Research and Exposition in Mathematics
Volume 24
E. Martín Peinador, J. Núñez García (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.
- 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.
- 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.
- 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].
- 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).
- 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).
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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).
- 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.