Automorphisms and derivations of a Fréchet algebra of locally integrable functions

We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra ${\displaystyle L{¹}_{loc}}$ of locally integrable functions on the half-line ${\displaystyle {ℝ}^{+}}$. We show, among other things, that every automorphism θ of ${\displaystyle L{¹}_{loc}}$ is of the form ${\displaystyle \theta ={\phi }_a{e}^{\lambda X}{e}^D}$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and ${\displaystyle {\phi }_a}$ is the dilation operator ${\displaystyle \left({\phi }_{a}f\right)\left(x\right)=af\left(ax\right)}$ (${\displaystyle f\in L{¹}_{loc}}$, ${\displaystyle x\in {ℝ}^{+}}$). 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.