Smooth topological groups
Abstract.
In this paper we study the Lie theoretic properties of a class of topological groups which carry a Banach manifold structure but whose multiplication is not smooth. If and are Banach–Lie groups and is a homomorphism defining a continuous action of on , then is a Banach manifold with a topological group structure for which the left multiplication maps are smooth, but the right multiplication maps need not to be. We show that these groups share surprisingly many properties with Banach–Lie groups: (a) for every regulated function the initial value problem , , has a solution and the corresponding evolution map from curves in to curves in is continuous; (b) every curve with and satisfies ; (c) the Trotter formula holds for oneparameter groups in ; (d) the subgroup of elements with smooth orbit maps in carries a natural Fréchet–Lie group structure for which the action is smooth; (e) the resulting Fréchet–Lie group is also regular in the sense of (a).
Key words and phrases:
Topological group, Lie group, regular Lie group2010 Mathematics Subject Classification:
22E65, 58B251. Introduction
The theory of infinite dimensional Lie groups can be developed very naturally in the context of Lie groups modelled on locally convex spaces, so called locally convex Lie groups. For more details on this theory, we recommend the survey article [Nee06] or the forthcoming monograph [GN]. The theory of locally convex Lie groups has, however, certain drawbacks, the most serious one being that the Inverse and Implicit Function Theorem fail beyond the class of Banach manifolds. In some situations one can still use the Nash–Moser Theorem, but this theorem is difficult to apply because its assumptions are often hard to verify.
It is for this reason that, early on in infinite dimensional Lie theory, people have tried to “approximate” Fréchet–Lie groups by certain Banach manifolds to work in a context where the analytic tools, such as existence of solutions of ODEs and inverse function results, can be applied, and then perform a passage to the Fréchet limit, which often is a projective limit of topological groups. The most prominent situation where this strategy has been applied with great success is the analysis of diffeomorphism groups of compact smooth manifolds . The group of smooth diffeomorphisms carries the structure of a Fréchet–Lie group, but the usual construction of charts also applies to the groups of diffeomorphisms for any . This provides the structure of a manifold structure on each of the topological groups , but neither multiplication nor inversion are smooth. Only the right multiplications are smooth maps. This kind of “weak Lie group structure” is usually dealt with in the context of ILB (inverse limit of Banach) Lie groups, which play an important role in geometric analysis (cf. [AK98], [EM99]). The Lie theory of these groups has been developed by H. Omori and his collaborators in a series of papers culminating in [MOKY85] (see also Omori’s monograph [Omo97]).
Another context where Lie theory leaves its natural “smooth context” is in the theory of group representations. There one studies representations of a Lie group on a Fréchet space, for which the action of on is continuous, but not in general smooth. This seemingly weak requirement is dictated by the applications where smoothness of the action of on would be much too strong. This phenomenon is wellknown from the theory of oneparameter semigroups on Banach spaces, where norm continuity is much too strong and strong continuity is the natural regularity assumption. Applying Lie theoretic methods to continuous representations can be a difficult task, but recently some quite effective tools to overcome these difficulties have been developed (see in particular [NS13]). To apply these tools, one has to assume that the addition in the Lie algebra of the Lie group under consideration is compatible with the topological group structure in the sense that has the Trotter property, i.e., for every ,
holds uniformly on compact subsets of .
All locally exponential Lie groups (i.e., groups for which the exponential function
is a local diffeomorphism in ) have this property, and this includes in
particular all Banach–Lie groups ([Nee06]). Beyond the Banach context,
the Trotter property is often hard to verify, but in [NS13] this is
done for diffeomorphism groups of compact manifolds and the Virasoro group.
Much of this has recently been facilitated by H. Glöckner’s new regularity
results [Glö15] which provide also solutions to differential
equations of the form , where
is not necessarily continuous. So could also be a Riemannian
step function, or a uniform limit of step functions, i.e., a regulated function.
In the present paper we pursue a more detailed analysis of a class of Banach manifolds which carry a topological group structure but which are not Lie groups, namely semidirect products , where both and are Banach–Lie groups, but the homomorphism only defines a continuous action . Then is a topological group and a smooth Banach manifold. The multiplication on , however, is in general not smooth as the right multiplication maps
are, in general, only continuous. On the other hand, the left multiplication maps are smooth. We call such a topological group, with a Banach manifold structure and smooth left multiplication maps, a smooth topological group. The semidirect products as above constitute an important class of examples of smooth topological groups because they are still rather wellbehaved but they also display many of the pathologies of smooth topological groups that are not Lie groups. Here already the case is very interesting. Another interesting class of examples arises for the group , a compact smooth manifold, a Banach–Lie group and , where the action of on comes from a smooth action of on . The aim of this paper is to understand to which extent the Lie theoretic properties of Banach–Lie groups survive in the framework of these smooth topological groups.
We now describe our main results in more detail. The first problem to investigate is the existence of an exponential function on the tangent space of at the identity, that is, of a map such that for each , the curve defined by is a solution to the initial value problem (IVP)
where we denoted for each by the action of the tangent map of at the identity on . More generally, for each continuous curve , one may ask whether there exists a solution to the IVP
(1.1) 
If a solution to (1.1) exists, then it is unique (see [Nee06, §II.3]), yielding an evolution map
If, moreover, is continuous, then the group is called regular. For instance, Banach–Lie groups are regular (see [Glö15, Theorem C]).
In [Glö15], H. Glöckner further defined a concept of regularity (which implies regularity), by replacing the space in the above definition with the space of regulated functions, that is, of functions in that are uniform limits of step functions (see §2.10 below for more details and the precise meaning of the IVP (1.1) in this context). H. Glöckner then proves that Banach–Lie groups are regular, and derives various important consequences.
Our first result implies in particular that the smooth topological group possesses a (continuous) exponential map (see §4.1):
Theorem A.
Let be Banach–Lie groups, and let define a continuous action. Then the smooth topological group is regular.
A second natural problem is to understand whether admits a Lie algebra structure, and whether such a structure can be, as in the classical case, reconstructed from the space of oneparameter subgroups of . We recall that for any Banach–Lie group with Lie algebra and exponential function , the Lie algebra structure on can be obtained by using the identification , and the fact that has the Trotter property, i.e., for all ,
holds uniformly for in compact subsets of , as well as the commutator property, i.e., for all ,
holds uniformly for in compact subsets of . Actually has the strong Trotter property (which implies both the Trotter and commutator properties within the class of locally convex Lie groups, see [Glö15, Theorem H]), that is, for each curve with ,
uniformly for in compact subsets of .
It turns out that the space carries, in general, no natural Lie bracket, so that one cannot speak of the “Lie algebra of ” (see §4.2). On the other hand, we show, as in [Glö15, Theorem I], that the regularity of implies that has the strong Trotter property.
Theorem B.
Let be Banach–Lie groups, and let define a continuous action. Then the smooth topological group has the strong Trotter property.
In our setting, the strong Trotter property does not immediately imply the Trotter property, as for two curves , the curve need not be . Nevertheless, with some extra work, we can show that has the Trotter property as well.
Theorem C.
Let be Banach–Lie groups, and let define a continuous action. Then the smooth topological group has the Trotter property.
We actually prove a stronger result, generalising both Theorems B and C (see Theorem 4.14 below for a precise statement). As a surprising side result, we further show in §4.3 that, if is abelian, then any continuous oneparameter subgroup of is conjugate to a smooth oneparameter subgroup, hence of the form for some , .
In representation theory, an important technique consists in the passage from a continuous representation on a Banach space to the subspace of smooth vectors, i.e., the elements with smooth orbit maps, on which the Lie algebra acts naturally. In this context, a third problem is to ask whether the subgroup
of smooth elements of for the action carries a natural Lie group structure. Building on results from [Nee10], where the above question is shown to have a positive answer when is a Banach space, we prove that has a Fréchet–Lie group structure for which the induced action is smooth. This implies in particular the following (see §5.1).
Theorem D.
Let be Banach–Lie groups and define a continuous action. Then carries a natural Fréchet–Lie group structure for which the action is smooth. In particular, the semidirect product is a Fréchet–Lie group.
Finally, we investigate in §5.2 the regularity of the Lie group .
Theorem E.
Let be Banach–Lie groups, and let define a continuous action. Then is regular with a smooth evolution map. In particular, has the strong Trotter and commutator properties.
Acknowledgement
The authors thank Helge Glöckner for enlightening discussions on the topic of measurable regularity properties.
2. Preliminaries
Notation.
Throughout this paper, denotes the set of positive integers, and the set of nonnegative integers.
We first recall the basic concepts pertaining to infinitedimensional Lie groups modelled on locally convex spaces, and their measurable regularity properties. The main references for this section are [Glö15] and [Nee06].
2.1. Lebesgue spaces ([Glö15, 1.7–1.13, 1.25, 1.31])
Let for some , which we view as a measure space for the (restriction of) the Lebesgue measure on . Let be a real Fréchet space, which we view as a measurable space with respect to its algebra of Borel sets. We write for the set of all continuous seminorms .
We let denote the space of all measurable functions with separable image (i.e. has a dense countable subset) such that
Similarly, we let denote the space of all measurable functions with separable and bounded image, so that
For , we equip with the (nonHausdorff) locally convex vector topology defined by the seminorms for .
Let be the space of all measurable maps with relatively compact image. We endow with the topology induced by . Finally, let be the space of functions that are the uniform limit of a sequence of step functions. We recall that is a step function if there exists a partition of such that is constant for all . A function is called regulated, and we endow with the topology induced by .
Given a measurable map , we write for the equivalence class of measurable maps such that for almost all . (When no confusion is possible, we will also simply write for its equivalence class .) We then define (resp. , , ) as the space of equivalence classes with in (resp. , , ). For , we equip with the locally convex vector topology defined by the seminorms , and we give and the induced topology, coming from the inclusions . Note that admits a basis of open neighbourhoods consisting of the sets for
where runs through a basis of open neighbourhoods in .
Finally, note that the map is injective; we will equip the space of continuous functions with the induced topology, given in this case by the seminorms for .
2.2. Integration ([Glö15, 1.16–1.28])
Let be a real locally convex space, be its topological dual (that is, the space of all continuous linear functionals ), and let be a function such that for each . We define the weak integral of , if it exists, as the unique element such that
and we write .
As usual, a map is called differentiable at if the limit exists in . We have the following version of the Fundamental Theorem of Calculus:
Lemma 2.1.
Let be a Fréchet space, and . Then the weak integrals needed to define
exist, and is a continuous function which is differentiable almost everywhere, with .
2.3. Differentiation ([Glö15, 1.49–1.52])
Let and be real locally convex spaces, be an open set and be a map. The derivative of at in the direction is defined as the limit
whenever it exists. We say that is if it is continuous. We say that is if is continuous, the derivatives exist in for all , and is continuous. Recursively, we say, for some integer , that is if is and is . Equivalently, is if and only if it is continuous and, for all positive integers , the iterated directional derivatives
exist for all and , and the map is continuous. We call smooth or if it is for all .
We record for future reference the following results.
Lemma 2.2 ([Glö15, 2.1]).
Let be Fréchet spaces, and let be a continuous map such that is linear for all . Let be a continuous function and . Then .
Lemma 2.3.
Let be Fréchet spaces and let be open. Let be a smooth map such that is linear for all . Then
is smooth.
Proof.
This follows from [Glö15, 2.2], since the natural injection is smooth. ∎
2.4. Manifolds ([Glö15, 1.53], [Nee06, Chapter I])
Since composition of smooth maps are smooth, one can define a smooth manifold modelled on a real locally convex space (or just manifold) by replacing the modelling space by in the classical definitions of manifolds (see [Nee06]). If is a Banach (resp. Fréchet) space, then is called a Banach (resp. Fréchet) manifold.
As usual, then denotes the tangent bundle of and the tangent space of at . Likewise, for a smooth map between smooth manifolds, (resp. ) is the corresponding tangent map (resp. tangent map at ). If is an open subset of a locally convex space , we identify with . For a smooth map from a smooth manifold to a locally convex space , we then write for the second component of . Note that §2.3 also yields a notion of maps between smooth manifolds for each .
2.5. Lie groups ([Nee06, Chapters II–IV])
A (locally convex) Lie group is a group with a smooth manifold structure modelled on a locally convex space, for which the group operations (multiplication and inversion) are smooth. We write (or simply if no confusion is possible) for the identity element of , and and for the left and right multiplication maps. For , we also write for the conjugation map.
For each , there is a unique left invariant vector field with , defined by . The Lie bracket on the space of left invariant vector fields then induces a continuous Lie bracket on , characterised by for . We let denote the functor from the category of locally convex Lie groups to the category of locally convex topological Lie algebras, which associates to a group its Lie algebra and to a Lie group morphism the corresponding tangent map at the identity is a Banach (resp. Fréchet) space, then is called a Banach (resp. Fréchet) Lie group. . If
The left multiplication and conjugation maps on induce smooth maps
Note that the adjoint action of on is by topological isomorphisms.
A map is called an exponential function if for any , the curve is a oneparameter subgroup with . If has an exponential function, then it is unique.
Assume now that is a Banach Lie group. Then has a smooth exponential function , and maps some open (convex) neighbourhood in diffeomorphically onto some open subset of . If is an open (convex) neighbourhood in such that
then one can define on the local multiplication
which is a smooth map satisfying for all and with ([Nee06, Example IV.2.4]).
Given some with , some locally convex spaces , some open subset and some map , we will use the exponential chart of to define the th derivative of in the first coordinate by setting
where and, recursively, for all . In other words, for all , and ,
(2.1) 
Note that is a map for all . For a map , we also define as above by viewing as a map .
2.6. Smooth topological groups
In this paper, we will consider the following generalisation of a Lie group (see also Section 3 below). Let be a locally convex space, and let be a smooth manifold modelled on . We call a (left) smooth topological group modelled on if admits a topological group structure (with respect to the manifold topology) such that all left multiplication maps are smooth.
2.7. Local Lie groups ([Nee06, ii.1.10])
Given a group with multiplication and identity , a quadruple consisting of a symmetric subset with and of a subset with and is a socalled local group. If is such a local group and, in addition, has a smooth manifold structure, is open, and the local multiplication and inversion maps and are smooth, then (or simply ) is called a local Lie group.
2.8. Absolutely continuous maps ([Glö15, 3.6–3.20, 4.2])
Let for some and let be a Fréchet space. Define as the space of continuous functions for which there exists such that
Then is unique by Lemma 2.1, and the map
is an isomorphism, which we use to define on a locally convex vector topology. The inclusion map is then continuous, and for any open subset , the set
is open in .
More generally, given a smooth manifold modelled on , one can define as the set of all continuous functions for which there is a partition of such that, for each , there exists a chart of with and .
If is a Fréchet–Lie group, then is a group under pointwise multiplication, and there is a unique Lie group structure on such that
is open in and
is a diffeomorphism for each chart of such that and .
2.9. Logarithmic derivative ([Glö15, 5.1–5.11])
Let be a Fréchet space. Let be either a smooth topological group or a local Lie group modelled on . Write . Let for some and let . We define the (left) logarithmic derivative of as follows.
Let be a partition of such that, for each , there exists a chart of with . Then
for all , and one can thus consider , which we write as for some . Define
via if with and . We then set
where
Note that if and are smooth Fréchet–Lie groups and is a smooth homomorphism, then for each and
(2.2) 
where if (see [Glö15, 5.2(b)]).
2.10. regularity ([Glö15, 5.14–5.26])
Let for some and let be a Fréchet space. Let also be open, and let be a map. A continuous function is called an Carathéodory solution to if , the map is in , and
Consider next a smooth manifold modelled on , and let and be a map. Then is an Carathéodory solution to the initial value problem (IVP)
if and for each , there exists such that for some chart of and is an Carathéodory solution to with
Assume now that . If is a smooth topological group modelled on , with Lie algebra , then is called semiregular if for each , there exists such that
(2.3) 
If it exists, then is uniquely determined. Note that, writing for some , the map satisfies (2.3) if and only if it is an Carathéodory solution to the IVP
with . We moreover say that is regular^{1}^{1}1Note that our notion of regularity is weaker than the one in [Glö15, 5.16]: regular in loc. cit. means regular with a smooth evolution map in this paper. if it is semiregular and the evolution map
is continuous. Note that if is semiregular, then it has an exponential function given by
(2.4) 
where is the constant function and
If is a local Lie group modelled on , with Lie algebra , we call locally semiregular if there exists an open neighbourhood such that for each , there exists such that
If it exists, then is uniquely determined. If, moreover, has a global chart, then is called locally regular if it is locally semiregular and can be chosen such that
is continuous.
We record for future reference the following results.
Lemma 2.4 ([Glö15, 5.20]).
Set , and let be an regular Fréchet–Lie group with Lie algebra . Then the evolution map is smooth if and only if it is smooth as a map .
Lemma 2.5 ([Glö15, 5.25]).
Let be a Fréchet–Lie group. Then

is semiregular if and only if is locally semiregular;

is regular with a smooth evolution map if and only if is locally regular with a smooth evolution map.
Lemma 2.6 ([Glö15, Theorems A and C]).
Every Banach–Lie group is regular, with a smooth evolution map.
Lemma 2.7.
Let be Banach Lie groups with respective Lie algebras , and let be a Lie group morphism. Then for any , we have
3. Smooth topological groups
In this section, we introduce in more detail the main protagonists of this paper, namely the smooth topological groups.
3.1. Definition and examples
Definition 3.1.
Let be a locally convex space and be a smooth manifold modelled on . We recall from §2.6 that is called a (left) smooth topological group modelled on if admits a topological group structure (with respect to the manifold topology) such that all left multiplication maps are smooth. In this paper, we will only consider smooth topological groups modelled on Banach spaces, and simply call them smooth topological groups.
Remark 3.2.
Note that one may replace left multiplication maps by right multiplication maps in Definition 3.1, to obtain a concept of right smooth topological group. If is a right smooth topological group, one can also define the concepts presented in §2.9 and §2.10, using right logarithmic derivatives. Of course, if denotes the multiplication in , then the group obtained by equipping with the multiplication instead of is a (left) smooth topological group.
There are several sources of examples of interest of smooth topological groups, and we now list some of them.
Example 3.3.
Let be Banach–Lie groups, and let be a continuous automorphic action of on , in the sense that the map
is continuous. Consider the topological group , with multiplication
Then all left multiplication maps are smooth, whereas the right multiplication maps are in general only continuous. In particular, is a smooth topological group.
Example 3.4.
Extended mapping groups are concrete classes of Example 3.3: here is the Banach–Lie group of maps () from the compact smooth manifold to some Banach–Lie group , and is a Banach–Lie group acting smoothly on , which yields a continuous action by An important special case arises for on which the circle group acts by rigid rotations (loop groups).
Example 3.5.
A second particular case of Example 3.3 is the following: let be a Banach–Lie group (e.g. ) and consider a continuous representation of on some Banach space . Then the affine group is a smooth topological group.
Example 3.6.
Example 3.7.
Let be a compact manifold and . Then the group of diffeomorphisms of is a right smooth topological group (see [Omo97, §VI.2]).