Research and Exposition in Mathematics

Volume 24

E. Martin Peinador, J. Nunez Garcia (eds.)
Nuclear Groups and Lie Groups
258 pages, soft cover, ISBN 3-88538-224-5, EUR 36.00, 2001

Contents with Abstracts

L. Aussenhofer: A Survey on Nuclear Groups, 1--30
The aim of this article is to give a survey on the theory of nuclear groups. We first point out common properties of locally compact abelian groups and of nuclear vector spaces and we will motivate afterwards the definition of nuclear groups. The class of nuclear groups forms a Hausdorff variety of abelian groups which contains the groups mentioned above. Finally we present some theorems known to be valid for nuclear groups.

W. Banaszczyk: Theorems of Bochner and Levy for Nuclear Groups, 31--44
The Bochner theorem on positive definite functions and the Levy continuity theorem remain valid for some abelian topological groups which are not locally compact, e.g. for products of LCA groups and for nuclear locally convex spaces. The article gives a survey of known results and outlines methods of proofs.

O. Blasco: Bilinear Maps and Convolutions, 45--56
Let X, Y, Z be Banach spaces and let u: X ´ Y --> Z be a bounded bilinear map. Given a locally compact abelian group G, and two functions f from L1(G, X) and g from L1(G, Y), we define the u-convolution of f and g as the Z-valued function f *u g(t) = Integral over G of u( f (t - s), g(s)) dmG(s), where dmG stands for the Haar measure on G. We define the concepts of vector-valued approximate identity and summability kernel associated to a bounded bilinear map, showing the corresponding approximation result in this setting. A Haussdorf-Young type result associated to a bounded bilinear map is also presented under certain assumptions on the Banach space X.

W. W. Comfort, S. Hernandez, D. Remus, F. J. Trigos-Arrieta: Some Open Questions on Topological Groups, 57--76
We comment here on some recent results and problems for the most part related to the Bohr compactification and Bohr topology of an arbitrary topological group. It will become clear in what follows that, though the subject relates closely to topological algebra, it leads quickly and naturally to questions of interest to researchers working in a variety of disciplines. Our principal goal is to summarize some of the recent progress concerning the Bohr topology of a topological group. Accordingly, we begin with a brief discussion of the Bohr topology on a discrete Abelian group. We consider also the Pontryagin-van Kampen duality theory of topological Abelian groups and the extension of this theory to non Abelian groups introduced by Chu. As will be shown, the two subjects have many points in common. Many of our questions, while intelligible for arbitrary topological groups, are unsolved even for locally compact groups (and even in certain cases, for LCA (=locally compact Abelian) groups). Thus our approach is transversal and perhaps complementary to the one adopted by V. Pestov in his survey "Topological groups: where to from here?" [Proc. 14th Summer Conference on General Topology and its Applications, C. W. Post College 1999, Topology Proc. (2001), to appear].

E. Corbacho Rosas, D. Dikranjan, V. Tarieladze: Absorption Adjunctable Semigroups, 77--104
We study the topological semigroups that admit the adjunction of a non-isolated absorbing element and the structure and permanence properties of the class AA of topological semigroups admitting this type of adjunctions. No precompact topological group can belong to the class AA. More generally, a subsemigroup X of a compact Hausdorff semigroup K belongs to AA iff X misses the Suskevic kernel of the closure cl(X) in K. Every non-torsion abelian group belongs to AA when equipped with the discrete topology. Our interest in the class AA stems from the question of quasi-uniformizability of semigroups (every topological AA-group, after the adjunction of a non-isolated absorbing element, gives rise to a non-quasi-uniformizable semigroup).

D. Dikranjan: The Lattice of Group Topologies and Compact Representations, 105--126
We discuss lattice theoretic properties of group topologies (maximal and minimal topologies, complementation etc.) as well as their relation to topologies generated by compact representations (homomorphisms in the circle group in the abelian case). A special emphasis is given to the submaximal topology (the infimum of all maximal topologies).

X. Dominguez, V. Tarieladze: GP-Nuclear Groups, 127--162
In a previous paper ["Nuclear and GP-nuclear groups", Acta Math. Hungar. 88 (2000) 301--322] we introduced GP-nuclear groups as those topological Abelian groups for which the groups of summable and absolutely summable sequences are the same algebraically and topologically. It is proved in the present paper that, any countable direct sum of GP-nuclear groups is GP-nuclear and this may not hold in the uncountable case. At the end we show that a locally convex vector group is a GP-nuclear group if and only if it is nuclear. This recovers Grothendieck-Pietsch theorem for locally convex spaces. All necessary definitions and auxiliary assertions are presented in the text; in particular, we give a survey of some known results about GP-nuclear groups (including the Riemann-Dirichlet theorem for real numbers, the Dvoretzky-Rogers theorem for normed spaces, the results of W. Banaszczyk in "Additive subgroups of topological vector spaces [Lecture Notes Math. Springer 1466 (1991)] for nuclear groups and the related results in the above-mentioned paper of the authors). Thus, the paper is self-contained.

H. Glöckner, K.-H. Neeb: Minimally Almost Periodic Abelian Groups and Commutative W*-Algebras, 163--186
An abelian topological group is called minimally almost periodic if it does not have a non-trivial continuous character. We say that an abelian topological group is reduced if its weakly continuous unitary representations separate points: these are the groups accessible by unitary representation theory. We characterize the commutative W*-algebras whose unitary groups are minimally almost periodic (when equipped with the weak topology), obtaining a new class of reduced minimally almost periodic abelian topological groups in this way. We show that an abelian topological group G is minimally almost periodic if and only if its universal enveloping W*-algebra W*(G) has precisely one minimal non-zero idempotent. An abelian topological group G is reduced and minimally almost periodic if and only if there exists a commutative W*-algebra A such that U(A) is minimally almost periodic, and an injective continuous homomorphism k: G --> U(A) such that every continuous unitary representation of G extends to a continuous unitary representation of U(A). In an appendix, we show that the predual W*(G)* of W*(G) can be made a semi-simple commutative unital Banach algebra in a natural way, and give various descriptions of its spectrum.

P. Matysiak: Non-Reflexive Closed Connected Subgroups of Banach Spaces, 187--196
Every Banach space, when treated as an additive topological group, satisfies the Pontriagin-van Kampen duality theorem [M. F. Smith, Ann. of Math. 56 (1952) 248--253]. Here we prove that every infinite-dimensional Banach space contains a closed connected additive subgroup which does not satisfy the duality theorem.

M. G. Megrelishvili: Operator Topologies and Reflexive Representability, 197--208
Using the concept of fragmentability, we show that weakly continuous group representations are frequently strongly continuous. We show that if a Banach (or, even, Frechet) space X has the Radon-Nikodym property RNP, then the weak and strong operator topologies coincide on every bounded (respectively, equicontinuous) subgroup G of GL(X). We also strengthen a result of Shtern on reflexive representability of topological groups.

M. G. Megrelishvili, V. G. Pestov, V. V. Uspenskij: A Note on the Precompactness of Weakly Almost Periodic Groups, 209--216
An action of a group G on a compact space X is called weakly almost periodic if the orbit of every continuous function on X is weakly relatively compact in C(X). We observe that for a topological group G the following are equivalent: (i) every continuous action of G on a compact space is weakly almost periodic; (ii) G is precompact. For monothetic groups the result was previously obtained by Akin and Glasner, while for locally compact groups it has been known for a long time.

R. Steglinski: Quotient Groups of Normed Spaces for which the Bochner Theorem Fails Completely, 217--225
It is proved that every infinite dimensional real normed space contains a discrete additive subgroup K such that the quotient group G = (span K) / K admits a non-trivial continuous positive definite function, but does not admit any non-trivial continuous characters. Consequently, G cannot satisfy any form of the Bochner theorem.

W. Wojtynski: An Introduction to String Lie Theory, 227--238
The theory of Lie groups in finite dimensions relies on the strong connection between the two structures of the group: the finite dimenional manifold structure and the topological group structure. The most important manifestation of this connection comes early in the course of the theory, when from the existence of a manifold structure on a given group G one derives the fact that the family L(G), of all continuous one-parameter subgroups of G is rich enough to determine G locally.
This connection breaks down completely when we pass to some infinite dimensional analogues of Lie groups. On the one hand, even under a reasonable differential assumption such as the presence of a compatible Cinfinity Frechet manifold structure on G, it is not kown whether G necessarily has a non-trivial one-parameter subgroup. On the other hand numerous groups are known which have a good supply of one-parameter subgroups, but for topological reasons are not able to carry a topological manifold structure (e.g. infinite products of compact Lie groups, in particular the countable dimensional torus Tinfinity).
Therefore, when thinking about the foundations of Lie group theory in general, it seems inevitable that one should make a choice between the two options:
(1) either to assume existence of a proper differential structure on G;
(2) or to demand the existence of a 'good' class of one-parameter subgroups of G.
We support here the second choice, calling the class of groups we are going to investigate 'string Lie groups'.
So roughly speaking string Lie theory aims to treat a topological group G in terms of the family L(G) of its one-parameter subgroups, i.e. to introduce a suitable topological Lie algebra structure on L(G) and to find procedures enabling one to recover (locally) G from  L(G).

M. Wüstner: A Short Survey on the Surjectivity of Exponential Lie Groups, 239--250
This is a survey on the surjectivity of the exponential function with particular emphasis on new results on the role of the centralizers of nilpotent elements. What this survey lacks in completeness is likely to be found in an earlier survey by D. Z. Dokovic and K. H. Hofmann ["The exponential in real Lie groups: A status report", J. Lie Theory 7 (1997) 171--191]. Where proofs are omitted, references will be given where they can be found.