Isomorphic sheaf representations of normal lattices

This article examines hereditary f-rings, after first characterizingthe hereditary von Neumann regular commutative rings as those for which the space of maximal ideals is hereditarily paracompact. It is shown that if C ( X , Z ) , the ring of integer-valued continuous functions on a zero-dimensional space X, is hereditary, then X is finite. This is shown two ways; once as a consequence of the following: if A is any singular archimedean f-ring, then A/2A is a boolean ring, and gd(A) >= gd(A/2A)+ 1 , where gd(A) stands for the global dimension of A. As a consequence of this it is also shown that if A is a singular archimedean f-ring and gd(A) <= 2, then Min(A), the space of minimal prime ideals is hereditarily paracompact. The paper concludes with a calculation of the global dimension of a mihereditary singular archimedean f-ring A, in which the cellularityof Min(A) is “much less” than |A| if finite, it is k+2, where Nk = |A|.

Weakly closed, commutative, symmetric rings of operators on a Hilbert space are examined and their structure is ex ploited to characterize normal operators in terms of the rings they generate. In the first chapter, notation is established, and the less common theorems from Hilbert space theory not to be used in later sections are briefly reviewed. In Chapter II, the structure of weakly closed, commu tative, symmetric rings with a cyclic vector is specified. This theorem is then used in conjunction with a strucure called a canonical decomposition system to establish that every weakly closed, commutative, symmetric ring or opera tors acting on a separable space and containing the iden tity is spatially isomorphic to a "divect sum" of L rings of a certain type. In Chapter III, two sets of necessary and sufficient conditions are given for two normal operators on a separ able Hilbert space to be unitarily equivalent. The concept of a spectral class of a normal operator is introduced -a natural generalization of the concept of an eigen value. Each spectral class is assigned a multiplicity, and it is shown that two normal operators are unitarily equivalent iff they have the same spectral classes with the same respective multiplicities. A multiplicity is iv also defined for weakly continuous positive functionals o n the symmetric ring generated by a normal operator in the weak topology, and it is shown that this multiplicity function determines the operator. This multiplicity theory is compared w i t h a previous theory. v CHAPTER I INTRODUCTION We shall be concerned in this paper with rings of operators on a Hilbert space, in particular, weakly closed, commutative, symmetric rings. All Hilbert spaces to be considered will be complex Hilbert spaces. If H is a Hilbert space and x,y e H, then (x(y) will denote the inner product of x and y, and 1 Jxi| will denote the norm of x. We shall consider only bounded linear operators in this paper, and therefore we shall use the word "operator" solely to mean a bounded linear operator. The set of all operators on the Hilbert space H will be denoted by B(H). If A s B(H), then A* will denote the adjoint of A, We employ the usual concepts of isometric, unitary, hermitian, normal, and positive definite operators. Knowledge of the weak, strong, strongest, and norm topologies for B(H) is assumed. The norm topology is the strongest of the four; hence, any set of operators closed in one of the topologies is closed in the norm topology. By a ring of operators we shall mean a set of opera tors forming a ring in the usual algebraic sense which is also a vector space over the complex numbers. A ring R of operators is symmetric if A e R whenever A e R. from the theory of Banach algebras, we know that any norm closed, 1 we define an operator A a e BCLg) by A ftf = af* Note that two Ly,, functions which differ only on a set of measure zero give rise to the same operator. The ring of operators thus obtained is a maximal, commutative, symmetric ring. If E is a weakly closed, commutative, symmetric ring with maximal ideal space M, then the fact that E is closed under sups gives rise to some strong topological properties of M. In this case, M is totally disconnected, and the closure of every open set is open. Thus, M contains many sets which are both open and closed, and such sets we call clopen. It is easy to see that projections in E correspond v i a the Gelfand transform to the characteristic functions of clopen sets. If U is a clopen subset of M, then we will let pu denote the corresponding projection in E. pu (m) -JtjjCm) = the characteristic function of U. Throughout the remainder of this discussion, E will denote a weakly closed commutative, symmetric subring of B(H) with identity, and M will denote the maximal ideal space of E. Corresponding to each vector 5 e H, there is a measure u on M defined by The topological properties of M carry strong implications for the resulting measure space. If S = M is measurable, then there exists a clopen set U such that m (S\UUU\S) * 0. If ? is cyclic for E ' , then u is supported on all of M, in ( A $ 0 = J A (m)du(m)

This paper is devoted to an investigation of a topological ring of analytic functions. Specifically, this ring, denoted by R, is the set of functions analytic on the unit disc with the usual addition and scalar multiplication, the Hadamard product for its ring multiplication, and the compact-open topology. The ring R is identified algebrai cally with a subring RA of the ring of continuous functions on the non-negative integers X. The operations in fT are the usual pointwise operations, ana the structure of R is determined by considering its isomorph iT. In Chapter I we are concerned with the problems of identifying the maximal ideal space of R and describing the maximal ideals intrinsically. We first show, using theorems on general rings of continuous functions, that the maximal ideals are in one-to-one correspondence with the points of the Stone-Cech compactification px of X. We next give an intrinsic description of the maximal ideals, using the properties of the power series expansions for analytic functions. Using this description we strengthen the prev ious theorem appreciably and show that the maximal ideal space with the hul1-kernel topology is homeomorphic to pX. Finally, the Hadamard product is used to give a simple iv characterization of the dual space of the topological linear space of analytic functions on the unit disc. This dual space is isomorphic to the set of functions in R whose radius of convergence exceeds one, which is exactly the intersection of the maximal ideals corresponding to points of pX -X (the dense maximal ideals of R) . In Chapter II we continue the investigation of the maximal ideals by studying the structure of their associated residue class rings. The complex number field 0 is isomorphically embedded in R/M, where M is a maximal ideal of R. If M corresponds to a point of X, then R/M and the isomorph 0* of 0 are identical; whereas, if M corresponds to a point of ^X -X, then R/M is a transcendental extension of 0* having transcendence degree c, the cardinality of the continuum. Moreover, we show, in the second case, that R/M is algebraically closed. Using theorems on transcendental extensions and algebraically closed fields, we show that, in either case, R/M and 0 are isomorphic fields. The two classes of maximal ideals are distinguished by the fact that their residue class rings admit radically different types of complex-valued isomorphisms. In Chapter III we are concerned primarily with the structure of the closed ideals of R. The basic tool used is the rotational completeness theorem for analytic v functions, which we proved using the methods and results of harmonic analysis. We show that the closure of every principal ideal is principal, give a necessary and suffi cient condition that a principal ideal be closed, and show that every closed ideal is a principal ideal generated by an idempotent element of R. Using these theorems we indicate connections with the general theory of dual rings, of which R is an example, and raise several questions for further i investigation in the direction of releasing some of the restrictions with which most of the results so far have been obtained.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Let A be a function algebra, X its Silov boundary, M its maximal ideal space. One of the outstanding problems in function theory is to find a structure of complex analytic space on M such that the functions of A are analytic for this analytic structure. In some cases (algebras with few representative measures, or polynomial hulls of regular curves in ~n) , such structures can be described. (See also the recent work of R. Basener [I] and of Gamelin, Sibony and Wermer, this conference.) The first counterexample to the existence of analytic structure was found by Stolzenberg [14] who exhibited a compact K c ~2 such that the polynomial hull h(K) of K is distinct from K , and there exis~ no analytic structure on h(K) K . More explicit examples were found by Wermer and Basener for rationally comvex hulls (see [2] and also [6] for related class of examples). Moreover, if an analytic structure exists, it would have to live on the Gleason parts of the maximal ideal space. As a consequence,if f is a non constant function in a Gleason part P , then f(P) would have to be interior point to f(M), by the fact that analytic functions are open mappings.~nd that assertion is completely false in general. The interest for the research of analytic structure is that it provides regularity theorems on the functions of A , and also theorems about representative measures, bounded point derivation, approximation theory, etc. But such theorems, in particular differentiability theorems about the functions of A , can be proved if one can find in the maximal ideal space, some kind of dynamical system along the trajectories of which one can study the properties of functions. The aim of this paper is to show how such generalized dynamical systems (brownian paths or even sheets) lives in spectrum of certain function algebras even if there is no analytic structure in these spectrumnand to obtain from that regularity theorems, theorems about Jensen measures, approximation and derivations, just as if there was analytic structure.

It is shown that the space of all regular maximal ideals in the Banach algebra H ∞ ( D ) H^{\infty }(\mathbb {D} ) with respect to the Hadamard product is isomorphic to N 0 . \mathbb {N} _{0}. The multiplicative functionals are exactly the evaluations at the n n -th Taylor coefficient. It is a consequence that for a given function f ( z ) = ∑ n = 0 ∞ a n z n f(z) =\sum _{n=0}^{\infty }a_{n} z^{n} in H ∞ ( D ) H^{\infty }(\mathbb {D} ) and for a function F ( z ) F(z) holomorphic in a neighborhood U U of 0 0 with F ( 0 ) = 0 F(0) =0 and a n ∈ U a_{n} \in U for all n ∈ N 0 n \in \mathbb {N}_{0} the function g ( z ) = ∑ n = 0 ∞ F ( a n ) z n g(z) =\sum _{n=0}^{\infty }F(a_{n} ) z^{n} is in H ∞ ( D ) . H^{\infty }(\mathbb {D} ) .

Let G denote a locally compact abelian topological group (an l.c.a.group) with dual group C\ We will denote by A7(G) the Banach algebra of bounded regular Borel measures on G under convolution multiplication and by L(G) the algebra of bounded measures absolutely continuous with respect to Haar measure on G (for discussions of these Banach algebras cf.[1], [2], and [5]).In this paper we shall be concerned with closed subalgebras 501 of M(G) with the following two properties:(1) if p e 50c and v is absolutely continuous with respect to p, then vs;(2) the maximal ideal space of 501 is G~, where the Gelfand transform p~ of P e 501 is given by p~(x)=J x dp for ^Ga; i.e., the Gelfand transform coincides with the Fourier-Stieltjes transform on 50.Any closed subalgebra of M(G) satisfying (1) will be called an L-subalgebra of M(G).It is well known thatL(G) satisfies ( 1) and (2) (cf.[5, Chapter 1]).In Theorem 1 we show that any L-subalgebra 50c of M(G), with L(G)<=50cc(L(G))1,2) also satisfies (2), where (L(G))112 is the intersection of all maximal ideals of M(G) containing L(G).We conjecture that the converse is also true; i.e., if 501 satisfies (1) and (2) then L(G)c 50c c(L(G))1/2.In Theorem 2 we prove that this is true provided G contains no copy of 7?" for > 1.The problem arises in the following way : in [6] we define the concept of abstract convolution measure algebra and prove that any such algebra 501, provided it is commutative and semisimple, may be represented as an L-subalgebra of M(S), where S is a compact topological semigroup called the structure semigroup of 5D.M(G) and L(G) are convolution measure algebras as is any L-subalgebra of the measure algebra on a semigroup.The map p-> ps which embeds 50c in M(S) is an isometry as well as an algebraic isomorphism and it preserves the order theoretic properties of 50 as a space of measures.The maximal ideal space of 50c may be identified as S~, the set of all semicharacters of S, where the Gelfand transform of p e 50c is given by p~(f) = $fdp.sfor fe S~ (a semicharacter of S is a continuous homomorphism of S into the unit disc which is not identically zero).Under pointwise multiplication S~ is a semigroup provided 50c has an approximate identity.

We study the cluster value problem for certain Banach algebras ofholomorphic functions defined on the unit ball of a complex Banachspace X. The main results are for spaces of the form X= C(K). 1 Preliminaries. A cluster value problem for a complex Banach space Xis a weak version ofthe corona problem for the open unit ball Bof X, which is a long-standingopen problem in complex analysis when Xhas dimension at least 2. Insteadof studying when B is dense in the spectrum of a uniform algebra H ofbounded analytic functions on Bin the weak topology induced by H(coronaproblem), the cluster value problem investigates the following situation:Let B¯ ∗∗ be the closed unit ball of the bidual X ∗∗ , and let M H be the spec-trum (i.e. maximal ideal space) of a uniform algebra Hof norm continuousfunctions on Bwith H⊃ X ∗ .Then π: M H → B¯ ∗∗ ,given by π(τ) = τ| X ∗ for τ∈ M H ,is surjective (as a consequence of the results in Chapter 2 and5 of [10]). For each x ∗∗ ∈ B¯ ∗∗ ,M x ∗∗ (B) = π −1 (x

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

The Sylvester equation in Banach algebras

We give some conditions under which on certain subsets of the maximal ideal space of a uniform algebra there exist structures of generalized-analytic coverings in the sense of R. Gunning and H. Rossi.

Let M +denote the Banach algebra of all complex Borel measures with support contained in [0,+8), with the usual addition and scalar multiplication, and with convolution *, and the norm being the total variation of μ. We show that the maximal ideal space X(M +) of M +, equipped with the Gelfand topology, is contractible as a topological space. In particular, it follows that every left invertible matrix with entries from M +can be completed to an invertible matrix, that is, the following statements are equivalent for f ∈ (M +)n×k, k < n: 1. There exists a matrix g ∈ M + k×n such that g * f = I k . 2. There exist matrices F,G ∈ M + n×n such that G * F = I n and F ij = f ij , 1 < i = n, 1 < j < k. We also show a similar result for all subalgebras of M + satisfying a mild condition.KeywordsContractibility of the maximal ideal spaceconvolution algebra of measuresHermite ringTolokonnikov’s lemmacoprime factorizationMathematics Subject Classification (2000)Primary 54C40Secondary 46J1032A3893D15

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.

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).

Some subalgebras of 𝐿^{∞}(𝑇) determined by their maximal ideal spaces