Osamu Hatori

Commutative Banach algebras which satisfy a Bochner-Schoenberg-Eberlein type-theorem

A class of commutative Banach algebras which satisfy a Bochner- Schoenberg-Eberlein-type inequality is introduced. Commutative C*-algebras, the disk algebra and the Hardy algebra on the open disk are examples.

ON A CHARACTERIZATION OF THE MAXIMAL IDEAL SPACES OF COMMUTATIVE C*-ALGEBRAS IN WHICH EVERY ELEMENT IS THE SQUARE OF ANOTHER

A topological condition is given for a locally connected compact Hausdorff space on which every complex-valued continuous function is the square of another. The condition need not be necessary nor sufficient unless the space is locally connected.

FREDHOLM COMPOSITION OPERATORS ON SPACES OF HOLOMORPHIC FUNCTIONS

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorCφ on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC2 boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if φ is a holomorphic automorphism.

A structure of ring homomorphisms on commutative Banach algebras

We give a structure theorem for a ring homomorphism of a commutative regular Banach algebra into another commutative Banach algebra. In particular, it is shown that: (i) A ring homomorphism of a commutative C∗-algebra onto another commutative C∗-algebra with connected infinite Gelfand space is either linear or anti-linear. (ii) A ring automorphism of L1(R ) is either linear or anti-linear. (iii) Cn([a, b]), L1(R ) and the disc algebra A(D) are neither ring homomorphic images of `1(S) nor Lp(G) (1 ≤ p <∞, G a compact abelian group). Let A and B be two commutative Banach algebras with Gelfand spaces ΦA and ΦB, respectively. Let ρ be a ring homomorphism of A into B such that {ρ(x)̂(ψ) : x ∈ A} = C, the complex field, (∗) for each ψ ∈ ΦB (“̂” denotes the Gelfand transform). This, of course, holds if ρ is onto. The purpose of this note is to show the following structure theorem of ρ applying the method which L. Molnar used in [5] to prove that a commutative semisimple Banach algebra which is the range of a ring homomorphism from a commutative C∗-algebra must be C∗-equivalent. Theorem 1. Suppose A is regular. Then there exist a continuous map ρ̂ of ΦB into ΦA and a division {ΦB,ΦB,ΦB} of ΦB such that ΦB and ΦB are closed, and for each a ∈ A, ρ(a)̂ = â ◦ ρ̂ on ΦB, ρ(a)̂ = ̄̂ a ◦ ρ̂ on ΦB and ρ(a)̂(ψ) = τψ(â(ρ̂(ψ))) for every ψ ∈ ΦB and for a certain discontinuous ring automorphism τψ of the complex field C. Moreover, if ρ is surjective, then ρ̂ is injective, and if A satisfies the following condition (#), then ρ̂(ΦB) is a finite set: (#) For any λn∈C with |λn|≤1/2n (n = 1, 2, . . . ) and any sequence {φ1, φ2, . . . } in ΦA such that each φn is an isolated point in {φ1, φ2, . . . }, there exists an element a ∈ A such that â(φn) = λn (n = 1, 2, . . . ). Received by the editors May 29, 1997 and, in revised form, October 27, 1997. 1991 Mathematics Subject Classification. Primary 46J05, 46E25.

Measures with natural spectra on locally compact abelian groups

Every bounded regular Borel measure on noncompact LCA groups is a sum of an absolutely continuous measure and a measure with natural spectrum. The set of bounded regular Borel measures with natural spectrum on a nondiscrete LCA group G whose Fourier-Stieltjes transforms vanish at infinity is closed under addition if and only if G is compact. Let G be a locally compact abelian group and F its dual group. Let M(G) be the measure algebra on G, and Mo(G) the subalgebra of M(G) which consists of measures whose Fourier-Stielties transforms vanish at infinity. For every ,ta E M(G) we denote by j4 the Gelfand transform of /,t. X denotes the maximal ideal space of M(G). We denote by Moo(G) the algebra of all ,t E M(G) whose -Gelfand transforms vanish off r. Note that L1(G) C Moo(G). Note also that F C X and ft = , on F for every ,t E M(G), where ft is the Fourier-Stieltjes transform of /,t. Let sp(/,t) denote the spectrum of /,t. Then sp(/,t) = ,f(X). We denote by NS(G) the set of all measures with natural spectra, that is, NS(G) = {p E M(G): Sp(1ta) = fl(F)}. Williamson [6] proved that NS(G) is a proper subset of M(G) for every non-discrete G. Put NSo(G) = NS(G) n Mo(G). Then Moo(G) C NSo(G) holds for every G. Neumann [3, Theorem 9] proved, as a generalization of a theorem of Zafran [7, Theorem 3.2], that NSo(G) = Moo(G) = Reg Mo(G) = Dec Mo(G) if G is compact, where Reg M0 (G) is the greatest regular closed subalgebra of Mo (G) and Dec M0 (G) is the Apostol algebra of MO (G). Rudin [4] for G = 1R and Varopoulos [5] for an arbitrary non-discrete G proved that NSO (G) is a proper subset of MO (G). Eschmeier, Laursen and Neumann [1] gave examples of measures in NSo (G) \ Moo (G) for certain non-compact G. In this note we show that for every non-compact G and for every ,t E M(G) (resp. Mo(G)) there exists an f E L1(G) such that ,t f E NS(G) (resp. NSo(G)). It follows that NSO (G) \ Moo (G)

Hermitian Operators and Isometries on Banach Algebras of Continuous Maps with Values in Unital Commutative -Algebras

/ 0 for every non-compact and non-discrete G, which is a solution to the question posed by Eschmeier, Laursen and Neumann [1, p.288]. Theorem 1. Let G be a non-compact locally compact abelian group. Then we have NS(G) + L1(G) = M(G). Received by the editors September 19, 1996 and, in revised form, January 20, 1997. 1991 Mathematics Subject Classification. Primary 43A10, 43A25.

A note on a class of Banach algebra-valued polynomials

We study isometries on algebras of the Lipschitz maps and the continuously differentiable maps with the values in a commutative unital -algebra. A precise proof of a theorem of Jarosz concerning isometries on spaces of continuous functions is exhibited.

Ring Derivations on Semi-Simple Commutative Banach Algebras

Let F be a Banach algebra. We give a necessary and sufficient condition for F to be finite dimensional, in terms of finite type n-homogeneous F-valued polynomials.