Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: F. Ghahramani
Journal: Trans. Amer. Math. Soc. 302 (1987), 647-659
MSC: Primary 46J35; Secondary 43A20
MathSciNet review: 891639
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $||| \cdot ||{|_p}$ 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 $||| \cdot ||{|_p}$ 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 ${l^1}(w)$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46J35, 43A20

Retrieve articles in all journals with MSC: 46J35, 43A20

Additional Information

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