Complex Unital Banach Algebra Research Articles

Numerical range submultiplicity

Let and be elements of a complex unital Banach algebra. The numerical range submultiplicity is investigated. As a consequence of the inclusions established, we confirm a conjecture that if and are self-adjoint operators such that is positive then the product is self-adjoint whenever it is normal.

Banach-valued holomorphic functions on the maximal ideal space of H ∞

We study Banach-valued holomorphic functions defined on open subsets of the maximal ideal space of the Banach algebra H ∞ of bounded holomorphic functions on the unit disk $\mathbb{D}\subset \mathbb{C}$ with pointwise multiplication and supremum norm. In particular, we establish vanishing cohomology for sheaves of germs of such functions and, solving a Banach-valued corona problem for H ∞, prove that the maximal ideal space of the algebra $H_{\mathrm{comp}}^{\infty}(A)$ of holomorphic functions on $\mathbb{D}$ with relatively compact images in a commutative unital complex Banach algebra A is homeomorphic to the direct product of maximal ideal spaces of H ∞ and A.

Wielandt’s theorem, spectral sets and Banach algebras

Let A be a complex unital Banach algebra and let a,b∈A. We give regions of the complex plane which contain the spectrum of a+b or ab using von Neumann spectral set theory. These results are a direct generalization of a theorem of Wielandt on the eigenvalues of the sum of two normal matrices.

Linear maps preserving pseudospectrum and condition spectrum

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

Solvability of the H ∞ Algebraic Riccati Equation in Banach Algebras

Let R be a commutative complex unital semisimple Banach algebra with the involution {Mathematical expression} . Sufficient conditions are given for the existence of a stabilizing solution to the H∞ Riccati equation when the matricial data has entries from R. Applications to spatially distributed systems are discussed. © 2011 Springer Basel AG.

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).

On operator-valued cosine sequences on UMD spaces

A two-sided sequence $(c_n)_{n\in\mathbb{Z}}$ with values in a complex unital Banach algebra is a cosine sequence if it satisfies $c_{n+m} + c_{n-m} = 2 c_n c_m$ for any $n,m \in \mathbb{Z}$ with $c_0$ equal to the unity of the alge

On the norm and spectral radius of Hermitian elements

Let \(\mathcal{A}\) be a complex unital Banach algebra. An element a ∈ \(\mathcal{A}\) is said to be Hermitian if ‖exp(ita)‖ = 1 for all t ∈ ℝ. In the case of the algebra of bounded linear operators in a Hilbert space, this Hermitian property agrees with the ordinary self-adjointness. If a ∈ \(\mathcal{A}\) is Hermitian, then ‖a‖ = |a|, where |a| denotes the spectral radius of a. A function F: ℝ → ℂ is called a universal symbol if ‖F(a)‖ = | F(a)| for every \(\mathcal{A}\) and all Hermitian a ∈ \(\mathcal{A}\). We characterize universal symbols in terms of positive-definite functions.

An example on ordered Banach algebras

Let B {\mathcal B} be a complex unital Banach algebra. We consider the Banach algebra A = B × C {\mathcal A}={\mathcal B} \times \mathbb C ordered by the algebra cone K = { ( a , ξ ) ∈ A : ‖ a ‖ ≤ ξ } K=\{(a,\xi ) \in {\mathcal A}: \|a\| \le \xi \} , and investigate the connection between results on ordered Banach algebras and the right bound of the numerical range of elements in B {\mathcal B} .

KOLIHA-DRAZIN INVERTIBLES FORM A REGULARITY

:In this paper we show that the set of all Koliha-Drazin invertible elements in a complex unit al Banach algebra forms a regularity as defined by Kordula and Müller, and we explore the properties of the set as a regularity. We also use this result to simplify the proof that the set of all Drazin invertible elements of a complex unital Banach algebra forms a regularity.

KOLIHA-DRAZIN INVERTIBLES FORM A REGULARITY

According to Kordula and Muller [5] and Muller [6, p. 50], a regularity in a complex unital Banach algebra A is defined as a set 1Z C ^4, such that (i) if a G A and n G N, then a G K ^=> an G Tl' (ii) if a, b are relatively prime elements of A, then ab G 1Z « * a G 1Z and b G 1Z (a, e are relatively prime if there exist x,y G A such that {a, 6, x, y} is a commuting set and ax + by = 1). The concept of regularity is at the heart of the Kordula-Muller axiomatic spectral theory, where the spectrum of a G A relative to the regularity 7Z is defined by:

On the S-transform over a Banach algebra

The S-transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B-valued Banach noncommutative probability space, for an arbitrary unital complex Banach algebra B. Also, a new proof of the additivity of the R-transform in this setting is given.

On a theorem of Vesentini

Let ${\mathcal A}$ be a Banach algebra over ${\mathbb C}$ with unit ${\bf 1}$ and $f: {\mathbb C} \to {\mathbb C}$ an entire function. Let ${\bf f}: {\mathcal A} \to {\mathcal A}$ be defined by $$ {\bf f} (a)={f}(a) \hskip 1em (a\in {\mathcal A}), $$

On commuting operator exponentials

Let A A , B B be bounded operators on a Banach space with 2 π i 2\pi i -congruence-free spectra such that e A e B = e B e A e^Ae^B=e^Be^A . E. M. E. Wermuth has shown that A B = B A AB=BA . Ch. Schmoeger later established this result, using inner derivations and, in a second paper, has shown that: for a , b a,b in a complex unital Banach algebra, if the spectrum of a + b a+b is 2 π i 2\pi i -congruence-free and e a e b = e a + b = e b e a e^ae^b=e^{a+b}=e^be^a , then a b = b a ab=ba (and thus, answering an open problem raised by E. M. E. Wermuth). In this paper we use the holomorphic functional calculus to give alternative simple proofs of both of these results. Moreover, we use the Borel functional calculus to give new proofs of recent results of Ch. Schmoeger concerning normal operator exponentials on a complex Hilbert space, under a weaker hypothesis on the spectra.

A spectral problem in ordered Banach algebras

We recall the definition and properties of an algebra cone C of a complex unital Banach algebra A. It can be shown that C induces on A an ordering which is compatible with the algebraic structure of A, and A is then called an ordered Banach algebra. The Banach algebra ℒ(E) of all bounded linear operators on a complex Banach lattice E is an example of an ordered Banach algebra, and an interesting aspect of research in ordered Banach algebras is that of investigating in an ordered Banach algebra-context certain problems that originated in ℒ(E). In this paper we investigate the problems of providing conditions under which (1) a positive element a with spectrum consisting of 1 only will necessarily be greater than or equal to 1, and (2) f (a) will be positive if a is positive, where f (a) is the element defined by the holomorphic functional calculus.

Propagation of normality along regular analytic Jordan arcs in analytic functions with values in a complex unital Banach algebra with continuous involution

Globevnik and Vidav have studied the propagation of normality from an open subset V V of a region D \mathcal {D} of the complex plane for analytic functions with values in the space L ( H ) \mathcal {L}(\mathcal {H}) of bounded linear operators on a Hilbert space H \mathcal {H} . We obtain a propagation of normality in the more general setting of a converging sequence located on a regular analytic Jordan arc in the complex plane for analytic functions with values in a complex unital Banach algebra with continuous involution. We show that in this more general setting, the propagation of normality does not imply functional commutativity anymore as it does in the case studied by Globevnik and Vidav. An immediate consequence of the Propagation of Normality Theorem is that the further generalization given by Wolf of Jamison’s generalization of Rellich’s theorem is equivalent to Jamison’s result. We obtain a propagation property within Banach subspaces for analytic Banach space-valued functions. The propagation of normality differs from the propagation within Banach subspaces since the set of all normal elements does not form a Banach subspace.

Hermitian powers: A Müntz theorem and extremal algebras

Given ${\mathbb S}\subset {\mathbb N}$, let $\widehat {{\mathbb S}}$ be the set of all positive integers $m$ for which $h^m$ is hermitian whenever $h$ is an element of a complex unital Banach algebra $A$ with $h^n$ hermitian for each $n\in {\mathbb S}$. W

Remarks on commuting exponentials in Banach algebras, II

Suppose that a a and b b are elements of a complex unital Banach algebra such that the spectrum of a a is 2 π i 2\pi i -congruence-free and e a e b = e b e a e^ae^b = e^be^a . We show that then a b − b a ab-ba is the sum of nilpotent elements. If r ( b ) r(b) denotes the spectral radius of b b , then we show that the additional assumption r ( b ) > 2 π r(b)>2 \pi implies that b ( a b − b a ) 2 = ( a b − b a ) 2 b . \begin{equation*} b (ab-ba)^2 = (ab-ba)^2 b. \end{equation*}

Lie Derivations on Banach Algebras

Let D be a Lie derivation on a unital complex Banach algebra A. Then for every primitive ideal P of A, except for a finite set of them which have finite codimension greater than one, there exist a derivation d from A/P to itself and a linear functional τ on A such that QPD(a)=d(a+P)+τ(a) for all a∈A (where QP denotes the quotient map from A onto A/P). Moreover the preceding decomposition holds for all primitive ideals in the case where D is continuous.

Remarks on commuting exponentials in Banach algebras

Suppose that a a and b b are elements of a complex unital Banach algebra such that the spectra of a a and b b are 2 π i 2\pi i -congruence-free. E.M.E. Wermuth has shown that then \[ e a e b = e b e a implies that a b = b a . e^a e^b = e^b e^a \quad \text {implies that} \quad ab = ba. \] In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth’s result.