## 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* ⋊_{π} *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 \( \overset{\cdot }{\gamma } \) (*t*) = *γ*(*t*)*ξ*(*t*), *γ*(0) = 1_{H}, has a solution and the corresponding evolution map from curves in 𝔥 to curves in *H* is continuous; (b) every *C*^{1}-curve *γ* with *γ*(0) = 1 and *γ′*(0) = *x* satisfies lim_{n→∞} *γ*(*t*/*n*)^{n} = exp(*tx*); (c) the Trotter formula holds for *C*^{1} one-parameter groups in *H*; (d) the subgroup *N*^{∞} of elements with smooth *G*-orbit maps in *N* carries a natural Fréchet–Lie group structure for which the *G*-action is smooth; (e) the resulting Fréchet–Lie group *H*^{∞} := *N*^{∞} ⋊ *G* is also regular in the sense of (a).

## Preview

Unable to display preview. Download preview PDF.

## References

- [AK98]V. I. Arnold, B. A. Khesin,
*Topological Methods in Hydrodynamics*, Applied Mathematical Sciences, Vol. 125, Springer-Verlag, New York, 1998.Google Scholar - [BR87]O. Bratteli, D. W. Robinson,
*Operator Algebras and Quantum Statistical Mechanics*. 1, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, New York, 1987.Google Scholar - [BV17]M. Bruveris, F-X. Vialard,
*On completeness of groups of diffeomorphisms*, J. Eur. Math. Soc.**19**(2017), no. 5, 1507–1544.MathSciNetCrossRefzbMATHGoogle Scholar - [EM99]D. G. Ebin, G. Misiołek,
*The exponential map on*\( {D}_{\mu}^s \), in:*The Arnoldfest*(Toronto, ON, 1997), Fields Inst. Commun., Vol. 24, Amer. Math. Soc., Providence, RI, 1999, pp. 153–163.Google Scholar - [Glö07]H. Glöckner,
*Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces*, preprint (2007).Google Scholar - [Glö15]H. Glöckner,
*Measurable regularity properties of infinite-dimensional Lie groups*, arXiv:1601.02568 (2015).Google Scholar - [GN]H. Glöckner, K-H. Neeb,
*Infinite-dimensional Lie Groups*, book in preparation.Google Scholar - [Go05]M. Gordina,
*Hilbert–Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry*, J. Funct. Anal.**227**(2005), no. 2, 245–272.MathSciNetCrossRefzbMATHGoogle Scholar - [Ha17]M. Hanusch,
*Regularity of Lie groups*, arXiv:1711.03508 (2017).Google Scholar - [Ha18]M. Hanusch,
*The strong Trotter property for locally μ-convex Lie groups*, arXiv:1802.08923 (2018).Google Scholar - [KMR15]A. Kriegl, P. W. Michor, A. Rainer,
*An exotic zoo of diffeomorphism groups on*ℝ^{n}, Ann. Global Anal. Geom.**47**(2015), no. 2, 179–222.MathSciNetCrossRefzbMATHGoogle Scholar - [MOKY85]Y. Maeda, H. Omori, O. Kobayashi, A. Yoshioka,
*On regular Fréchet–Lie groups*. VIII.*Primordial operators and Fourier integral operators*, Tokyo J. Math.**8**(1985), no. 1, 1–47.MathSciNetCrossRefzbMATHGoogle Scholar - [Nee06]K-H. Neeb,
*Towards a Lie theory of locally convex groups*, Japan. J. Math.**1**(2006), no. 2, 291–468.MathSciNetCrossRefzbMATHGoogle Scholar - [Nee10]K-H. Neeb,
*On differentiable vectors for representations of infinite-dimensional lie groups*, J. Funct. Analysis**259**(2010), no. 11, 2814–2855.MathSciNetCrossRefzbMATHGoogle Scholar - [NS13]K-H. Neeb, H. Salmasian,
*Differentiable vectors and unitary representations of Fréchet–Lie supergroups*, Math. Z.**275**(2013), no. 1–2, 419–451.MathSciNetCrossRefzbMATHGoogle Scholar - [Omo97]H. Omori,
*Infinite-dimensional Lie Groups*, Translations of Mathematical Monographs, Vol. 158, American Mathematical Society, Providence, RI, 1997.Google Scholar