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

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 ${l^1}(w)$.