Maximal Ideal Space Research Articles

TOPOLOGICAL CHARACTERIZATION OF DUALLY NORMAL ALMOST DISTRIBUTIVE LATTICES

A dually normal almost distributive lattice is characterized topologically in terms of its maximal ideals and prime ideals. Some necessary and sufficient conditions for the space of maximal ideals to be dually normal are obtained.

On The Structure of A Commutative Banach Algebra Generated By Toeplitz Operators With Quasi-Radial Quasi-Homogeneous Symbols

Let \({\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}\) denote the standard weighted Bergman space over the unit ball \({\mathbb{B}^n}\) in \({\mathbb{C}^n}\) . New classes of commutative Banach algebras \({\mathcal{T}(\lambda)}\) which are generated by Toeplitz operators on \({\mathcal{A}_{\lambda}^2(\mathbb{B}^n)}\) have been recently discovered in Vasilevski (Integr Equ Oper Theory 66(1):141–152, 2010). These algebras are induced by the action of the quasi-elliptic group of biholomorphisms of \({\mathbb{B}^n}\) . In the present paper we analyze in detail the internal structure of such an algebra in the lowest dimensional case n = 2. We explicitly describe the maximal ideal space and the Gelfand map of \({\mathcal{T}(\lambda)}\) . Since \({\mathcal{T}(\lambda)}\) is not invariant under the *-operation of \({\mathcal{L}(\mathcal{A}_{\lambda}^2(\mathbb{B}^n))}\) its inverse closedness is not obvious and is proved. We remark that the algebra \({\mathcal{T}(\lambda)}\) is not semi-simple and we derive its radical. Several applications of our results are given and, in particular, we conclude that the essential spectrum of elements in \({\mathcal{T}(\lambda)}\) is always connected.

Normed algebras of infinitely differentiable Lipschitz functions

Let X be a perfect compact plane set, ω ={w n } be a weight sequence of positive numbers, and 0 < α ≤ 1. Then Lip(X, α, ω) denotes the algebra of infinitely differentiable functions f on X such that f (n), for n = 0, 1, …, satisfies the Lipschitz condition of order α, and such that ,where ||·||α is the Lipschitz norm on X. Using some formulae from combinatorial analysis, we show that, under certain conditions on ω, if f ∈ Lip(X, α, ω) and f (z)≠0 for every z ∈ X, then 1/f ∈ Lip(X, α, ω). We then conclude that if, moreover, the maximal ideal space of (X, α, ω), the uniform closure of Lip(X, α, ω), equals X then every non-zero continuous complex homomorphism on Lip(X, α, ω) is an evaluation character at some point of X.

On the maximal ideal space of extended polynomial and rational uniform algebras

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.

A note on τ-convergence, τ-convergent algebra and applications

To extract a unital commutative C⁎-algebra from the space lac∞(N,C) of almost convergent sequences, the concept of multiplier is introduced and characterized by τ-convergence. For applications, it is proved that the maximal ideal space τN of τ-convergent algebra Mτ is a compactification of N, which includes βN as a closed subset, and Mτ provides an example of non-C⁎-reflexive C⁎-algebra according to Frank–Manuilov–Troitskyʼs criterion.

A Note on Some Uniform Algebra Generated by Smooth Functions in the Plane

We determine, via classroom proofs, the maximal ideal space, the Bass stable rank as well as the topological and dense stable rank of the uniform closure of all complex-valued functions continuously differentiable on neighborhoods of a compact planar set and holomorphic in the interior of . In this spirit, we also give elementary approaches to the calculation of these stable ranks for some classical function algebras on .

Stone duality for real-valued multisets

In joint work with E. Dubuc and D. Mundici, the first author extended Stone duality for boolean algebras to locally finite MV-algebras. On the topological side of the duality, one has to assign to each point of a boolean space a generalized natural number by way of a multiplicity, so as to make the assignment continuous with respect to the Scott topology of the lattice of generalized natural numbers. The continuous maps between such generalized multisets are to be multiplicity-decreasing with respect to the divisibility order of generalized natural numbers. In this paper we extend these results to the class of MV-algebras that are locally weakly finite, i.e., such that all their finitely generated subalgebras split into a finite direct product of simple MV-algebras. Using the Scott topology on the lattice of subalgebras of the real unit interval (regarded with its natural MV-algebraic structure), we construct a `real-valued multiset' over the (boolean) space of maximal ideals of a locally weakly finite MV-algebra. Building on this, we obtain a duality for locally weakly finite MV-algebras that includes as a special case the above-mentioned duality for locally finite MV-algebras. We give an example that shows that the duality established in this paper via the Scott topology cannot be extended, without non-trivial modifications, to larger classes of algebras.

Maximal abelian subalgebras and projections in two Banach algebras associated with a topological dynamical system

If $\Sigma=(X,\sigma)$ is a topological dynamical system, where $X$ is a compact Hausdorff space and $\sigma$ is a homeomorphism of $X$, then a crossed product Banach $\sp{*}$-algebra $\ell^1(\Sigma)$ is naturally associated with these data. If $X$ consists of one point, then $\ell^1(\Sigma)$ is the group algebra of the integers. The commutant $C(X)'_1$ of $C(X)$ in $\ell^1(\Sigma)$ is known to be a maximal abelian subalgebra which has non-zero intersection with each non-zero closed ideal, and the same holds for the commutant $C(X)'_*$ of $C(X)$ in $C^*(\Sigma)$, the enveloping $C^*$-algebra of $\ell^1(\Sigma)$. This intersection property has proven to be a valuable tool in investigating these algebras. Motivated by this pivotal role, we study $C(X)'_1$ and $C(X)'_*$ in detail in the present paper. The maximal ideal space of $C(X)'_1$ is described explicitly, and is seen to coincide with its pure state space and to be a topological quotient of $X\times\mathbb{T}$. We show that $C(X)'_1$ is hermitian and semisimple, and that its enveloping $C^*$-algebra is $C(X)'_*$. Furthermore, we establish necessary and sufficient conditions for projections onto $C(X)'_1$ and $C(X)'_*$ to exist, and give explicit formulas for such projections, which we show to be unique. In the appendix, topological results for the periodic points of a homeomorphism of a locally compact Hausdorff space are given.

Maximal ideal space of certain alpha-Lipschitz operator algebras

Maximal ideal space of certain alpha-Lipschitz operator algebras

The corona theorem for the Drury–Arveson Hardy space and other holomorphic Besov–Sobolev spaces on the unit ball in ℂn

We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^{\sigma}$ has the "baby corona property" for all $\sigma \geq 0$ and $1<p<\infty $. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

A harmonic analysis approach to essential normality of principal submodules

Guo and the second author have shown that the closure [ I ] in the Drury–Arveson space of a homogeneous principal ideal I in C [ z 1 , … , z n ] is essentially normal. In this note, the authors extend this result to the closure of any principal polynomial ideal in the Bergman space. In particular, the commutators and cross-commutators of the restrictions of the multiplication operators are shown to be in the Schatten p-class for p > n . The same is true for modules generated by polynomials with vector-valued coefficients. Further, the maximal ideal space X I of the resulting C ⁎ -algebra for the quotient module is shown to be contained in Z ( I ) ∩ ∂ B n , where Z ( I ) is the zero variety for I, and to contain all points in ∂ B n that are limit points of Z ( I ) ∩ B n . Finally, the techniques introduced enable one to study a certain class of weight Bergman spaces on the ball.

Finitely 1-convex f-rings

This paper investigates f-rings that can be constructed in a finite number of steps where every step consists of taking the fibre product of two f-rings, both being either a 1-convex f-ring or a fibre product obtained in an earlier step of the construction. These are the f-rings that satisfy the algebraic property that rings of continuous functions possess when the underlying topological space is finitely an F-space (i.e. has a Stone–Čech compactification that is a finite union of compact F-spaces). These f-rings are shown to be SV f-rings with bounded inversion and finite rank and, when constructed from semisimple f-rings, their maximal ideal space under the hull-kernel topology contains a dense open set of maximal ideals containing a unique minimal prime ideal. For a large class of these rings, the sum of prime, semiprime, primary and z-ideals are shown to be prime, semiprime, primary and z-ideals respectively.

Uniform approximation on manifolds

It is shown that if A is a uniform algebra generated by a family of complex-valued C 1 functions on a compact C 1 manifold-with-boundary M, the maximal ideal space of A is M, and E is the set of points where the dierentials of the functions in fail to span the complexied cotangent space to M, then A contains every continuous function on M that vanishes on E. This answers a 45-year-old question of Michael Freeman who proved the special case in which the manifold M is two-dimensional. More general forms of the theorem are also established. The results presented strengthen results due to several mathematicians.

Real linear isometries between function algebras. II

AbstractWe describe the general form of isometries between uniformly closed function algebras on locally compact Hausdorff spaces in a continuation of the study by Miura. We can actually obtain the form on the Shilov boundary, rather than just on the Choquet boundary. We also give an example showing that the form cannot be extended to the whole maximal ideal space.

Compact homomorphisms of regular banach algebras

Let A be a complex commutative Banach algebra and let MA be the maximal ideal space of A. We say that A has the bounded separating property if there exists a constant C > 0 such that for every two distinct points ϕ1, ϕ2 ∈ MA, there is an element a ∈ A for which , and ‖a‖ ⩽ C, where is the Gelfand transform of a ∈ A. We show that if A is a strongly regular Banach algebra with the bounded separating property, then every compact homomorphism from A into another Banach algebra is of finite dimensional range. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Factorization of Blaschke products and ideal theory in [formula omitted

Let H ∞ be the Banach algebra of bounded analytic functions on the open unit disk D . Let G be the union set of all nontrivial Gleason parts in the maximal ideal space of H ∞ . Let E be a nonvoid compact and totally disconnected subset of G and n E be a bounded numbering function on E. We characterize n E for which there is a closed ideal I in H ∞ such that Z ( I ) = E and ord ( I , x ) = n E ( x ) for every x ∈ E . Let I 1 , I 2 , … , I k be closed ideals in H ∞ satisfying Z ( I i ) ⊂ G for 1 ⩽ i ⩽ k . We prove that ∏ i = 1 k I i = { ∏ i = 1 k f i : f i ∈ I i , 1 ⩽ i ⩽ k } is a closed ideal. A local ideal theory in H ∞ plays an important role to prove our results.

Transitivity in spaces of vector-valued functions

AbstractWe exhibit a real Banach space M such that C(K,M) is almost transitive if K is the Cantor set, the growth of the integers in its Stone–Čech compactification or the maximal ideal space of L∞. For finite K, the space C(K,M) = M|K| is even transitive.

Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras

Abstract Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r(a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r(T(a) + T(b)) = r(a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that $$ \widehat{T\left( a \right)}\left( y \right) = \left\{ \begin{gathered} \widehat{T\left( e \right)}\left( y \right)\hat a\left( {\phi \left( y \right)} \right) y \in K \hfill \\ \widehat{T\left( e \right)}\left( y \right)\overline {\hat a\left( {\phi \left( y \right)} \right)} y \in M_\mathcal{B} \backslash K \hfill \\ \end{gathered} \right. $$ for all a ∈ A, where e is unit element of A. If, in addition, $$ \widehat{T\left( e \right)} = 1 $$ and $$ \widehat{T\left( {ie} \right)} = i $$ on M B, then T is an algebra isomorphism.

REPRESENTATIONS OF REAL BANACH ALGEBRAS

AbstractA commutative complex unital Banach algebra can be represented as a space of continuous complex-valued functions on a compact Hausdorff space via the Gelfand transform. However, in general it is not possible to represent a commutative real unital Banach algebra as a space of continuous real-valued functions on some compact Hausdorff space, and for this to happen some additional conditions are needed. In this note we represent a commutative real Banach algebra on a part of its state space and show connections with representations on the maximal ideal space of the algebra (whose existence one has to prove first).

Projective free algebras of continuous functions on compact abelian groups

It is proved that the Wiener algebra of functions on a connected compact abelian group whose Bohr–Fourier spectra are contained in a fixed subsemigroup of the (additive) dual group, is projective free. The semigroup is assumed to contain zero and have the property that it does not contain both a nonzero element and its opposite. The projective free property is proved also for the algebra of continuous functions with the same condition on their Bohr–Fourier spectra. As an application, the connected components of the set of factorable matrices are described. The proofs are based on a key result on homotopies of continuous maps on the maximal ideal spaces of the algebras under consideration.