Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

Marion, Jean Luc; Robart, Thierry

doi:10.1007/bf02255990

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…( 1 ) For involutive ILB-algebras, which are certain countable projective limits of unital involutive Banach algebras, a similar result had been obtained before in [26].…”

Section: Preliminaries From Differential Calculussupporting

confidence: 54%

Algebras whose groups of units are Lie groups

Glöckner

2002

Studia Math.

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Abstract. Let A be a locally convex, unital topological algebra whose group of units A × is open and such that inversion ι :Then inversion is analytic, and thus A × is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then A × has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group A × is an analytic Lie group without a globally defined exponential function. We also discuss generalizations in the setting of "convenient differential calculus", and describe various examples.

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“…( 1 ) For involutive ILB-algebras, which are certain countable projective limits of unital involutive Banach algebras, a similar result had been obtained before in [26].…”

Section: Preliminaries From Differential Calculussupporting

confidence: 54%

Algebras whose groups of units are Lie groups

Glöckner

2002

Studia Math.

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“…These groups are examples of so called Baker-Campbell-Hausdorff Lie groups (e.g. [20], [21], [22], [50], [38], [49], [57], [58]). Let us mention here that the question of whether the exponential map is a local diffeomorphism into an infinite-dimensional Lie group has a long history.…”

Section: Introductionmentioning

confidence: 99%

“…Note that in the terminology of Banach-Lie groups it means that we prove that the groups we consider are Baker-Campbell-Hausdorff Lie groups (e.g. [20], [21], [22], [50], [38], [49], [57], [58]). Some of these articles also address the issue of completeness of the space over which a Lie group is modeled.…”

Section: Introductionmentioning

confidence: 99%

Hilbert–Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

Gordina

2005

Journal of Functional Analysis

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We describe the exponential map from an infinite-dimensional Lie algebra to an infinitedimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is −∞.

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Towards a Lie theory of locally convex groups

2006

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In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.

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Regular Fréchet-Lie groups of invertible elements in some inverse limits of unital involutive Banach algebras

Cited by 3 publications

References 11 publications

Algebras whose groups of units are Lie groups

Algebras whose groups of units are Lie groups

Hilbert–Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

Towards a Lie theory of locally convex groups

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