Unital Commutative Banach Algebra Research Articles

A Characterization of Polynomially Convex Sets in Banach Spaces

Let E be a Banach space and \({\triangle ^{\!*}}\) be the closed unit ball of the dual space \(E^*\). For a compact set K in E, we prove that K is polynomially convex in E if and only if there exist a unital commutative Banach algebra A and a continuous function \(f:{\triangle ^{\!*}}\rightarrow A\) such that (i) A is generated by \(f({\triangle ^{\!*}})\), (ii) the character space of A is homeomorphic to K, and (iii) \(K=\vec {\textsc {sp}}(f)\) the joint spectrum of f. In case \(E=\mathscr {C}(X)\), where X is a compact Hausdorff space, we will see that \({\triangle ^{\!*}}\) can be replaced by X.

Peripherally multiplicative operators on unital commutative Banach algebras

Let $T: A \longrightarrow B$ be a surjective operator between two unital semisimple commutative Banach algebras $A$ and $B$ with $T1=1$. We show that if $T$ satisfies the peripheral multiplicativity condition $\sigma_\pi(Tf.Tg) = \sigma_\pi (f.g)$ for all $f$ and $g$ in $A$, where $\sigma_\pi(f)$ shows the peripheral spectrum of $f$, then $T$ is a composition operator in modulus on the $\check{S}$ilov boundary of $A$ in the sense that $|f(x)|=|Tf(\tau(x))|,$ for each $f\in A$ and $x\in \partial(A)$ where $\tau: \partial (A) \longrightarrow \partial (B)$ is a homeomorphism between $\check{S}$ilov boundaries of $A$ and $B$.

The Structure of Maximal Ideal Space of Certain Banach Algebras of Vector-valued Functions

Let X be a compact metric space, B be a unital commutative Banach algebra and fi 2 (0;1). In this paper, we flrst deflne the vector-valued (B-valued) fi-Lipschitz operator algebra Lipfi(X;B) and then study its structure and characterize of its maximal ideal space.

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.

Closed ideals with countable hull in algebras of analytic functions smooth up to the boundary

We denote by $\bbt$ the unit circle and by $\bbd$ the unit disc. Let $\calb$ be a semi-simple unital commutative Banach algebra of functions holomorphic in $\bbd$ and continuous on $\overline{\bbd}$, endowed with the pointwise product. We assume that $\calb$ is continously imbedded in the disc algebra and satisfies the following conditions: (H1) The space of polynomials is a dense subset of $\calb$. (H2) $\lim_{n\to +\infty}\|z^n\|_{\calb}^{1/ n}=1$. (H3) There exist $k \geq 0$ and $C > 0$ such that \begin{eqnarray*} \big| 1- |\lambda| \big|^{k} \big\| f \big\|_{\calb} \leq C \big\| (z-\lambda) f \big\|_{\calb}, \quad (f \in \calb, |\lambda| < 2) \end{eqnarray*} When $\calb$ satisfies in addition the analytic Ditkin condition, we give a complete characterisation of closed ideals $I$ of $\calb$ with countable hull $h(I)$, where $$ h(I) = \big\{ z \in \overline{\bbd} : \, f(z) = 0, \quad (f \in I) \big\}. $$ Then, we apply this result to many algebras for which the structure of all closed ideals is unknown. We consider, in particular, the weighted algebras $\ell^1(\omega$) and $L^1(\bbr^{+},\omega)$.

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.

Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ ( f g ) of the product of any two elements f and g in A coincides with the spectrum σ ( T f T g ) . Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and non-multiplicative unital map from a commutative C*-algebra into itself such that σ ( T f T g ) = σ ( f g ) holds for every f , g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ ( T f T g ) ⊂ σ ( f g ) holds for every f , g .

ON SUBSPECTRA GENERATED IN SUBALGEBRAS

Let B be a unital commutative Banach algebra, and let A be an arbitrary subalgebra of B. Suppose that an ideal I ⊂ A consists of elements that are non-invertible in B. Then there exists an ideal J in A that: (i) contains I, (ii) also consists of elements non-invertible in B, and (iii) is of codimension one in A. This result is used in the study of a class of subspectra on A. 2000 Mathematics Subject Classification 46J20.

Unital strongly harmonic commutative Banach algebras

A unital commutative Banach algebraA is spectrally separable if for any two distinct non-zero multiplicative linear functionals ' and on it there exist a and b in A such that ab = 0 and '(a) (b)6 0: Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra A is spectrally separable if there are enough elements in A such that the corresponding multiplication operators onA have the decomposition property ( ): On the other hand, if A is spectrally separable, then for each a 2 A and each Banach left A-module X the corresponding multiplication operator La on X is super-decomposable. These two statements improve an earlier result of Baskakov.

Local Cohomology for Commutative Banach Algebras

The purpose of this paper is to introduce a local cohomology theory in the unital commutative Banach algebras context and to describe a connection between the local cohomology functor and direct limit of hom functors.