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The connectedness of the group of automorphisms of $ L\sp 1(0,1)$

Author: F. Ghahramani
Journal: Trans. Amer. Math. Soc. 302 (1987), 647-659
MSC: Primary 46J35; Secondary 43A20
MathSciNet review: 891639
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Abstract: For each of the radical Banach algebras $ {L^1}(0,1)$ and $ {L^1}(w)$ an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of $ {L^1}(0,1)$ and $ {L^1}(w)$ endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if $ \vert\vert\vert \cdot \vert\vert{\vert _p}$ denotes the norm of $ B({L^p}(0,1)$, $ {L^1}(0,1))$, $ 1 < p \leq \infty $, then the group of automorphisms of $ {L^1}(0,1)$ topologized by $ \vert\vert\vert \cdot \vert\vert{\vert _p}$ is arcwise connected. It is shown that every automorphism $ \theta $ of $ {L^1}(0,1)$ is of the form $ \theta = {e^{\lambda d}}{\operatorname{lim}}{e^{qn}}({\text{BSO}})$, where each $ {q_n}$ is a quasinilpotent derivation. It is shown that the group of principal automorphisms of $ {l^1}(w)$ under operator norm topology is arcwise connected, and every automorphism has the form $ {e^{i\alpha d}}{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$, where $ \alpha \in {\mathbf{R}}$, $ \lambda > 0$, and $ D$ is a derivation, and where $ {({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$ denotes the extension by continuity of $ {e^{\lambda d}}{e^D}{e^{ - \lambda d}}$ from a dense subalgebra of $ {l^1}(w)$ to $ {l^1}(w)$.

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Keywords: Automorphism, derivation, connectedness, quasinilpotent derivation
Article copyright: © Copyright 1987 American Mathematical Society

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