The theory of infinite-dimensional lie groups and its applications

The purpose of this paper is to survey the theory of regular Fréchet-Lie groups developed in [1–10]. Such groups appear and are useful in symplectic geometry and the theory of primitive infinite groups of Lie and Cartan [11]. From the group theoretical standpoint, general relativistic mechanics is a more closed system than Newtonian mechanics. Quantized objects of these classical groups are closely related to the group of Fourier integral operators [12]. These can also be managed as regular Fréchet-Lie groups. However, there are many Fréchet-Lie algebras which are not the Lie algebras of regular Fréchet-Lie groups [13]. Thus, the enlargeability of the Poisson algebra is discussed in detail in this paper. Enlargeability is relevant to the global hypoellipticity [14, 15] of second-order differential operators.     

Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

We consider a wide class of unital involutive topological algebras provided with aC ^*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebra sare taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet-Lie groups of Campbell-Baker-Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.     

Spectral invariance of smooth crossed products, and rapid decay locally compact groups

We show that all rapid-decay locally compact groups are unimodular and that the set of rapid-decay functions on a locally compact rapidly decaying group forms a dense and spectral invariant Fréchet *-subalgebra of the reduced group C ^*-algebra. In general, the set of rapid-decay functions on a locally compact strongly rapid-decay group with values in a commutative C ^*-algebra forms a dense and spectral invariant Fréchet *-subalgebra of the twisted crossed product C ^*-algebra. The spectral invariance property implies that the K-theories of both algebras are naturally isomorphic under inclusion.This project is supported in part by the National Science Foundation Grant #DMS 92-04005.     

On compact multipliers of topological algebras

It is shown that if the maximal ideal space (A) of a semisimple commutative complete metrizable locally convex algebra contains no isolated points, then every compact multiplies is trivial. In particular, compact multipliers on semisimple commutative Fréchet algebras whose maximal ideal space has no isolated points are identically zero.     

Universal Central Extensions of Lie Groups

We call a central Z-extension of a group G weakly universal for an Abelian group A if the correspondence assigning to a homomorphism ZA the corresponding A-extension yields a bijection of extension classes. The main problem discussed in this paper is the existence of central Lie group extensions of a connected Lie group G which is weakly universal for all Abelian Lie groups whose identity components are quotients of vector spaces by discrete subgroups. We call these Abelian groups regular. In the first part of the paper we deal with the corresponding question in the context of topological, Fréchet, and Banach–Lie algebras, and in the second part we turn to the groups. Here we start with a discussion of the weak universality for discrete Abelian groups and then turn to regular Lie groups A. The main results are a Recognition and a Characterization Theorem for weakly universal central extensions.     

The property (DN) and the exponential representation of holomorphic functions

It is shown that a nuclear Fréchet spaceE has the property (DN) if and only if every holomorphic function onE ^*, the strongly dual space ofE, with values in the strongly dual space of a Fréchet spaceF having the property ( ) can be represented in the exponential form. Moreover, it is shown that the space of holomorphic functions onC , the space of all complex number sequences, has a linearly absolutely exponential representation system. But the space of holomorphic functions onE ^* does not have such a system whenE is a nuclear Fréchet space that does not have the property (DN).Supported by the State Program for Fundamental Researches in Natural Sciences     

Metric regularity and subdifferential calculus in Banach spaces

In this paper we give verifiable conditions in terms of limiting Fréchet subdifferentials ensuring the metric regularity of a multivalued functionF(x)=–g(x)+D. We apply our results to the study of the limiting Fréchet subdifferential of a composite function defined on a Banach space.     

Unique maximal rings of functions

For several classes of groups G, we characterize when the near-ring M₀(G) of 0-preserving selfmaps on G contains a unique maximal ring. Definitive results are obtained for finite Abelian, finite nilpotent, and finite permutation groups. As an application, we determine those finite groups G such that all rings in M₀(G) are commutative.     

Exponential objects and Cartesian closedness in the constructPrtop

We give an internal characterization of the exponential objects in the constructPrtop and investigate Cartesian closedness for coreflective or topological full subconstructs ofPrtop. If $ is the set {0} {1/n;n 1} endowed with the topology induced by the real line, we show that there is no full coreflective subconstruct ofPrtop containing $ and which is Cartesian closed. With regard to topological full subconstructs ofPrtop we give an example of a Cartesian closed one that is large enough to contain all topological Fréchet spaces and allT ₁ pretopological Fréchet spaces.Aspirant NFWO     

On Primitive and Realisable Classes

Let S be a scheme, and let G be a finite, flat, commutative group scheme over S. In this paper we show that (subject to a mild technical assumption) every primitive class in Pic(G) is realisable. This gives an affirmative answer to a question of Waterhouse. We also discuss applications to locally free classgroups and to Selmer groups of Abelian varieties.     

Reduction of discontinuity for derivations on Frechet algebras

We consider strictly irreducible representations with whichthe discontinuity of a derivation on a (locally multiplicativelyconvex) Fréchet algebra must be associated. Only thosestrictly irreducible representations which are compatible withthe topology of the algebra are considered. The main resultsshow that when consideration is fixed upon each seminorm, theexceptional set of primitive ideals supporting the discontinuitymust be a finite set, with each ideal being the kernel of somefinite-dimensional irreducible representation. This result isthe best possible, as can be seen by considering the radicalFréchet algebra constructed by Charles Read with identityadjoined which has a derivation with separating ideal that isthe entire algebra, and one could take (countable) Fréchetproducts of his counterexample. It is also proved that derivationson commutative Fréchet algebras, the structure spacesof which are compact metric in the weak* topology, have onlyfinitely many such exceptional points overall.     

On the stability of Jensen’s functional equation on groups

In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and T (2, K), where K is an arbitrary commutative field with characteristic different from two. We also prove that any group A can be embedded into some group G such that the Jensen functional equation is stable on G.     

Analysis on the Crown Domain

This paper is a further development of complex methods in harmonic analysis on semi-simple Lie groups [AG], [BeR], [KrS1,2]. We study the growth behaviour of the holomorphic extension of the orbit map of the spherical vector of an irreducible spherical representation of a real reductive group G when approaching the boundary of the crown domain of the Riemannian symmetric space G/K. As an application, we prove that Maa? cusp forms have exponential decay. Received: August 2006, Revision: June 2007, Accepted: June 2007     

Dubrovin valuation properties of skew group rings and crossed products

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G _T=<1>,where G _T is the inertial group of R.     

Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.     

Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ? is a prime different from p, and for some difference closed field (?, σ) the ?-primary torsion of G(?) is contained in a modular group definable in (?, σ), then it is contained in a group of the form {x∈G(?) :σ(x) =[a](x) } with a∈ℕ\p ^ℕ. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall. Received: 28 May 1998 / Revised version: 20 December 1998     

Galois Structure of K-Groups of Rings of Integers

N/Kbe a Galois extension of number fields with finite Galois group G.We describe a new approach for constructing invariants of the G-module structure of the K groups of the ring of integers of N in the Grothendieck group of finitely generated projective Z[G]modules. In various cases we can relate these classes, and their function field counterparts, to the root number class of Fröhlich and Cassou-Noguès.     

Polarity and transitive parabolic unitals in translation planes of odd order

A parabolic unital of a translation plane is called transitive, if the collineation group G fixing fixes the point at infinity of and acts transitively on the affine points of . It has been conjectured that if a transitive parabolic unital consists of the absolute points of a unitary polarity in a commutative semi-field plane, then the sharply transitive normal subgroupK of G is not commutative. So far, this has been proved for commutative twisted field planes of odd square order, see [1],[5]. Here we prove this conjecture for commutative Dickson planes. Received 14 May 2001.