Springer Science+Business Media New York (2018)
91058 Erlangen, Germany
91058 Erlangen, Germany
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 G and N are Banach–Lie groups and π : G → Aut(N) is a homomorphism deﬁning
a continuous action of G on N, then H := N
G is a Banach manifold with a
topological group structure for which the left multiplication maps are smooth, but the
right multiplication maps need not be. We show that these groups share surprisingly
many properties with Banach–Lie groups: (a) for every regulated function ξ : [0, 1] → h
the initial value problem ˙γ(t) = γ(t)ξ(t), γ(0) = 1
, has a solution and the corresponding
evolution map from curves in h to curves in H is continuous; (b) every C
-curve γ with
γ(0) = 1 and γ
(0) = x satisﬁes lim
= exp(tx); (c) the Trotter formula
holds for C
one-parameter groups in H; (d) the subgroup N
of elements with smooth
G-orbit maps in N carries a natural Fr´echet–Lie group structure for which the G-action
is smooth; (e) the resulting Fr´echet–Lie group H
G is also regular in the
sense of (a).
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 fails 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” Fr´echet–Lie groups by certain Banach manifolds to
work in a context where the analytic tools, such as the existence of solutions of
ODEs and inverse function results, can be applied, and then perform a passage
Supported by a Marie Curie Intra-European Fellowship.
Supported by DFG-grant NE 413/9-1, “Invariante Konvexit¨at in unendlich-dimen-
Received July 26, 2016. Accepted April 15, 2018.
Corresponding Author: T. Marquis, e-mail: firstname.lastname@example.org
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