely generated -modules and with , it is shown that . Let be a finitely generated module over a local ring such that . Using the above result and the notion of connectedness dimension, it is proved that Here denotes the connectedness dimension of the topological space . Finally, as a consequence of this inequality, two previously known generalizations of Faltings' connectedness theorem are improved.

     

A note concerning the index of the shift

Let be a finite, positive Borel measure with support in such that - the closure of the polynomials in - is irreducible and each point in is a bounded point evaluation for . We show that if 0$">and there is a nontrivial subarc of such that

-\infty,\end{displaymath}">

then for each nontrivial closed invariant subspace for the shift on .

     

Small volume on big -spheres

We consider Riemannian metrics on the -sphere for such that the distance between any pair of antipodal points is bounded below by 1. We show that the volume can be arbitrarily small. This is in contrast to the -dimensional case where Berger has shown that .

     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

On modules of finite projective dimension over complete intersections

Recently Avramov and Miller proved that over a local complete intersection ring in characteristic 0$">, a finitely generated module has finite projective dimension if for some 0$"> and for some 0$">, being the frobenius map repeated times. They used the notion of ``complexity' and several related theorems. Here we offer a very simple proof of the above theorem without using ``complexity' at all.

     

Numerical invariants for bundles on blow-ups

We suggest an effective procedure to calculate numerical invariants for rank two bundles over blown-up surfaces. We study the moduli spaces of rank two bundles on the blown-up plane splitting over the exceptional divisor as We use the numerical invariants to give a topological decomposition of

     

Complemented isometric copies of in dual Banach spaces

Let be a real or complex Banach space and . Then contains a -complemented, isometric copy of if and only if contains a -complemented, isometric copy of if and only if contains a subspace -asymptotic to .

     

Codimension growth of two-dimensional non-associative algebras

Let be a field of characteristic zero and let be a two- dimensional non-associative algebra over . We prove that the sequence of codimensions of is either bounded by or grows exponentially as . We also construct a family of two-dimensional algebras indexed by rational numbers with distinct T-ideals of polynomial identities and whose codimension sequence is , .

     

Hierarchical structure of the family of curves with maximal genus verifying flag conditions

Fix integers such that and , and let be the set of all integral, projective and nondegenerate curves of degree in the projective space , such that, for all , does not lie on any integral, projective and nondegenerate variety of dimension and degree . We say that a curve satisfies the flag condition if belongs to . Define where denotes the arithmetic genus of . In the present paper, under the hypothesis , we prove that a curve satisfying the flag condition and of maximal arithmetic genus must lie on a unique flag such as , where, for any , denotes an integral projective subvariety of of degree and dimension , such that its general linear curve section satisfies the flag condition and has maximal arithmetic genus . This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.

     

On syzygies of Segre embeddings

We study the syzygies of the ideals of the Segre embeddings. Let , ; we prove that the line bundle on the ( copies) satisfies Property of Green-Lazarsfeld if and only if . Besides we prove that if we have a projective variety not satisfying Property for some , then the product of it with any other projective variety does not satisfy Property . From this we also deduce other corollaries about syzygies of Segre embeddings.

     

Algebraic structures determined by 3 by 3 matrix geometry

Using a ``3 by 3 matrix trick' we show that multiplication (an algebraic structure) in a *-algebra is determined by the geometry of the *-algebra of the 3 by 3 matrices with entries from , . This is an example of an algebra-geometry duality which, we claim, has applications.

     

Transversals for strongly almost disjoint families

For a family of sets , and a set , is said to be a transversal of if and for each . is said to be a Bernstein set for if for each . Erdos and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets is said to admit a -transversal if can be written as such that each admits a transversal. We study the question of when an almost disjoint family admits a -transversal and related questions.

     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

On the index and spectrum of differential operators on

If is an system of differential operators on having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that is Fredholm from to for all or no value We prove that both the index (when defined) and the spectrum of are independent of