Finite-dimensional Lie subalgebras of algebras with continuous inversion

Beltiţă, Daniel; Neeb, Karl-Hermann

doi:10.4064/sm185-3-3

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…It is a natural question whether the linearity condition on a connected finite-dimensional Lie group becomes weaker if we only require that it is a Lie subgroup of the unit group of some Banach algebra or even a CIA. According to the following theorem, this is not the case ([BelNe06]). Its Banach version is due toLuminet and Valette ([LV94]).…”

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confidence: 78%

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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confidence: 78%

Towards a Lie theory of locally convex groups

2006

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“…It should be also be mentioned that, according to [31] (see also [5]), a connected (finite-dimensional, real) Lie group G has a continuous faithful embedding into the group of invertible elements of some Banach algebra with its norm topology if and only if G is a linear Lie group.…”

Section: Nss Sequences In Lie Groupsmentioning

confidence: 99%

Escaping a Neighborhood Along a Prescribed Sequence in Lie Groups and Banach Algebras

Badea

Devinck²,

Grivaux³

2019

Can. Math. Bull.

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It is shown that Jamison sequences, introduced in [C. Badea and S. Grivaux, Unimodular eigenvalues, uniformly distributed sequences and linear dynamics, Adv. Math. 211 (2007), no. 2, 766-793], arise naturally in the study of topological groups with no small subgroups, of Banach algebra elements whose powers are close to identity along subsequences, and in characterizations of (self-adjoint) positive operators by the accretiveness of some of its powers. The common core of these results is a description of those sequences for which non-identity elements in Lie groups or Banach algebras escape an arbitrary small neighborhood of the identity in a number of steps belonging to the given sequence. Several spectral characterizations of Jamison sequences are given and other related results are proved.2010 Mathematics Subject Classification. 47A10, 47A12, 47A60, 22E15.

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“…If, in addition, A is complete, then the same arguments as for Banach algebras lead to a holomorphic functional calculus ([Wa67], [Gl02]). Since completeness is in general not inherited by quotients ([Ko69], §31.6), it is natural to consider for CIAs the weaker condition that they are FC-complete in the sense that they are closed under holomorphic functional calculus (see [BlN08]). This means that for a ∈ A, any open neighborhood U of σ(a), each holomorphic function f ∈ O(U ) and any contour Γ around σ(a) in U , the integral…”

Section: Definitions and Examplesmentioning

confidence: 99%

Geometric Characterization of Hermitian Algebras With Continuous Inversion

Beltiţă¹,

Neeb²

2009

Bull. Aust. Math. Soc.

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A hermitian algebra is a unital associative C-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in R. In the case of an algebra A endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra A is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups U n (A) on each flag manifold is transitive.2000 Mathematics subject classification: primary 46K05; secondary 22E65, 58B10, 46H30.

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Finite-dimensional Lie subalgebras of algebras with continuous inversion

Cited by 4 publications

References 13 publications

Towards a Lie theory of locally convex groups

Towards a Lie theory of locally convex groups

Escaping a Neighborhood Along a Prescribed Sequence in Lie Groups and Banach Algebras

Geometric Characterization of Hermitian Algebras With Continuous Inversion

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