The Connectedness of the Group of Automorphisms of L 1 (0, 1)

F. Ghahramani
1987 Transactions of the American Mathematical Society  
For each of the radical Banach algebras Ll (0,1) and Ll (w) an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of Ll (O,I) and Ll(W) endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if III· IIi!, denotes the norm of B(LP(O, 1), Ll(O,I)), 1 < p::; 00, then the group of automorphisms of Ll(O, 1) topologized by 111·lllp is arcwise connected. It is shown that every automorphism (J of Ll
more » ... tomorphism (J of Ll (0, 1) is of the form (J = e>.d lim e qn (BSO), where each qn is a quasinilpotent derivation. It is shown that the group of principal automorphisms of 11 (w) under operator norm topology is arcwise connected, and every automorphism has the form eil>d(e>.deDe->.d)-, where a E R, A > 0, and D is a derivation, and where (e>.deDe->.d)-denotes the extension by continuity of e>'deDe->'d from a dense subalgebra of 11(w) to 11(w). O. Introduction. Suppose in the Banach space L 1 (0, 1) we define the product * by gEL 1 (0,1) , a.e. xE (0,1)). With this "convolution" product L1(0, 1) becomes a radical Banach algebra [12] . In [12] Kamowitz and Scheinberg studied the derivations and automorphisms of L1(0, 1), and asked whether the group of automorphisms is connected. We prove that this group is connected in the bounded strong operator topology (BSO) and for topologies induced by the norm of B(LP(O, 1), L1(0, 1)), where 1 < P ~ 00. The class of weighted convolution algebras has been studied by several authors from different viewpoints [1, 3, 4 and 5]. Suppose w is a continuous and positive function on R+ with w(O) = 1 and w(s + t) ~ w(s)w(t), and let L1(W) be the Banach space of all equivalence classes of Lebesgue measurable functions f with is a Banach algebra. We prove that the group of automorphisms of L1 (w) endowed with the topology (BSO) is connected. In the following lemma, for J. L 1:-0, a(J.L) denotes the infimum of the support of J.L. It follows from Titchmarsh's convolution theorem that if J. L 1:-0, v 1:-0, and a(J.L) + a(v) < 1, then a(J.L * v) = a(J.L) + a(v). License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
doi:10.2307/2000861 fatcat:bmfmy3a4qvaubkly6zeqcwzk3y