Derivations and automorphisms of $L\sp{1}\,(0,\,1)$

Herbert Kamowitz, Stephen Scheinberg
1969 Transactions of the American Mathematical Society  
In this paper we investigate the derivations and automorphisms of the radical algebra L\0, 1), in which the product off and g is given by Recall that if X is an algebra, a linear map D is a derivation provided D(xy) =xD(y) + D(x)y for all x, ye X. All automorphisms and derivations in this paper are assumed to be bounded. In §1 we show that every derivation has the form Df=xf* p, where \p\[0, t] = 0(1/(1 -t)) as t -*■ 1 ~ ; D is quasinilpotent if and only if p has no mass at 0. We also determine
more » ... . We also determine necessary and sufficient conditions for two derivations to commute. In §2 we prove a converse to the well-known theorem that the exponential of a derivation is an automorphism. We show that if A is an automorphism of a Banach algebra X and the series for log A converges, then log A is a derivation on X. In particular, if ||i-A || < 1 or if I-A is quasinilpotent, then A is the exponential of a derivation. §3 is devoted to the study of the automorphism group s4 of L^O, 1) and its relationship to the space of derivations. We find that s/" the component of the identity in sJ, consists of automorphisms of the form eXxe", where q is a quasinilpotent derivation and A is a constant. Finally we determine those A's for which eKxeq = eD has a solution D for arbitrary q and also those A's for which D is unique. We leave open the question of whether si, is all of si.
doi:10.1090/s0002-9947-1969-0233210-4 fatcat:zidsif25bfhf3mj5c3wjtlsr6e