Wigner's theorem in Hilbert -algebras of compact operators

Let be a Hilbert -module over the -algebra of all compact operators on a Hilbert space. It is proved that any function which preserves the absolute value of the -valued inner product is of the form , where is a phase function and is an -linear isometry. The result generalizes Molnár's extension of Wigner's classical unitary-antiunitary theorem.

     

On individual stability of semigroups

Let be a -semigroup with generator on a Banach space . Let be a fixed element. We prove the following individual stability results.

(i) Suppose is an ordered Banach space with weakly normal closed cone and assume there exists such that for all . If the local resolvent admits a bounded analytic extension to the right half-plane 0\}$">, then for all and we have

(ii) Suppose is a rearrangement invariant Banach function space over with order continuous norm. If is an element such that defines an element of , then for all and we have

     

Some finiteness conditions on the set of overrings of an integral domain

Let be an integral domain with quotient field and integral closure . An overring of is a subring of containing , and denotes the set of overrings of . We consider primarily two finiteness conditions on : (FO), which states that is finite, and (FC), the condition that each chain of distinct elements of is finite. (FO) is strictly stronger than (FC), but if , each of (FO) and (FC) is equivalent to the condition that is a Prüfer domain with finite prime spectrum. In general satisfies (FC) iff satisfies (FC) and all chains of subrings of containing have finite length. The corresponding statement for (FO) is also valid.

     

Five degrees of separation

If is an infinite Abelian group, , then can be transformed in five steps of type or into a predetermined subset of the diagonal (depending on ).

     

On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body

We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality which relates the quermassintegrals of a convex body to those of an arbitrary hyperplane projection of . A consequence is the following fact: for any convex body , for any -dimensional subspace of and any 0$">,

where denotes the Euclidean unit ball and denotes volume in the appropriate dimension.

     

Vanishing of cohomology over Gorenstein rings of small codimension

We prove that if , are finite modules over a Gorenstein local ring of codimension at most , then the vanishing of for is equivalent to the vanishing of for . Furthermore, if has no embedded deformation, then such vanishing occurs if and only if or has finite projective dimension.

     

Sur une question de capitulation

Let and be prime numbers such that and . Let , , and let be the 2-Hilbert class field of , the 2-Hilbert class field of and the Galois group of . The 2-part of the class group of is of type , so contains three extensions . Our goal is to study the problem of capitulation of the 2-classes of in , and to determine the structure of .

RSESUM´E. Soient et deux nombres premiers tels que et , , , , le 2-corps de classes de Hilbert de , le 2-corps de classes de Hilbert de et le groupe de Galois de . La 2-partie du groupe de classes de est de type , par suite contient trois extensions . On s'intéresse au problème de capitulation des 2-classes de dans , et à déterminer la structure de .

     

Helgason-Marchaud inversion formulas for Radon transforms

Let be either the hyperbolic space or the unit sphere , and let be the set of all -dimensional totally geodesic submanifolds of . For and , the totally geodesic Radon transform is studied. By averaging over all at a distance from , and applying Riemann-Liouville fractional differentiation in , S. Helgason has recovered . We show that in the hyperbolic case this method blows up if does not decrease sufficiently fast. The situation can be saved if one employs Marchaud's fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for , are obtained.

     

Character degree sets that do not bound the class of a -group

Suppose that we are given a set of powers of a prime and that . A technique is presented that enables the construction of a -group of specified nilpotence class such that its set of irreducible character degrees is exactly . If , then this can be done for and if , then the only requirement is .

     

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 .

     

Every set of finite Hausdorff measure is a countable union of sets whose Hausdorff measure and content coincide

A set is -straight if has finite Hausdorff -measure equal to its Hausdorff -content, where is continuous and non-decreasing with . Here, if satisfies the standard doubling condition, then every set of finite Hausdorff -measure in is shown to be a countable union of -straight sets. This also settles a conjecture of Foran that when , every set of finite -measure is a countable union of -straight sets.

     

On Schwarz type inequalities

We show Schwarz type inequalities and consider their converses. A continuous function is said to be semi-operator monotone on if is operator monotone on . Let be a bounded linear operator on a complex Hilbert space and be the polar decomposition of . Let and for . (1) If a non-zero function is semi-operator monotone on , then for , where . (2) If are semi-operator monotone on , then for . Also, we show converses of these inequalities, which imply that semi-operator monotonicity is necessary.

     

Roots of complex polynomials and Weyl-Heisenberg frame sets

A Weyl-Heisenberg frame for is a frame consisting of modulates and translates , , of a fixed function , for . A fundamental question is to explicitly represent the families so that is a frame for . We will show an interesting connection between this question and a classical problem of Littlewood in complex function theory. In particular, we show that classifying the characteristic functions for which is a frame for is equivalent to classifying the integer sets so that does not have any zeroes on the unit circle in the plane.

     

A proof of Weinberg's conjecture on lattice-ordered matrix algebras

Let be a subfield of the field of real numbers and let () be the matrix algebra over . It is shown that if is a lattice-ordered algebra over in which the identity matrix 1 is positive, then is isomorphic to the lattice-ordered algebra with the usual lattice order. In particular, Weinberg's conjecture is true.

     

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.

     

Isomorphic -subspaces in Orlicz-Lorentz sequence spaces

Given a decreasing weight and an Orlicz function satisfying the -condition at zero, we show that the Orlicz-Lorentz sequence space contains an -isomorphic copy of , if and only if the Orlicz sequence space does, that is, if , where and are the Matuszewska-Orlicz lower and upper indices of , respectively. If does not satisfy the -condition, then a similar result holds true for order continuous subspaces and of and , respectively.

     

The depth of an ideal with a given Hilbert function

Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

     

Conditional weak laws in Banach spaces

Let be a separable Banach space. Let be centered i.i.d. random vectors taking values on with law , , and let Under suitable conditions it is shown for every open and convex set that \varepsilon \Big\vert\frac{{\displaystyle S_n}}{\displaystyle n}\in D\right)$"> converges to zero (exponentially), where is the dominating point of As applications we give a different conditional weak law of large numbers, and prove a limiting aposteriori structure to a specific Gibbs twisted measure (in the direction determined solely by the same dominating point).

     

Maximum principle on unbounded domains for sub-Laplacians: A potential theory approach

The maximum principle on a wide class of unbounded domains is proved for solutions to the partial differential inequality , where and is a real sub-Laplacian. A potential theory approach is followed.

     

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).