Fine potential and function algebras

A complete description is given of the analytic structure of maximal dimension in the spectra of a wide class of concrete function algebras generated by analytic functions.A connection is also given with point derivations on such algebras.I. Introduction.Let B be a function algebra acting on its spectrum (maximal ideal space) Sp B, with Shilov boundary TB.According to a wellknown result of Rossi [15], the functions in B obey a weak maximum modulus principle on Sp B -TB.One of the principal themes in the theory of function algebras is the "explanation" of this analytic-type behavior by the existence of analytic structure in Sp B. More precisely, let us say that an analytic variety (of dimension n) in Sp B is a pair (X, r) where X is an analytic variety (of pure dimension n), r: X -► Sp B is a continuous map, and b ° r is analytic for each b e B. Then the question is: what are the nontrivial analytic varieties in SpS?Although Stolzenberg's example [16] shows that one does not have the existence of nontrivial analytic structure in all situations, a number of global and local conditions which imply the existence of analytic structure have been found by Gleason [11], Browder [8], Clayton [9] and others.A different approach, due to Bishop [3], which depends upon the existence of functions in B with finite fibers, leads to the construction of 1-dimensional analytic structure in Sp B. Bishop's ideas have been expanded and exploited by Stolzenberg [17], Bjork [5],[6], Alexander [1], [2], the author [20], and others, and have proven very useful in dealing with questions of approximation.In this paper, we shall be concerned with the classification of high-dimensional analytic structure.For a fairly wide class of concrete function algebras, those which contain a sufficiently rich collection of analytic functions, we are able to completely describe the analytic structure of maximal dimension which

Point derivations and analytic structure in the spectrum of a Banach algebra

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

In this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

Let X be a compact nowhere dense subset of the complex plane ℂ, let C(X) be the linear space of all continuous functions on X endowed with the uniform norm, and let dA denote two-dimensional Lebesgue (or area) measure in ℂ. Denote by R(X) the closure in C(X) of the set of all rational functions having no poles on X. It is well known that if X is sufficiently massive, then the functions in R(X) can inherit many of the properties usually associated with the analytic functions, such as unlimited degrees of differentiability and even the uniqueness property itself. Here we shall examine the extent to which some of those properties are inherited by the larger algebra H ∞ (X), which by definition is the weak-* closure of R(X) in L∞(X) = L∞(X, dA).

Let A \mathfrak {A} be a UHF algebra and A ( D ) \mathcal {A}({\mathbf {D}}) the disk algebra. If A = [ ∪ n ≥ 1 A n ] − \mathfrak {A} = {\left [ {{ \cup _{n \geq 1}}{\mathfrak {A}_n}} \right ]^ - } and α \alpha is a product-type automorphism of A \mathfrak {A} which leaves each A n {\mathfrak {A}_n} invariant, then α \alpha defines an embedding \[ A n ⊗ A ( D ) ↪ ı n A n + 1 ⊗ A ( D ) \mathfrak {A}_n \otimes \mathcal {A}({\mathbf {D}}) \stackrel {\imath _n}{\hookrightarrow } {\mathfrak {A}_{n + 1}} \otimes \mathcal {A}({\mathbf {D}}) \] . The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum Γ ( α ) \Gamma (\alpha ) is finite.

Inspired by the notion of convex hull of a compact set in R n \mathbb {R}^{n} ,one defines the notion of the polynomial hull of a compact set K K in C n \mathbb {C}^{n} , denoted K ^ \widehat {K} . The complementary set K ^ ∖ K \widehat {K} \setminus K is of interest in many problems, and has been studied since the 1950’s. In this paper we consider the question: Fix an integer n n and fix a smooth compact n n -dimensional manifold L L in C n \mathbb {C}^{n} . Consider a set S S in C n \mathbb {C}^{n} and fix a point z 0 z^{0} in S S . We say that S S has analytic structure at z 0 z^{0} if there exists a Riemann surface contained in S S and passing through the point z 0 z^{0} . Question: When does the manifold L L (introduced above) contain analytic structure? When the answer is Yes, how can we describe this analytic structure? We answer this question explicitly for certain cases and pose some open problems.Further, we report on work by H. Alexander (1996) in this area. Alexander bases his proofs on work by G. Gromov in complex geometry.

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

SynopsisA classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.

Decision making about healthcare–related tests and diagnostic test strategies: what “The death of Ivan Ilyich” teaches us

We study an interesting class of Banach function algebras of in nitely di erentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of in nitely di erentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)( n!Mn) for m; n 2 N [ f0g and 2 (0; 1]. Let d = lim sup( n!Mn)1n and Xd = fz 2 C :dist(z;X)  dg. Let LipP;d(X;M; )[LipR;d(X;M; )] be the subalgebra of all f 2 Lip(X;M; )that can be approximated by the restriction to Xd of polynomials [rational functions with poles o Xd]. We show that the maximal ideal space of LipP;d(X;M; ) is cXd, the polynomially convex hullof Xd, and the maximal ideal space of LipR;d(X;M; ) is Xd, for certain compact plane sets.. Usingsome formulae from combinatorial analysis, we nd the maximal ideal space of certain subalgebrasof Lipschitz algebras of in nitely di erentiable functions.

Projective free algebras of continuous functions on compact abelian groups

Let K and X be compact plane sets such that K X. Let P (K) be the uniform closure of polynomials on K. Let R(K) be the closure of rational functions K with poles o K. Dene P (X;K) and R(X;K) to be the uniform algebras of functions in C(X) whose restriction to K belongs to P (K) and R(K), respectively. Let CZ(X;K) be the Banach algebra of functions f in C(X) such that fjK = 0. In this paper, we show that every nonzero complex homomorphism ’ on CZ(X;K) is an evaluation homomorphism ez for some z in XnK. By considering this fact, we characterize the maximal ideal space of the uniform algebra P (X;K). Moreover, we show that the uniform algebra R(X;K) is natural.

We discuss two types of `regularity point', points of continuity and R-points for Banach function algebras, which were introduced by the first author and Somerset in an earlier paper on non-regularity for Banach function algebras. We show that, even for the natural uniform algebras $R(X)$ (for compact plane sets X), these two types of regularity point can be different. We then give a new method for constructing Swiss cheese sets $X$ such that $R(X)$ is not regular, but such that $R(X)$ has no non-trivial Jensen measures. The original construction appears in the first author's previous work. Our new approach to constructing such sets is more general, and allows us to obtain additional properties. In particular, we use our construction to give an example of such a Swiss cheese set $X$ with the property that the set of points of discontinuity for $R(X)$ has positive area.

Matthews [2] presented a note in which he considers Hermite interpolation with the aid of an n-dimensional Haar subspace X of $C^1 [a,b]$ which has an $(n - 1)$-dimensional Haar subspace of $C^0 [a,b]$ as the space of all first derivatives of elements of X. This implies that X contains the constant functions. In this note we assume that X is an n-dimensional Haar subspace of $C^1 [a,b]$, and that the space of all first derivatives of elements of X is either an n-dimensional Haar subspace of $C^0 [a,b]$, too, or an $(n - 1)$-dimensional Haar subspace as in Matthews’ note. Thus we can include, for example, Haar subspaces which consist of certain exponential sums. We can show that there exists a unique element $x \in X$ which interpolates a given, $f \in C^1 [a,b]$ and its first derivative at given points of the interval $(a,b)$. In the case that X is obtained as a restriction to $[a,b]$ of an n-dimensional Haar subspace $\tilde X$ of $C^1 [a - \varepsilon ,b + \varepsilon ],\, \varepsilon > 0$, and $\tilde Y$, the space of all derivatives of functions $F \in \tilde X$, is an n-dimensional Haar subspace of $C^0 [a - \varepsilon ,b + \varepsilon ]$, too, then one even can interpolate at given points of $[a,b]$. Our proof makes use of the distinction of simple and double zeros of an $f \in C^1 [a,b]$ (cf. Karlin and Studden [1]). A similar technique has only been used in approximation theory (Chebyshev’s alternation theorem), and then it was not used in connection with Hermite interpolation and differentiable functions.