John Wermer

Banach algebras and analytic functions

Introduction 1. TheAlgebrasP(X) 52 2. The GeneralCase 55 3. Maximal Subalgebras 60 4. Dirichlet Algebras 63 5. Examples of Dirichlet Algebras 68 6. Boundaries 70 7. Polynomial Approximation in the Plane 74 8. Use of the Theory of Several Complex Variables ..... 78 9. Special Spaces 83 10. Analytic Structure in Maximal Ideal Spaces ........ 87 11. Bounded Analytic Functions 90 12. The Real Functions in A 92 Notes 93 References 98

On algebras of continuous functions

Let C denote the algebra of functions continuous on the unit circle. With norm Jjfll =sup,xi-, jf(X) J, C is a Banach algebra. Let A denote the set of all f in C which are boundary values of functions analytic in Izl 0. In [1] the question was raised whether if +(X) is any function in C and not in A, the closed algebra generated by 4 and A is all of C. It was shown in [1 ] that if 4 is real or if 4 satisfies a Lipschitz condition, then the algebra generated by A and 4 does equal C. In the following theorem the question is answered in the affirmative.

The isometries of some function spaces

H2 is a Hilbert space, and thus has many isometries; most of these do not preserve the additional structure which H2 has as a consequence of the fact that it is a space of analytic functions. For H1 and H,., however, the opposite is true: here the isometries are induced by conformal maps of the unit disc. We shall prove the following: THEOREM 1. Every linear isometry of H,, onto H,, is of the form

Pick conditions on a uniform algebra and von Neumann inequalities

Let A be a uniform algebra and let M1,…, Mn be points in the maximal ideal space of A. We describe the set D in Cn of all points (w1,…, wn) such that given e > 0 there exists f ϵ A with f(Mα) = wα, 1 ⩽ α ⩽ n, and ∥f∥ < 1 + e. This problem was solved for the disk algebra by G. Pick in 1916. We generalize Picks condition, and we also study the related problem of generalizations of von Neumanns inequality for contraction operators on Hilbert space. In our work a certain gometric notion defined in the Introduction, Definition 1, of a hyperconvex set in Cn, plays a key role.

Bounded point derivations on certain Banach algebras

Let now X be a compact set in the z-plane. Denote by R,(X) the algebra of all rational functions having no poles on X. Let R(X) be the closure of R,(X) in the uniform norm on X. R(X) is then a Banach algebra in this norm. It is well known that the homomorphisms of R(X) into the scalars are all of the formf-+f( x ), w h ere x is some point in X. We identify the point x with the homomorphism it induces. Fix x in X. It is easy to verify that there exists a bounded point derivation at x if and only if

Pick interpolation, Von Neumann inequalities, and hyperconvex sets

In 1916 Georg Pick started the study of interpolation by bounded analytic functions in the unit disk. In 1951 John Von Neumann proved an inequality for contractions on a Hilbert space. These two subjects are in fact closely connected, as was shown by Donald Sarason in 1967. We continue the study of this relationship, from the point of view of representations of uniform algebras by operators on a Hilbert space. Our work leads us to define and investigate certain convex bodies in Cn which we call hyperconvex sets.

Capacity and uniform algebras

Let A be a uniform algebra on a compact space X, let M be the maximal ideal space of A, and consider an element ƒ of A. Choose a component W of C⧹ƒ(X). In 1963 Bishop showed that {y in M ¦ ƒ(y) ϵ W} can be made into a one-dimensional complex analytic space provided there is a subset G of W having positive area such that for each λ in G {y in M ¦ ƒ(y) = λ} is finite. We show that the hypothesis of “positive area” may be replaced by “positive exterior capacity” and that no weaker condition will suffice.

Polynomial approximation on three-dimensional real-analytic submanifolds of ⁿ

It was once conjectured that if A is a uniform algebra on its maximal ideal space X and if each point of X is a peak point for A, then A = C(X). This peak point conjecture was disproved by Brian Cole in 1968. However, it was recently shown by Anderson and Izzo that the peak point conjecture does hold for uniform algebras generated by smooth functions on smooth twomanifolds with boundary. Although the corresponding assertion for smooth three-manifolds is false, we establish a peak point theorem for real-analytic three-manifolds with boundary.

The argument principle and boundaries of analytic varieties

Let M be a smooth, compact, oriented manifold in the space of n complex variables. When does there exist a complex-analytic subvariety of ℂ n ∖ M having M as its boundary in the sense of currents? R. Harvey and B. Lawson posed and solved this problem in 1975. Recently Herbert Alexander and the author gave a different solution based on Linking Numbers, in “Linking numbers and boundaries of varieties”, Ann. of Math., 151 (2000).

Isometries of certain operator algebras

To a given basis φ1, . . . , φn on an n-dimensional Hilbert space H, we associate the algebra A of all linear operators on H having every φj as an eigenvector. So, A is commutative, semisimple, and n-dimensional. Given two algebras of this type, A and B, there is a natural algebraic isomorphism τ of A and B. We study the question: When does τ preserve the operator norm? 0. Introduction We consider an n-dimensional Hilbert spaceH and fix an ordered basis φ1, . . . , φn in H. For each λ = (λ1, . . . , λn) in C, we denote by Tλ the operator in B(H) with Tλφj = λjφj , 1 ≤ j ≤ n. We put A = {Tλ|λ ∈ C}. Thus A is an n-dimensional commutative semisimple operator algebra on H. Given two such algebras A and B on Hilbert spaces H and H′, there is a natural map τ of A on B, described as follows: Let us denote by Pj the idempotent operator in A such that Pjφj = φj , Pjφk = 0 if k 6= j, 1 ≤ j, k ≤ n. Then, for each λ,