The Group of Automorphisms of L 1 (0, 1) is Connected

Abstract

It is shown that the group of the automorphisms of the radical algebra LI (0, 1) is connected in the operator norm topology, and thus every automorphism is of the form eAdeq , where A is a complex number, d is the derivation df(x) = xf(x) and q is a quasinilpotent derivation. Suppose in the Banach space L1(0, 1) we define the convolution product * by x (f * g)(x) =J f(x y) g(y) dy (f,~g Ei L (0, 1), a.e. x E (0, 1)). Then V = L1(0, 1) with this product becomes a radical Banach algebra [6], called the Volterra algebra. Kamowitz and Scheinberg in [6] investigated the structure of the automorphisms and derivations on V. There was one problem left open: is the automorphism group of V connected, in the operator norm topology? We answer this question in the affirmative. We will use the fact that the aufomorphisms and the derivations on V are continuous [4, Remark (3a)]. Every automorphism of V has an extension to an automorphism of the measure algebra M[0, 1), which we will denote by the same symbol [5, ?8]. On the space B(V) of all bounded linear operators on V, we consider strong operator topology (SO) defined by: a net (T ) of operators tends to an operator T in (SO) if, and only if, Taf -, Tf in norm, for every f E V . Since M[O, 1) can be identified with the multiplier algebra of V [6, Remark 10] the topology (SO) induces to M[0, 1). We denote the induced topology by (so). Let Co[0, 1) be the space of continuous functions f on [0, 1) with limx 1f(x) = 0. Then M[0, 1) = C0[0, 1)* . Let w* = a[M[0, 1), C0[0, 1)]. Then if (/a) is a (so) W bounded net and (SO) , then U u [1, Lemma 1-1]. In [1] we have shown that if an automorphism 0 of V is extended to M[0, 1), then there exists a complex number z, such that for every x E [0, 1 ), (1) 0(6$x) = ezxtx + AX x where a(,ux) > x and 4ux({x}) = 0 (for every measure ,u, we denote the infimum of the support of It by a(,)). Following the terminology used by Received by the editors November 11, 1987 and, in revised form, March 3, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 46H99, 46J99; Secondary 43A20, 43A22.