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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 $T: A \longrightarrow B$ be a surjective operator between two unital semisimple commutative Banach algebras $A$ and $B$ with $T1=1$. We show that if $T$ satisfies the peripheral multiplicativity condition $\sigma_\pi(Tf.Tg) = \sigma_\pi (f.g)$ for all $f$ and $g$ in $A$, where $\sigma_\pi(f)$ shows the peripheral spectrum of $f$, then $T$ is a composition operator in modulus on the $\check{S}$ilov boundary of $A$ in the sense that $|f(x)|=|Tf(\tau(x))|,$ for each $f\in A$ and $x\in \partial(A)$ where $\tau: \partial (A) \longrightarrow \partial (B)$ is a homeomorphism between $\check{S}$ilov boundaries of $A$ and $B$.

In 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that $ \mathrm{tsr}(A) \geq \mathrm{tsr}(C(\Phi_A))$ whenever $ A$ is approximately regular.

In this paper, we shall study contractive and pointwise contractive Banach function algebras, in which each maximal modular ideal has a contractive or pointwise contractive approximate identity, respectively, and we shall seek to characterize these al- gebras. We shall give many examples, including uniform algebras, that distinguish between contractive and pointwise contractive Banach function algebras. We shall describe a con- tractive Banach function algebra which is not equivalent to a uniform algebra. We shall also obtain results about Banach sequence algebras and Banach function algebras that are ideals in their second duals.

Let $A$ be a unital Banach function algebra with character space $\Phi_A$. For $x\in \Phi_A$, let $M_x$ and $J_x$ be the ideals of functions vanishing at $x$, and in a neighbourhood of $x$, respectively. It is shown that the hull of $J_x$ is connected, and that if $x$ does not belong to the Shilov boundary of $A$ then the set $\{y\in\Phi_A: M_x\supseteq J_y\}$ has an infinite, connected subset. Various related results are given.

We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity, such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a `large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0., Comment: 28 pages, to appear Transactions American Math. Society

In this note we answer a question of W. G. Bade by showing that if a normal, unital Banach function algebra A is strongly regular at one of its characters φ has a bounded approximate identity, then A has bounded relative units at φ. In particular, every strong Ditkin algebra has bounded relative units at all points of its character space. There need not, however, be a global bound available for the relative units.

In this note we use Gelfand theory to show that the validity of a spectral mapping theorem for a given representation Φ:M → L(X) of a Banach function algebra M on a bounded open set in Cn implies the validity of one-sided spectral mapping theorems for all subspectra. In particular, a spectral mapping theorem for the Taylor spectrum yields at least a one-sided spectral mapping theorem for the essential Taylor spectrum. In our main exampleMis the multiplier algebra of a Banach space X of analytic functions and F is the canonical representation of M on X. In this case, we show that interpolating sequences for M, or suitably defined Berezin transforms, can sometimes be used to obtain the missing inclusion for the essential Taylor spectrum or its parts.

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.

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.

We construct certain Banach algebras of infinitely differentiable functions on compact plane sets such that the algebras are quasianalytic, and we use these algebras to construct examples of Banach algebras defined on their maximal ideal spaces which, first, have only countably many peak points and, second, have the property that a discontinuous function operates on the algebra. We show that any function defined on an open subset of the plane which operates on a Banach function algebra is necessarily continuous on a dense open subset of its domain.

In this paper we study the functions which operate on Re A (the space of real parts of elements of a Banach function algebra A). We prove as our main result that if A is uniform or if A is ultraseparating, then only linear functions operate boundedly. We finally obtain a dichotomy for symbolic calculus on C̃(K), where K is a compact subset of the unit circle T andC̃(K) denotes the space of those continuous functions ƒ on K which admit a continuous extension to T whose Fourier conjugate is continuous.