Maximal Ideal Space Research Articles

Contractibility of the maximal ideal space of algebras of measures in a half-space

Let H (n) be the canonical half space in R n , that is, H (n) = {(t1,...,tn) ∈ R n {0} | ∀j, (tj 6 0 and t1 = t2 = � � � = tj 1 = 0) ⇒ tj > 0} ∪ {0}. Let M(H (n) ) denote the Banach algebra of all complex Borel measures with support contained in H (n) , with the usual addition and scalar multiplication, and with convolution ∗, and the norm being the total variation of µ. It is shown that the maximal ideal space X(M(H (n) )) of M(H (n) ), equipped with the Gelfand topology, is contractible as a topological space. In particular, it follows that M(H (n) ) is a projective free ring. In fact, for all subalgebras R of M(H (n) ) that satisfy a certain mild condition, it is shown that the maximal ideal space X(R) of R is contractible. Several examples of such subalgebras are also given.

Uniform algebras invariant under transitive group actions

It is shown that C (X ) is the only uniform algebra on a compact Hausdorff space X that is invariant under a transitive action on its maximal ideal space by a locally compact group that can be approximated by Lie groups.

Some algebraic properties of theWiener–Laplace algebra

We denote by W +(ℂ +) the set of all complex-valued functions defined in the closed right half plane ℂ + D {s ∈ ℂ | Re(s) ≥ 0} that differ from the Laplace transform of functions from L 1(0, ∞) by a constant. Equipped with pointwise operations, W +(C +) forms a ring. It is known that W +(ℂ +) is a pre-Bezout ring. The following properties are shown for W +(ℂ +): W +(ℂ +) is not a GCD domain, that is, there exist functions F 1, F 2 in W +(ℂ +) that do not possess a greatest common divisor in W +(ℂ +). W +(ℂ +) is not coherent, and in fact, we give an example of two principal ideals whose intersection is not finitely generated. We will also observe that W +(ℂ +) is a Hermite ring, by showing that the maximal ideal space of W +(ℂ +), equipped with the Gelfand topology, is contractible.

Prüfer’s ideal numbers as Gelfand’s maximal ideals

Polyadic arithmetics is a branch of mathematics related to p-adic theory. The authors suggest two non-classical models for the Prufer profinite completion Z of the ring Z. Firstly, letA be the algebra of all periodic functions on Z, then Z can be defined as the ring of all non-zero morphisms A → C with convolution ring operations equipped with some natural topology. Secondly, let (E, μ) be the maximal ideal space for the Banach algebra of almost periodic functions on Z with the Gelfand topology μ. One can define rings operations in (E, μ) which turn it into a topological ring isomorphic to Z.

The convolution algebra H1(R)

H1(R) is a Banach algebra which has better mapping properties under singular integrals than L1(R) . We show that its approximate identity sequences are unbounded by constructing one unbounded approximate identity sequence {vn}. We introduce a Banach algebra Q that properly lies between H1 and L1, and use it to show that c(1 + ln n) ≤ ||vn|| ≤ Cn1/2. We identify the maximal ideal space of H1 and give the appropriate version of Wiener′s Tauberian theorem.

AUTOMORPHISMS OF NONSELFADJOINT DIRECTED GRAPH OPERATOR ALGEBRAS

AbstractWe analyze the automorphism group for the norm closed quiver algebras 𝒯+(Q). We begin by focusing on two normal subgroups of the automorphism group which are characterized by their actions on the maximal ideal space of 𝒯+(Q). To further discuss arbitrary automorphisms we factor automorphism through subalgebras for which the automorphism group can be better understood. This allows us to classify a large number of noninner automorphisms. We suggest a candidate for the group of inner automorphisms.

Sufficient conditions for the projective freeness of Banach algebras

Let R be a unital semi-simple commutative complex Banach algebra, and let M ( R ) denote its maximal ideal space, equipped with the Gelfand topology. Sufficient topological conditions are given on M ( R ) for R to be a projective free ring, that is, a ring in which every finitely generated projective R-module is free. Several examples are included, notably the Hardy algebra H ∞ ( X ) of bounded holomorphic functions on a Riemann surface of finite type, and also some algebras of stable transfer functions arising in control theory.

Uniform algebras on the sphere invariant under group actions

It is shown under certain conditions that a uniform algebra on the unit sphere S in C2 that is invariant under the action of the 2-torus must be C(S). Contrasting with this, an example is presented showing that the statement becomes false when 2 is replaced by n > 2. It is also shown that C(M) is the only uniform algebra on a smooth manifold M that is invariant under a transitive Lie group action on its maximal ideal space. The results presented answer a question raised by Ronald Douglas in connection with a conjecture of William Arveson.

Compact failure of multiplicativity for linear maps between Banach algebras

We introduce notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a normed space, and prove some general results about these notions. We then consider linear maps $T:A\rightarrow B$ between Banach algebras that are ``close to multiplicative'' in the following senses: the failure of multiplicativity, defined by $S_T(a,b)=T(a)T(b)-T(ab)$ $(a,b\in A)$, is compact [respectively weakly compact]. We call such maps cf-homomorphisms [respectively wcf-homomorphisms]. We also introduce a number of other, related definitions. We state and prove some general theorems about these maps when they are bounded, showing that they form categories and are closed under inversion of mappings and we give a variety of examples. We then turn our attention to commutative $C^*$-algebras and show that the behaviour of the various types of ``close-to-multiplicative'' maps depends on the existence of isolated points in the maximal ideal space. Finally, we look at the splitting of Banach extensions when considered in the category of Banach algebras with bounded cf-homomorphisms [respectively wcf-homomorphisms] as the arrows. This relates to the (weak) compactness of 2-cocycles in the Hochschild-Kamowitz cohomology complex. We prove ``compact'' analogues of a number of established results in the Hochschild-Kamowitz cohomology theory.

Reversible extensions of irreversible dynamical systems: the C*-method

A construction of a reversible extension of irreversible dynamical systems is presented. It is based on calculating the maximal ideal spaces of the C*-algebras generated by these systems and the corresponding reversible extensions of endomorphisms. Connections between the objects that arise and dynamical systems of Smale horseshoe and other types are revealed.Bibliography: 20 titles.

Banach algebras and rational homotopy theory

Let A be a unital commutative Banach algebra with maximal ideal space Max(A). We determine the rational H-type of GLn(A), the group of invertible n × n matrices with coefficients in A, in terms of the rational cohomology of Max(A). We also address an old problem of J. L. Taylor. Let Lcn(A) denote the space of “last columns” of GLn(A). We construct a natural isomorphism Ȟ(Max(A);Q) ∼= π2n−1−s(Lcn(A))⊗ Q for n > 1 2 s+1 which shows that the rational cohomology groups of Max(A) are determined by a topological invariant associated to A. As part of our analysis, we determine the rational H-type of certain gauge groups F (X,G) for G a Lie group or, more generally, a rational H-space.

Peak point theorems for uniform algebras on smooth manifolds

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, Anderson and Izzo showed that the peak point conjecture does hold for uniform algebras generated by smooth functions on smooth two-manifolds with boundary. The corresponding assertion for smooth three-manifolds is false, but Anderson, Izzo, and Wermer established a peak point theorem for polynomial approximation on real-analytic three-manifolds with boundary. Here we establish a more general peak point theorem for real-analytic three-manifolds with boundary analogous to the two-dimensional result. We also show that if A is a counterexample to the peak point conjecture generated by smooth functions on a manifold of arbitrary dimension, then the essential set for A has empty interior.

On the maximal ideal space of Dales-Davie algebras of infinitely differentiable functions

Let X be a perfect, compact plane set, and let M=(Mn) be a sequence of positive numbers such that M0=1 and M n / M k M n − k ⩾ ( ) for all n∈ℕ and k=0, 1, 2, …, n. We consider a remarkable class of Banach function algebras of infinitely differentiable functions f on X such that ∑ n = 0 ∞ | | f ( n ) | | X / M n < ∞ These algebras, called Dales–Davie algebras, are denoted by D(X, M), and they are complete under certain conditions on X. The main aim of this work is to find conditions on the sequence M=(Mn) to guarantee that D(X, M) is natural; that is, its maximal ideal space is identified with X. We present a general result on the naturality of D(X, M) using some formulas from combinatorial analysis. In particular, it is shown that if X is uniformly regular, and if the sequence (Pn)=(Mn/n!) satisfies any one of the following conditions, then D(X, M) is natural: (i) sup {PiPj/Pi+j−1:i, j∈ℕ}<∞; (ii) P n 2 ⩽ P n − 1 P n + 1 for all n∈ℕ; and (iii) Bn=max {PkPn−k/Pn: 1⩽k⩽n−1}→0 as n→∞.

A sequence algebra related to H ∞ †

We present results that concern the properties of the maximal ideal space of the algebra , the algebra of bounded sequences of functions belonging to the disc algebra . We show that the structure of this maximal ideal space resembles the structure of the maximal ideal space of the algebra H ∞ . Following the construction used by Hoffman for the case of H ∞ , and using the properties of the sequences of finite Blaschke products, we obtain a description of the analytic structure of the maximal ideal space of . At the same time, the results obtained offer a glimpse into the topological structure of this maximal ideal space. This work comprises a portion of the author's doctoral dissertation written under the direction of Professor Stuart Sidney.

On the maximal ideal space of extended analytic lipschitz algebras

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, α), lip(X, K, α), consisting of those continuous functions f on X such that f|K ∈ Lip(K, α), lip(K, α), and their analytic subalgebras LipA(X, K, α) = Lip(X, K, α) ∩ A(X, K) and lipA(X, K, α) = lip(X, K, α)∩ A(X, K), where 0 < α ≤ 1 and A(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.

An example of a maximal unimodular Dirichlet algebra

We construct an example of a maximal unimodular Dirichlet algebra whose one-point Gleason parts are dense in the maximal ideal space. The main statements were announced in [1] and now we provide their complete proofs.

PROPERTIES OF CERTAIN SUBALGEBRAS OF DALES-DAVIE ALGEBRAS

AbstractWe study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by H. G. Dales and A. M. Davie in 1973, called Dales-Davie algebras and denoted by D(X, M), where X is a perfect, compact plane set and M = {Mn}∞n = 0 is a sequence of positive numbers such that M0 = 1 and (m + n)!/Mm+n ≤ (m!/Mm)(n!/Mn) for m, n ∈ N. Let d = lim sup(n!/Mn)1/n and Xd = {z ∈ C : dist(z, X) ≤ d}. We show that, under certain conditions on X, every f ∈ D(X, M) has an analytic extension to Xd. Let DP [DR]) be the subalgebra of all f ∈ D(X, M) that can be approximated by the restriction to X of polynomials [rational functions with poles off X]. We show that the maximal ideal space of DP is $X^_d$, the polynomial convex hull of Xd, and the maximal ideal space of DR is Xd. Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Dales-Davie algebras.

Common zero sets of equivalent singular inner functions II

We study connected components of a common zero set of equivalent singular inner functions in the maximal ideal space of the Banach algebra of bounded analytic functions on the open unit disk. To study topological properties of zero sets of inner functions

Toeplitz algebras on the disk

Let B be a Douglas algebra and let B be the algebra on the disk generated by the harmonic extensions of the functions in B. In this paper we show that B is generated by H ∞ ( D ) and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in B. Every element S in the Toeplitz algebra T B generated by Toeplitz operators (on the Bergman space) with symbols in B has a canonical decomposition S = T S ˜ + R for some R in the commutator ideal C T B ; and S is in C T B iff the Berezin transform S ˜ vanishes identically on the union of the maximal ideal space of the Douglas algebra B and the set M 1 of trivial Gleason parts.

Weighted L p -algebras on groups

The space Lp(G), 1 > p < ∞, on a locally compact group G is known to be closed under convolution only if G is compact. However, the weighted spaces Lp(G, w) are Banach algebras with respect to convolution and natural norm under certain conditions on the weight. In the present paper, sufficient conditions for a weight defining a convolution algebra are stated in general form. These conditions are well known in some special cases. The spectrum (the maximal ideal space) of the algebra Lp(G,w) on an Abelian group G is described. It is shown that all algebras of this type are semisimple.