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Let (Formula presented.) be a commutative, unital Banach algebra. We consider the number of different non-commutative, unital Banach algebras (Formula presented.) such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.). For example, we shall prove that, in the case where (Formula presented.) is an infinite-dimensional, unital Banach function algebra, (Formula presented.) is a maximal abelian subalgebra in infinitely-many closed subalgebras of (Formula presented.) such that no two distinct subalgebras are isomorphic; the same result holds for certain examples (Formula presented.) that are local algebras. We shall also give examples of uniform algebras of the form (Formula presented.), where (Formula presented.) is a compact space, with the property that there exists a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras (Formula presented.) such that each (Formula presented.) contains (Formula presented.) as a closed subalgebra and is such that (Formula presented.) is a maximal abelian subalgebra in (Formula presented.).

Let X be a hemicompact space with () as an admissible exhaustion, and for each n N, a Banach function algebra on with respect to such that and for all f, We consider the subalgebra A = { f C(X) : of C(X) as a frechet function algebra and give a result related to its spectrum when each is natural. We also show that if X is moreover noncompact, then any closed subalgebra of A cannot be topologized as a regular Frechet Q-algebra. As an application, the Lipschitzalgebra of infinitely differentiable functions is considered.d.

Functions in Op ( I D , Re ⁡ A + L ) {\text {Op}}\left ( {{I_D},\operatorname {Re} A + L} \right ) are harmonic on D D for a closed subalgebra A A of C 0 ( Y ) {C_0}\left ( Y \right ) , an ideal I I of A A and a linear subspace L L of finite dimension in C 0 , R ( Y ) {C_0}{,_R}\left ( Y \right ) unless the uniform closure of I I is selfadjoint.

Let A be a Banach function algebra on a compact Hausdorff space X. In this paper we consider some relations between the maximal ideal space, the Shilov boundary and the Choquet boundary of A and its uniform closure A. As an application we determine the maximal ideal space, the Shilov boundary and the Choquet boundary of algebras of infinitely differentiable functions which were introduced by Dales and Davie in 1973. For some notations, definitions, elementary and known results, one can refer to [2] and [3]. Let X be a compact Hausdorff space and let C(X) denote the space of all continuous complex valued functions on X. A function algebra on X is a subalgebra of C(X) which contains the constants and separates the points of X. If there is an algebra norm on A so that Ilf gII < IlfH *HgII for all f, g E A, then A is called a normed function algebra. A complete normed function algebra on X is called a Banach function algebra on X. If the norm of a Banach function algebra is the uniform norm on X; i.e. Ilf IX = supxEX If(x)I, it is called a uniform algebra on X. If A is a function algebra on X, then A, the uniform closure of A, is a uniform algebra on X. If (A, 11 11) is a Banach function algebra on X, for every x E X the map ox: A -* C, defined by Ox(f) = f(x), is a nonzero continuous complex homomorphism on A and so Ox E MA, where MA is the maximal ideal space of A. We call ox the evaluation homomorphism at x. Clearly for every || fKx = sup If(x)I = sup IO'(f)I < sup 10(f)l = 1flM ? llfll xEX XEX OEMA where E is the Gelfand transform of f. The Banach function algebra A on X is called natural, if every 0 E MA is given by an evaluation homomorphism ox at some x E X; or, in other Received by the editors February 9, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46J10; Secondary 46J20.

We give an example of a natural Banach function algebra on the unit disc such that a smaller disc is a peak set for the algebra, but it does not contain any peak point. A Banach function algebra on a compact Hausdorff space X is a Banach algebra A consisting of continuous functions on X, such that A separates points of X and contains the constant functions. If the norm of the algebra A coincides with the sup norm on X , it is called a uniform algebra. If any linear and multiplicative functional on A is of the form f 7→ f(x) for some x ∈ X, the algebra is called natural. A subset K of X is a peak set for A if there is an f ∈ A such that f ≡ 1 on K and |f(x)| < 1 for x / ∈ K; if K = {x0} , then x0 is a peak point. If no proper subset of K is a peak set we call it a minimal peak set for A. It is well known [2] that if A is a uniform algebra on a metrizable set X then any peak set contains a peak point. In [1] H. G. Dales constructed a natural Banach function algebra on a compact subset of C having a peak set without any peak point. T. G. Honary [3] provided an example in R, however his algebra is not natural. In this note we give a very simple example of a natural Banach function algebra on the unit disc with peak sets not containing any peak point. Let K be an open nonempty subset of the complex plane C. By C(K) we denote the algebra of all bounded complex valued functions on K with continuous and bounded first order partial derivatives on K. C(K) is a Banach function algebra on K if equipped with the norm ‖f‖ = ‖f‖∞ + ‖fx‖∞ + ‖fy‖∞ , where ‖·‖∞ is the sup norm. By A(K) we denote the uniform algebra of all continuous functions on K which are analytic on K. We put Dr = {z ∈ C : |z| < r} and C− = {z : Re z < 0}. Theorem 1. Let A be a subalgebra of C(D1) consisting of all functions which are analytic on D 1 2 . Then D 1 2 is a minimal peak set for A; in particular D 1 2 does not contain any peak point. Lemma 2. There is no h ∈ A(D1) such that h(1) = 0 and |1 + (z − 1)h(z)| < 1 for z 6= 1. Proof of the lemma. Assume that such a function does exist. If we compose 1 + (z − 1)h(z) with a suitable fractional linear transformation we get an f ∈ A (C−) Received by the editors November 8, 1995. 1991 Mathematics Subject Classification. Primary 46J10. c ©1997 American Mathematical Society

Let X be a compact subset of the complex plane C, and let 0 < a < 1 . We show that the maximal ideal space of liPA (X, a) is X. Let F be a closed nonempty set in RI, and take ca with 0 < ca < 1. Then Lip(F, ca) is the algebra of bounded complex-valued functions f on F such that pa(f) = SUp {I f(x) jf(Y) I: XYEF.x Y} is finite, and lip(F, ca) is the subalgebra of functions f such that IfPx) f(Y) I,0 as Ix yJ '?. For f E Lip(X, c), set lIfila = i1f lIF +Pa(f), where I lF is the uniform norm on the closed set F . Then (Lip(F, ca), I I *a) is a Banach function algebra on F, and lip(F, ca) is a closed subalgebra of Lip(F, ca). These Lipschitz algebras were first studied by Sherbert [3]. Let X be a compact subset of the complex plane C. We follow established custom in denoting by C(X) the algebra of all continuous complex-valued functions on X and denoting by A(X) the subalgebra of functions analytic on int(X). We further define LiPA(X, ca) = Lip(X, ca) n A(X) and lipA(X, ca) = lip(X, ca) n A(X), so that LiPA(X, ca) and lipA(X, ca) are closed subalgebras of Lip(X, ca). This paper concerns the maximal ideal space of the algebra liPA(X, ca). It is clear that when the interior of X is empty, lipA(X, ca) is the algebra lip(X, ca) and its maximal ideal space is X [3]. So suppose that the interior of X is nonempty. We shall show that the maximal ideal space of lipA(X, ca) is X. The proof of this result depends on an extension theorem and some approximation lemmas. An interesting extension theorem for the Lipschitz algebra Lip(X, ca) is given in [4, Chapter VI, Theorem 3]. Here we modify the proof of that result to prove the following extension theorem for lip(X, ca). Received by the editors December 9, 1992. 1991 Mathematics Subject Classification. Primary 46J 10; Secondary 46J1 5, 46J20.

We study the range transformations Op ⁡ ( A D , Re ⁡ B ) \operatorname {Op} ({A_{D,}}\operatorname {Re} B) and Op ⁡ ( A D , B ) \operatorname {Op} ({A_D},B) for Banach function algebras A A and B B . As a special instance, the harmonicity of functions in Op ⁡ ( A D , Re ⁡ A ) \operatorname {Op} ({A_D},\operatorname {Re} A) for a nontrivial function algebra A A is established and is compared with previous investigations of Op ⁡ ( A D , A ) \operatorname {Op} ({A_D},A) and Op ⁡ ( ( Re ⁡ A ) I , ( Re ⁡ A ) ) \operatorname {Op} ({(\operatorname {Re} A)_I},(\operatorname {Re} A)) for an interval I I . In § 2 \S 2 we present some results on Op ⁡ ( A D , B ) \operatorname {Op} ({A_D},B) and use them to show that functions in Op C ( A D , B ) {\operatorname {Op} ^C}({A_D},B) are analytic for certain Banach function algebras.

In this paper, localization for ultraseparability is introduced and a local version of Bernard's lemma is proven. By using these results it is shown that a function in Op(ID, Re A) is harmonic near the origin for a uniformly closed subalgebra A of CΌ(F) and an ideal I of A unless the uniform closure cl / of / is self-adjoint; in particular, it is shown that cl / is self-adjoint if Re / Re / c Re A9 which is not true when / is merely a closed subalgebra of A.

The symbolic calculus on Banach algebras of continuous functions and related spaces is studied. In particular, functions operating on the real part of the algebra are considered. The main tool in this paper is an ultraseparation argument. As a consequence it is shown, for example, that tp on [0, 1) for any p with 0 < p < 1 does not operate on the real part of a Banach function algebra on a compact Hausdorff space unless the algebra contains every continuous function.

H. G. Dales has proved in [1] that if A A is a Banach function algebra on a compact metrizable space X X , then S ¯ 0 ( A , X ) = Γ ( A , X ) {\bar S_0}(A,X) = \Gamma (A,X) , where S 0 ( A , X ) {S_0}(A,X) is the set of peak points of A A (w.r.t. X X ) and Γ ( A , X ) \Gamma (A,X) is the Shilov boundary of A A (w.r.t. X X ). Here, by considering the relation between peak sets and peak points of a Banach function algebra A A and its uniform closure A ¯ \bar A , we present an elementary and constructive proof of the density of peak points in the Shilov boundary.