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This thesis presents three pieces of work. Within the first two thirds of the thesis, we study Arens regularity of Banach algebras. We first study Arens regularity of weighted semigroup algebras that arise from totally ordered semilattices. This is a natural continuation of [24], where they focus on studying Arens regularity of the unweighted case. We provide a sufficient condition for when a weighted semigroup algebra is not strongly Arens irregular and a characterization of Arens regularity of the weighted semigroup algebra. We then focus on three specific totally ordered semilattices, the natural numbers with the minimum operation, the natural numbers with the maximum operation and the integers with the maximum operation to obtain stronger results than those obtained for a generic totally ordered semilattice. Later on, we focus on two different Banach sequence algebras, the James p-algebra and the Feinstein algebra. Amongst other properties, we prove that the Feinstein algebra is Arens regular, which provides a second example of an Arens regular natural Banach sequence algebra that is not an ideal in its bidual, the first one being the remarkable example obtained in [7]. We study whether the James p-algebra is a BSE algebra with a BSE norm, for 1 In the final part of the thesis, we focus on Blaschke products. We study the decomposability of a finite Blaschke product B of degree a potency of 2 into n degree-2 Blaschke products, examining the connections between Blaschke products, the elliptical range theorem, Poncelet theorem, and the monodromy group. We show that if the numerical range of the compression of the shift operator of B a Blaschke product of degree n is an ellipse, then B can be written as a composition of lower-degree Blaschke products that correspond to a factorization of the integer n. We also show that a Blaschke product of degree 2^n with an elliptical Blaschke curve has at most n distinct critical values, and we use this to examine the monodromy group associated with a regularized Blaschke product B. We prove that if B can be decomposed into n degree-2 Blaschke products, then the monodromy group associated with B is the wreath product of n cyclic groups of order 2. Lastly, we study the group of invariants of a Blaschke product B of order 2^n when B is a composition of n Blaschke products of order 2.

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.).

For a Banach function algebra A, we consider the problem of representing a continuous d-homogeneous polynomial P : A → X , where X is an arbitrary Banach space, that satisfies the property P ( f + g ) = P ( f ) + P ( g ) whenever f , g ∈ A are such that supp ( f ) ∩ supp ( g ) = ∅ . We show that such a polynomial can be represented as P ( f ) = T ( f d ) ( f ∈ A ) for some continuous linear map T : A → X for a variety of Banach function algebras such as the algebra of continuous functions C 0 ( Ω ) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lip α ( K ) for any compact metric space K and α ∈ ] 0 , 1 [ , the Figa–Talamanca–Herz algebra A p ( G ) for some locally compact groups G and p ∈ ] 1 , + ∞ [ , the algebras A C ( [ a , b ] ) and B V C ( [ a , b ] ) of absolutely continuous functions and of continuous functions of bounded variation on the interval [ a , b ] . In the case where A = C n ( [ a , b ] ) , P can be represented as P ( f ) = ∑ T ( n 1 , … , n d ) ( f ( n 1 ) ⋯ f ( n d ) ) , where the sum is taken over ( n 1 , … , n d ) ∈ Z d with 0 ≤ n 1 ≤ … ≤ n d ≤ n , for appropriate continuous linear maps T ( n 1 , … , n d ) : C n − n d ( [ a , b ] ) → X .

Let $\alpha _0\in \mathbb {C} \setminus \{0\}$, $A$ and $B$ be Banach function algebras. Also, let $\rho _1:\Omega _1 \rightarrow A$, $\rho _2:\Omega _2 \rightarrow A$, $\tau _1: \Omega _1 \rightarrow B$ and $\tau _2:\Omega _2 \rightarrow B$ be surjections such that $\|\rho _1(\omega _1)\rho _2(\omega _2)+\alpha _0\|_\infty =\|\tau _1(\omega _1)\tau _2(\omega _2)+\alpha _0\|_\infty $ for all $\omega _1\in \Omega _1, \omega _2\in \Omega _2$, where $\Omega _1$, $\Omega _2$ are two non-empty sets. Motivated by recent investigations on such maps between unital Banach function algebras, in this paper we characterize these maps for certain non-unital Banach function algebras including pointed Lipschitz algebras and abstract Segal algebras of the Talamanca-Herz algebras when the underlying groups are first countable. Moreover, sufficient conditions are given to guarantee such maps induce weighted composition operators.

Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by considering $\mathcal{F}$-differentiable functions on $X$. In this paper, we investigate composition, the chain rule, and the quotient rule for this notion of differentiability. We give an example where the chain rule fails, and give a number of sufficient conditions for the chain rule to hold. Where the chain rule holds, we observe that the Fa\'a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of $\mathcal{F}$-differentiable functions.

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 $B$.

Let $A$ and $B$ be Banach function algebras on compact Hausdorff spaces $X$ and $Y$, respectively. Given a non-zero scalar $\alpha$and $s,t\in \Bbb N$ we characterize the general form of suitable powers of surjective maps $T, T': A \longrightarrow B$ satisfying $\|(Tf)^s (T'g)^t-\alpha\|_Y=\|f^s g^t-\alpha \|_X$, for all $f,g \in A$, where $\|\cdot \|_X$ and $\|\cdot \|_Y$ denote the supremum norms on $X$ and $Y$, respectively. A similar result is given for the case where $T=T'$ and $T$ is defined between certain subsets of $A$ and $B$. We also show that if $T: A\longrightarrow B$ is a surjective map satisfying the stronger condition$R_\pi((Tf)^{s}(Tg)^{t}-\alpha)\cap R_\pi(f^{s}g^{t}-\alpha)\neq\varnothing $ for all $f,g \in A$, where $R_\pi(\cdot)$ denotes the peripheral range of the algebra elements, then there exists a homeomorphism $\varphi$ from the Choquet boundary $c(B)$ of $B$ onto the Choquet boundary $c(A)$ of $A$ such that $(Tf)^{d}(y)=(T1)^{d}(y)\,(f \circ \varphi(y))^{d}$ for all $f\in A$ and $y\in c(B)$,where $d$ is the greatest common divisor of $s$ and $t$.

Let 𝑋 and 𝐾 be compact plane sets with 𝐾 ⊆ 𝑋 . We define 𝐴 ( 𝑋 , 𝐾 ) = { 𝑓 ∈ 𝐶 ( 𝑋 ) ∶ 𝑓 | 𝐾 ∈ 𝐴 ( 𝐾 ) } , where 𝐴 ( 𝐾 ) = { 𝑔 ∈ 𝐶 ( 𝑋 ) ∶ 𝑔 is analytic on i n t ( 𝐾 ) } . For 𝛼 ∈ ( 0 , 1 ] , we define L i p ( 𝑋 , 𝐾 , 𝛼 ) = { 𝑓 ∈ 𝐶 ( 𝑋 ) ∶ 𝑝 𝛼 , 𝐾 ( 𝑓 ) = s u p { | 𝑓 ( 𝑧 ) − 𝑓 ( 𝑤 ) | / | 𝑧 − 𝑤 | 𝛼 ∶ 𝑧 , 𝑤 ∈ 𝐾 , 𝑧 ≠ 𝑤 } < ∞ } and L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) = 𝐴 ( 𝑋 , 𝐾 ) ∩ L i p ( 𝑋 , 𝐾 , 𝛼 ) . It is known that L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) is a natural Banach function algebra on 𝑋 under the norm | | 𝑓 | | L i p ( 𝑋 , 𝐾 , 𝛼 ) = | | 𝑓 | | 𝑋 + 𝑝 𝛼 , 𝐾 ( 𝑓 ) , where | | 𝑓 | | 𝑋 = s u p { | 𝑓 ( 𝑥 ) | ∶ 𝑥 ∈ 𝑋 } . These algebras are called extended analytic Lipschitz algebras. In this paper we study unital homomorphisms from natural Banach function subalgebras of L i p 𝐴 ( 𝑋 1 , 𝐾 1 , 𝛼 1 ) to natural Banach function subalgebras of L i p 𝐴 ( 𝑋 2 , 𝐾 2 , 𝛼 2 ) and investigate necessary and sufficient conditions for which these homomorphisms are compact. We also determine the spectrum of unital compact endomorphisms of L i p 𝐴 ( 𝑋 , 𝐾 , 𝛼 ) .

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y, f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A, $$ Tf\left( y \right) = \left\{ \begin{gathered} \eta \left( y \right)f\left( {\phi \left( y \right)} \right) y \in K, \hfill \\ - \frac{\alpha } {{\left| \alpha \right|}}\eta \left( y \right)\overline {f\left( {\phi \left( y \right)} \right)} y \in c\left( B \right)\backslash K \hfill \\ \end{gathered} \right. $$ . In particular, if T satisfies the stronger condition Rπ(fg + α) = Rπ(Tf Tg + α), where Rπ(.) denotes the peripheral range of algebra elements, then Tf(y) = T1(y)f(φ(y)), y ∈ c(B), for some homeomorphism φ: c(B) → c(A). At the end of the paper, we consider the case where X and Y are locally compact Hausdorff spaces and show that if A and B are Banach function algebras on X and Y, respectively, then every surjective map T: A → B satisfying ‖Tf Tg‖Y = ‖fg‖, f, g ∈ A, induces a homeomorphism between quotient spaces of particular subsets of X and Y by some equivalence relations.

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 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.

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.

Characterizations of the space CR (X) of all real-valued continuous functions on a compact Hausdorff space X among its subspaces are investigated under the circumstances of operating functions. One of the main purpose in this paper is to disprove the following conjecture: if a non-affine function operates on an ultraseparating real Banach function space E on X, then E = CR (X). A positive answer is given in the case that E satisfies a stronger separation axiom than ultraseparation one, which the real part of an ultraseparating Banach function algebra satisfies. For the original conjecture a counterexample is given; there is an ultraseparating real Banach function space on a compact metric space Y on which the function |·| operates, but it does not coincide with CR (Y). A characterization is given for non-affine functions which operate only on CR (X) among ultraseparating real Banach function spaces on X. By using these results the symbolic calculus on real Banach function spaces is investigated without extra hypothesis of ultraseparation. Several non-local-Lipschitz functions are shown not to operate on a real Banach function space E on X unless E = CR (X). In particular, the function tp defined on [0,1) for a p with 0 < p < 1 or the standard Cantor function on [0, 1] never operates on a real Banach function space E on X unless E = CR (X). Functions which operate on the real part of a real function algebra are also investigated. A positive answer is given for the conjecture that only affine functions operate on the real part of a non-trivial real function algebra.

We investigate the completeness and completions of the normed algebras (D(1)(X),∥•∥) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D(1)(X),∥•∥) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D(1)(X),∥•∥) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ. In an earlier paper of Bland and Feinstein, the notion of an F-derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D ((1))/F corresponding to the normed algebras (D(1)(X),∥•∥). In the present paper, we obtain stronger results concerning the questions when (D(1)(X),∥•∥) and D ((1))/F (X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘F-regular'. An example of Bishop shows that the completion of (D(1)(X),∥•∥) need not be semisimple. We show that the completion of (D(1)(X),∥•∥) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X. We prove that the character space of D(1)(X) is equal to X for all perfect, compact plane sets X, whether or not (D(1)(X),∥•∥) is complete. In particular, characters on the normed algebras (D(1)(X),∥•∥) are automatically continuous.

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.

Let X be a compact, plane set and let K be a compact subset of X . We introduce new classes of Lipschitz algebras Lip( X , K , &#945 ), lip( X , K , &#945 ), consisting of those continuous functions f on X such that f|K &#949 Lip( K , &#945 ), lip( K , &#945 ), and their analytic subalgebras Lip A ( X , K , &#945 ) = Lip( X , K , &#945 ) &#8745 A ( X , K ) and lip A ( X , K , &#945 ) = lip( X , K , &#945 ) ∩ A( X , K ), where 0 X , K ) is the algebra of all continuous complex-valued functions on X , which are analytic on the interior of K . We show that the maximal ideal spaces of these extended Lipschitz algebras coincide with X . Quaestiones Mathematicae 30(2007), 349–353

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

The classical problem of existence of non-local function algebras was settled in the affirmative by Eva Kallin in the early sixties by her well-known example [17], (see also [6, p. 170] and [22, p. 83, Example]. A few years later R. G. Blumenthal [3, 4] remarked that Kallin’s example was simply a particular case of a type of algebras studied by S. J. Sidney in his dissertation (see [21]). The previous results were obtained within the standard context of Banach function algebra theory. On the other hand, working within the general framework of Topological Algebras, not necessarily normed ones (we refer to A. Mallios [18] for the relevant terminology), we have already considered in [12] the spectrum of Sidney’s algebra. More precisely, we looked at it, as a ”gluing space” of the spectra of two factor tensor product algebras, whose sum constituted, by definition, the algebra of Sidney. In point of fact, it was Blumenthal (loc. cit.), who actually defined the spectrum of Sidney’s algebra, as a gluing space, his result being thus subsumed into ours [12, Theorem 5.2]. Now, continuing herewith our previous work in [12], we further obtain a general existence theorem for non-local topological algebras (a la Blumenthal; see Theorem 3.2). Furthermore, based on a recent article of R. D. Mehta [19], still within the Banach function algebra theory, we consider the Choquet boundary of the (generalized) algebra of Sidney (cf. Theorem 4.1 in the sequel). Indeed, by changing the hypotheses, appropriately, we are able to have the same boundary in a more concrete form, than that one in [12]. Yet, following in the preceding general set-up A. Mallios [18] (see Lemma 4.1 below), we also obtain the Silov boundary of the same Sidney’s algebra, as above.

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.

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.

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.

In this paper we study the symbolic calculus for a Banach function algebra with certain conditions. First we define a class of Banach function algebras which contains the class of all function algebras and the class of all ultraseparating Banach function algebras. Our purpose is to prove the theorem asserting that if A is a non-trivial Banach function algebra in the class, then only analytic functions can operate on A. It is a generalization of theorems of de Leeuw and Katznelson [6], Bernard [2] and Bernard and Dufresnoy [3].