Complex Unital Banach Algebra Research Articles

Description of Local Derivations on Jordan Algebras of Dimension Five

In the present paper we investigate local derivations on finite dimensional Jordan algebras. The Gleason--Kahane--\.{Z}elazko theorem, which is a fundamental contribution in the theory of Banach algebras, asserts that every unital linear functional $F$ on a complex unital Banach algebra $A$, such that $F(a)$ belongs to the spectrum $\sigma(a)$ of $a$ for every $a\in A,$ is multiplicative. In modern terminology this is equivalent to the following condition: every unital linear local homomorphism from a unital complex Banach algebra $A$ into ${\Bbb C}$ is multiplicative. We recall that a linear map $T$ from a Banach algebra $A$ into a~Banach algebra $B$ is said to be a~local homomorphism if, for every $a$ in $A$, there exists a homomorphism $\Phi_a : A\to B$, depending on $a$, such that $T(a)=\Phi_a(a)$. A similar notion was introduced and studied to give a characterization of derivations on operator algebras. Namely, the concept of local derivations was introduced by R.~Kadison and D.~Larson, A.~Sourour independently in 1990. R.~Kadison gave the description of all continuous local derivations from a von Neumann algebra into its dual Banach bemodule. B. Jonson extends the result of R. Kadison by proving that every local derivation from a $C^*$-algebra into its Banach bimodule is a derivation. It is known that, every local derivation on a JB-algebra is a derivation. In particular, every local derivation on a finite dimensional semisimple Jordan algebra is a derivation. In the present paper we investigate derivations and local derivations on five-dimensional nilpotent non-associative Jordan algebras. The description of local derivations of nilpotent Jordan algebras is an open problem. In~the~present paper we give the description of local derivations on five-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of characteristic $\neq 2$, $3$. We also give a~criterion of a~linear operator on Jordan algebras of dimension five to be a local derivation.

Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum

Abstract Let 𝒜 {\mathcal{A}} be a complex unital Banach algebra and let R ⊆ 𝒜 {R\subseteq\mathcal{A}} be a non-empty set. This paper defines the property such that R is closed for idempotent decomposition (in short, (CID) property) to explore the spectral decomposition relation. Further, for an upper semiregularity R with (CID) property, R D {R^{D}} is constructed as an extension of R to axiomatically study the accumulation of σ R ⁢ ( a ) {\sigma_{R}(a)} for any a ∈ 𝒜 {a\in\mathcal{A}} . At last, several illustrative examples on Banach algebra and operator algebra are provided.

$$L^\infty $$ estimates for the Banach-valued $${{\bar{\partial }}}$$-problem in a disk

We study the differential equation $$\frac{\partial G}{\partial {{\bar{z}}}}=g$$ with an unbounded Banach-valued Bochner measurable function g on the open unit disk $${\mathbb {D}}\subset {{\mathbb {C}}}$$ . We prove that under some conditions on the growth and essential support of g such equation has a bounded solution given by a continuous linear operator. The obtained results are applicable to the Banach-valued corona problem for the algebra of bounded holomorphic functions on $${\mathbb {D}}$$ with values in a complex commutative unital Banach algebra.

$$G_{1}$$-class elements in a Banach algebra

Let A be a complex unital Banach algebra with unit 1. An element $$a\in A$$ is said to be of $$G_{1}$$ -class if $$\begin{aligned} \Vert (z-a)^{-1}\Vert =\frac{1}{\text {d}(z,\sigma (a))} \quad \forall z\in {\mathbb {C}}\setminus \sigma (a). \end{aligned}$$ Here $$d(z, \sigma (a))$$ denotes the distance between z and the spectrum $$\sigma (a)$$ of a. Some examples of such elements are given and also some properties are proved. It is shown that a $$G_1$$ -class element is a scalar multiple of the unit 1 if and only if its spectrum is a singleton set consisting of that scalar. It is proved that if T is a $$G_1$$ class operator on a Banach space X, then every isolated point of $$\sigma (T)$$ is an eigenvalue of T. If, in addition, $$\sigma (T)$$ is finite, then X is a direct sum of eigenspaces of T.

Spectrum of banach valued holomorphic functions and polar sets

Let be a complex unital Banach algebra and D be a non-empty open connected subset of Let f be a holomorphic -valued function on D. We set: where is the set of all invertible elements in . In this paper, we give a description of using the classical spectrum of . Moreover, within the framework of this paper, we give a partial positive answer to the following problem which was posed by B. Aupetit: If the usual spectrum is polar for all is it true that is polar?

A multiplicative Gleason-Kahane-Żelazko theorem for C⋆-algebras

If A is a unital complex Banach algebra, and if σ(a) denotes the spectrum of an element a∈A, then the famous Gleason-Kahane-Żelazko Theorem says that any linear functional ϕ:A→C satisfying ϕ(a)∈σ(a) for each a∈A, is multiplicative and continuous. In this paper we establish a multiplicative Gleason-Kahane-Żelazko theorem for the case where A is a C⋆-algebra. Specifically, if A is a C⋆-algebra, then any continuous multiplicative functional ϕ:A→C satisfying ϕ(a)∈σ(a) for each a∈A, is linear and hence a character of A.

A weaker Gleason–Kahane–Żelazko theorem for modules and applications to Hardy spaces

Let A be a complex unital Banach algebra and M be a left A-module. Let Ʌ: M→ℂ be a map that is not necessarily linear. We establish conditions for Ʌ to be linear and of multiplicative kind, from its behavior on a small subset of M. We do not assume Ʌ to be continuous throughout. As an application, we give a characterization of weighted composition operators on the Hardy space H.

On the spectra of products and linear combinations of idempotents

Let $${\mathcal {A}}$$ be a unital complex Banach algebra and let $$p,q\in {\mathcal {A}}$$ be two idempotents. In this paper, we study the relationships between different spectra of pq and those of $$\alpha p+\beta q-\gamma pq$$ , where $$\alpha ,\beta ,\gamma \in {\mathbb {C}}$$ .

Gelfand theory for real Banach algebras

We are concerned with the development of the more general real case of the classical theorem of Gelfand on representation of a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space. To that end, we use only intrinsic methods which do not depend on the complexification of the algebra, and obtain two representation theorems for commutative unital real Banach algebras as algebras of continuous real (respectively, complex) functions on the compact space of real-valued (respectively, complex-valued) $$\mathbb R$$ -algebra homomorphisms.

Condition pseudospectral radius of bounded linear operators

ABSTRACT In this article, we consider the condition pseudospectrum of bounded linear operators on a Banach space. The condition pseudospectrum of normal matrices and Jordan blocks are characterized and condition pseudospectral radius of the classes are found. Sub-additivity and sub-multiplicativity of the condition pseudospectral radius for commuting pairs of bounded linear operators are proved. It is shown that the condition pseudospectral radius becomes a complete algebra norm in a commutative complex unital Banach algebra. Certain examples are given to illustrate the results. The results developed are also extended to a general setting.

Spectrum and related sets: a survey

In order to understand the behaviour of a square matrix or a bounded linear operator on a Banach space or more generally an element of a Banach algebra, some subsets of the complex plane are associated with such an object. Most popular among these sets is the spectrum $$\sigma (a) $$ of an element a in a complex unital Banach algebra A with unit 1 defined as follows: $$\begin{aligned} \sigma (a) := \{\lambda \in \mathbb {C}: \lambda - a \,\, \text{ is } \text{ not } \text{ invertible } \text{ in }\, A\}. \end{aligned}$$Here and also in what follows, we identify $$\lambda .1$$ with $$\lambda $$. Also quite popular is Numerical range V(a) of a. This is defined as follows: $$\begin{aligned} V(a) := \{ \phi (a): \phi \,\, \text{ is } \text{ a } \text{ continuous } \text{ linear } \text{ functional } \text{ on } \,\, A \, \, \text{ satisfying } \, \, \Vert \phi \Vert = 1 = \phi (1) \}. \end{aligned}$$Then there are many generalizations, modifications, approximations etc. of the spectrum. Let $$\epsilon > 0$$ and n a nonnegative integer. These include $$\epsilon -$$ condition spectrum $$\sigma _{\epsilon }(a)$$, $$\epsilon -$$pseudospectrum $$\Lambda _{\epsilon }(a)$$ and $$(n, \epsilon )-$$pseudospectrum $$\Lambda _{n, \epsilon }(a)$$. These are defined as follows: $$\begin{aligned} \sigma _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert \lambda - a\Vert \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \end{aligned}$$In this and the following definitions we follow the convention : $$ \Vert (\lambda -a)^{-1}\Vert = \infty $$ if $$\lambda - a$$ is not invertible. $$\begin{aligned}&\Lambda _{\epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-1}\Vert \ge \frac{1}{\epsilon } \right\} \\&\Lambda _{n, \epsilon }(a) := \left\{ \lambda \in \mathbb {C}: \Vert (\lambda -a)^{-2^n}\Vert ^{ 1/2^n } \ge \frac{1}{\epsilon } \right\}. \end{aligned}$$ In this survey article, we shall review some basic properties of these sets, relations among these sets and also discuss the effects of perturbations on these sets and the question of determining the properties of the element a from the knowledge of these sets.

CONTINUITY OF A CONDITION SPECTRUM AND ITS LEVEL SETS

Let ${\mathcal{A}}$ be a complex unital Banach algebra, let $a$ be an element in it and let $0<\unicode[STIX]{x1D716}<1$. In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the $\unicode[STIX]{x1D716}$-level set of a condition spectrum at a given $(\unicode[STIX]{x1D716},a)$ plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices.

Continuity of the $$\varvec{(n,\epsilon )}$$-Pseudospectrum in Banach Algebras

Let $$\epsilon >0$$ , n a non-negative integer, and A a complex unital Banach algebra. Define $$\gamma _n: A\times {\mathbb {C}}\rightarrow [0,\infty ]$$ by $$\begin{aligned} \gamma _n(a,z)={\left\{ \begin{array}{ll} \Vert (z -a)^{-2^n}\Vert ^{-1/2^n}, &{}\quad \text {if } (z-a) \text{ is } \text{ invertible }\\ 0, &{}\quad \text {if } (z-a) \text { is not invertible}. \end{array}\right. } \end{aligned}$$ The $$(n,\epsilon )$$ -pseudospectrum $$\Lambda _{n,\epsilon }(a)$$ of an element $$a\in A$$ is defined by $$\Lambda _{n,\epsilon }(a):= \{\lambda \in {\mathbb {C}}:\gamma _n(a,\lambda )\le \epsilon \}$$ . We show that $$\gamma _0$$ is Lipschitz on $$A\times {\mathbb {C}}$$ , $$\gamma _n$$ is uniformly continuous on bounded subsets of $$A\times {\mathbb {C}}$$ for $$n\ge 1$$ , and $$\gamma _n$$ is Lipschitz on some particular domains for $$n\ge 1$$ . Using these properties, we establish that the map $$(\epsilon ,a)\mapsto \Lambda _{n,\epsilon }(a)$$ is continuous at $$(\epsilon _0,a_0)$$ if and only if the level set $$\{\lambda \in {\mathbb {C}}: \gamma _n(a_0,\lambda )= \epsilon _0 \}$$ does not contain any non-empty open set. In particular, this happens when a is a compact operator on a Banach space or a bounded operator on a Hilbert space or on an $$L^p $$ space with $$1\le p\le \infty $$ . We also give examples of operators where this condition is not satisfied, and consequently, the map is not continuous.

The Drazin and generalized Drazin invertibility of linear combinations of idempotents

Let A be a unital complex Banach algebra and p, q be two idempotents in A. In this paper, we prove that the Drazin (resp. generalized Drazin) invertibility of αp+βq−γpq is independent of the choice of scalars α,β∈C∖{0} and γ∈C, we also give a simple representation formula for the Drazin inverse of αp+βq. As an application, we study the left (resp. right) Drazin invertibility and the property of being Browder of linear combinations of two idempotent operators on a Banach space.

Fréchet Derivative and Analytic Functional Calculus

We calculate Frechet derivatives of the mapping $$a\mapsto f(a)$$, where a is a member of a complex unital Banach algebra and f is a complex analytic function in a neighborhood of the spectrum of a. Thus, the connection between the Frechet derivative and the analytic functional calculus is established.

Berkovich spectra of elements in Banach rings

Extending the notion of the spectrum Σf of an element f in an ultrametric Banach algebra over a complete valued field (as defined by Berkovich), we introduce and briefly study the Berkovich spectrum σR,SBer(a) of an element a in a unital Banach algebra R over a commutative unital Banach ring S (in particular, when R is a Banach ring). This spectrum is a compact subset of the affine analytic line AS1 over S, and the latter can be identified with the suitable equivalence classes of all elements in all complete valued fields that are Banach S-algebras. If R is generated by a as a unital Banach S-algebra, then σR,SBer(a) coincides with the spectrum M(R) of R. If R is a unital complex Banach algebra but we regard it as a Banach ring only, then σR,Z1Ber(a) is the quotient of the ordinary spectrum σRC(a) by complex conjugation.For a non-Archimedean complete valued field k and an infinite dimensional ultrametric Banach k-space E with an orthogonal basis, if u∈L(E) is a completely continuous operator, we show that several different ways to define the spectrum of u give the same compact set σL(E),kBer(u). As an application, we give a relation between σL(E),kBer(u) and zeros of the Fredholm determinant det⁡(1−tu) (as defined by Serre) in complete valued field extensions of k.

The (n,ϵ)-pseudospectrum of an element of a Banach algebra

Let A be a complex unital Banach algebra, a∈A, n∈Z+ and ϵ>0. The (n,ϵ)-pseudospectrum Λn,ϵ(a) of a is defined asΛn,ϵ(a):=σ(a)∪{λ∉σ(a):‖(λ−a)−2n‖1/2n≥1ϵ}. Here σ(a) denotes the spectrum of a. The usual pseudospectrum Λϵ(a) of a is a special case of this, namely Λ0,ϵ(a). It is proved that (n,ϵ)-pseudospectrum approximates the closed ϵ-neighbourhood of spectrum for large n. Further, it has been shown that (n,ϵ)-pseudospectrum has no isolated points, has a finite number of connected components and each component contains an element from σ(a). Some examples are given to illustrate these results.

On the Factorization of Matrices Over Commutative Banach Algebras

Let A be a commutative unital complex Banach algebra and let $$GL_n(A)$$ be the group of invertible $$n\times n$$ matrices with entries in A. In this paper we study the problem of the representation of matrices in $$GL_n(A)$$ by finite products of upper and lower triangular matrices.

Level sets of the condition spectrum

For 0<ϵ≤1 and an element a of a complex unital Banach algebra A, we prove the following two topological properties about the level sets of the condition spectrum. (1) If ϵ=1, then the 1-level set of the condition spectrum of a has an empty interior unless a is a scalar multiple of the unity. (2) If 0<ϵ<1, then the ϵ-level set of the condition spectrum of a has an empty interior in the unbounded component of the resolvent set of a. Further, we show that, if the Banach space X is complex uniformly convex or if X∗ is complex uniformly convex, then, for any operator T acting on X, the level set of the ϵ-condition spectrum of T has an empty interior.

On the open ball centered at an invertible element of a Banach algebra

Let A be a complex unital Banach algebra. Since the set of invertible elements is open, there is an open ball around every invertible element. In this article, we investigate the Banach algebras for which the radius given by the Neumann series is optimal.