Algebras whose groups of units are Lie groups

Helge Glöckner. "Algebras whose groups of units are Lie groups." Studia Mathematica 153.2 (2002): 147-177. <http://eudml.org/doc/284755>.

@article{HelgeGlöckner2002,
abstract = {Let A be a locally convex, unital topological algebra whose group of units $A^\{×\}$ is open and such that inversion $ι : A^\{×\}→ A^\{×\}$ is continuous. 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.},
author = {Helge Glöckner},
journal = {Studia Mathematica},
keywords = {group of units; continuous inverse algebra; invertible elements; root-graded Lie groups; functional calculus},
language = {eng},
number = {2},
pages = {147-177},
title = {Algebras whose groups of units are Lie groups},
url = {http://eudml.org/doc/284755},
volume = {153},
year = {2002},
}

TY - JOUR
AU - Helge Glöckner
TI - Algebras whose groups of units are Lie groups
JO - Studia Mathematica
PY - 2002
VL - 153
IS - 2
SP - 147
EP - 177
AB - Let A be a locally convex, unital topological algebra whose group of units $A^{×}$ is open and such that inversion $ι : A^{×}→ A^{×}$ is continuous. 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.
LA - eng
KW - group of units; continuous inverse algebra; invertible elements; root-graded Lie groups; functional calculus
UR - http://eudml.org/doc/284755
ER -