Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform

Detlev Poguntke. "Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform." Colloquium Mathematicae 118.1 (2010): 283-298. <http://eudml.org/doc/286390>.

@article{DetlevPoguntke2010,
abstract = {For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group. As a consequence one sees that both algebras are commutative (which is not immediate from the definition), 𝔅 is C*-dense in ℭ, and 𝔅 is a completely regular symmetric Wiener algebra. As a byproduct of our approach we give another proof of the injectivity of Harish-Chandra's spherical Fourier transform, which is based on a theorem on C*-algebras of solvable Lie groups (due to N. V. Pedersen). The article closes with some open questions for more general solvable Lie groups. To some extent the article is written with a view to these questions, that is, we try to apply, as much as possible (at the moment), methods which work also outside the semisimple context.},
author = {Detlev Poguntke},
journal = {Colloquium Mathematicae},
keywords = {Lie group; Iwasawa decomposition; group algebra; semisimple Lie group; Laplacian; spherical Fourier transform},
language = {eng},
number = {1},
pages = {283-298},
title = {Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform},
url = {http://eudml.org/doc/286390},
volume = {118},
year = {2010},
}

TY - JOUR
AU - Detlev Poguntke
TI - Banach algebras associated with Laplacians on solvable Lie groups and injectivity of the Harish-Chandra transform
JO - Colloquium Mathematicae
PY - 2010
VL - 118
IS - 1
SP - 283
EP - 298
AB - For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group. As a consequence one sees that both algebras are commutative (which is not immediate from the definition), 𝔅 is C*-dense in ℭ, and 𝔅 is a completely regular symmetric Wiener algebra. As a byproduct of our approach we give another proof of the injectivity of Harish-Chandra's spherical Fourier transform, which is based on a theorem on C*-algebras of solvable Lie groups (due to N. V. Pedersen). The article closes with some open questions for more general solvable Lie groups. To some extent the article is written with a view to these questions, that is, we try to apply, as much as possible (at the moment), methods which work also outside the semisimple context.
LA - eng
KW - Lie group; Iwasawa decomposition; group algebra; semisimple Lie group; Laplacian; spherical Fourier transform
UR - http://eudml.org/doc/286390
ER -