# Automorphisms and derivations of a Fréchet algebra of locally integrable functions

Studia Mathematica (1992)

• Volume: 103, Issue: 1, page 51-69
• ISSN: 0039-3223

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## Abstract

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We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L{¹}_{loc}$ of locally integrable functions on the half-line ${ℝ}^{+}$. We show, among other things, that every automorphism θ of $L{¹}_{loc}$ is of the form $\theta ={\phi }_{a}{e}^{\lambda X}{e}^{D}$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and ${\phi }_{a}$ is the dilation operator $\left({\phi }_{a}f\right)\left(x\right)=af\left(ax\right)$ ($f\in L{¹}_{loc}$, $x\in {ℝ}^{+}$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded sets and is the semidirect product of a connected subgroup and a discrete group which is isomorphic to the discrete group of real numbers.

## How to cite

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Ghahramani, F., and McClure, J.. "Automorphisms and derivations of a Fréchet algebra of locally integrable functions." Studia Mathematica 103.1 (1992): 51-69. <http://eudml.org/doc/215935>.

@article{Ghahramani1992,
abstract = {We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L¹_\{loc\}$ of locally integrable functions on the half-line $ℝ^+$. We show, among other things, that every automorphism θ of $L¹_\{loc\}$ is of the form $θ = φ _ae^\{λX\}e^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and $φ_a$ is the dilation operator $(φ_af)(x) = af(ax)$ ($f ∈ L¹_\{loc\}$, $x ∈ ℝ^+$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded sets and is the semidirect product of a connected subgroup and a discrete group which is isomorphic to the discrete group of real numbers.},
author = {Ghahramani, F., McClure, J.},
journal = {Studia Mathematica},
keywords = {representations; automorphisms; derivations; multipliers; Fréchet algebra; locally integrable functions on the half-line; dilation operator; automorphism group; topology of uniform convergence on bounded sets; semidirect product of a connected subgroup and a discrete group},
language = {eng},
number = {1},
pages = {51-69},
title = {Automorphisms and derivations of a Fréchet algebra of locally integrable functions},
url = {http://eudml.org/doc/215935},
volume = {103},
year = {1992},
}

TY - JOUR
AU - Ghahramani, F.
AU - McClure, J.
TI - Automorphisms and derivations of a Fréchet algebra of locally integrable functions
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 1
SP - 51
EP - 69
AB - We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L¹_{loc}$ of locally integrable functions on the half-line $ℝ^+$. We show, among other things, that every automorphism θ of $L¹_{loc}$ is of the form $θ = φ _ae^{λX}e^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and $φ_a$ is the dilation operator $(φ_af)(x) = af(ax)$ ($f ∈ L¹_{loc}$, $x ∈ ℝ^+$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded sets and is the semidirect product of a connected subgroup and a discrete group which is isomorphic to the discrete group of real numbers.
LA - eng
KW - representations; automorphisms; derivations; multipliers; Fréchet algebra; locally integrable functions on the half-line; dilation operator; automorphism group; topology of uniform convergence on bounded sets; semidirect product of a connected subgroup and a discrete group
UR - http://eudml.org/doc/215935
ER -

## References

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