Isometries in semisimple, commutative Banach algebras

Abstract

We show that for any semisimple, commutative, complex Banach algebra A with unit there are norms on A, which we call natural norms, equivalent to the original norm on A with the following property: Let (A, II IA, eA) and (B, II -II B, eB) are commutative, semisimple Banach algebras with units and natural norms. Assume T is a linear isometry from (A, I1I IA) onto (B, I I IIB) with TeA = eB. Then T is an isomorphism in the category of Banach algebras. For a fairly large class of algebras, for example, for uniform algebras, for algebras of the form Ck(X), Lip(X), AC(X), the natural norm we have defined coincides with a usual norm. 0. Introduction. Let A and B be uniform algebras, i.e., Banach algebras with units such that IIf211 = If 112 for all f E A (for all f E B, respectively). The wellknown Nagasawa theorem [4] states that A and B are isometric if and only if they are isomorphic in the category of algebras, and that every linear isometry from A onto B which preserves units is an isomorphism of algebras. In [1, 2, 5-7] it has been proved that the Nagasawa theorem remains true for some other Banach algebras. In 1965 M. Cambern [1] proved that if both A and B are equal to C' [0, 1] or to AC[0, 1], then any isometry of A onto B is induced by a homeomorphism of the unit interval [0,1]. Here C1 [0, 1] is the algebra of complex-valued, continuously differentiable functions on [0,1] with norm IIf I1 = 0max (If(t) I + If'(t) |) for f E C' [0, 1], and AC[0, 1] is an algebra of complex-valued, absolutely continuous functions on [0,1] with norm IIf = Ilf loo + llf'lli = ma If(t)l + If'(t)I dt for f E AC[0, 1]. In 1971 N. V. Rao and A. K. Roy [7] proved that this holds for algebras of Lipschitz functions and continuously differentiable functions both with norm IIf 1 = li 10+ lif 11100 In 1981 M. Cambern and V. Pathak [2] proved it for Co (X), where X is a closed subset of the real line containing no isolated points. Here Co (X) is a Banach space of continuously differentiable functions f on X which are such that f and f' are zero at infinity with norm defined as for C' [0, 1]. Received by the editors January 9, 1984 and, in revised form, April 23, 1984. 1980 Mathematics Subject CaZasfication. Primary 46J05; Secondary 46B20, 46B25. (?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page 65 This content downloaded from 207.46.13.82 on Mon, 17 Oct 2016 05:03:53 UTC All use subject to http://about.jstor.org/terms