Noncommutative uniform algebras

In this paper we derive the explicit, closed-form, recursion-free formulae for the arbitrary-order Fr?chet derivatives of the exponential and logarithmic functions in unital Banach algebras (complex or real). These computations are obtained via the Bochner integrals for the Banach algebra valued functions, with respect to the standard Lebesgue measure. As an application, we utilize our results in the approximation schemes of the solutions to stochastic functional differential equations.

Invariants of algebraic derivations and automorphisms in Banach algebras

Based on the complex case, we present for commutative real Banach algebras $$\mathcal {R}$$ several non-algebraic versions of the Arens-Royden theorem and the Shilov idempotent theorem. It will be shown that the Gelfand transform induces a group isomorphism of $$\mathcal {R}^{-1}/\exp \mathcal {R}$$ onto $$C(X(\mathcal {R}), \tau )^{-1}/ \exp C(X(\mathcal {R}), \tau )$$, where $$C(X(\mathcal {R}),\tau )$$ is the algebra of $$\tau $$-symmetric complex-valued functions on the character space $$X(\mathcal {R} )$$ of $$\mathcal {R}$$ for some specific involution $$\tau $$. We will also prove that for any $$\tau $$-symmetric closed-open subset of $$X(\mathcal {R})$$ there is an idempotent e in $$\mathcal {R}$$ whose Gelfand transform coincides with the characteristic function of E. We apply the real Arens-Royden theorem to show that the Bezout equation $$uf+vg=1$$ has a solution in $$\mathcal {R}$$ with u invertible if (and only if) it has such a solution in $$C(X(\mathcal {R}), \tau )$$.

Let B be a strictly real commutative real Banach algebra with the carrier space ΦB. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B.

LetA be a real Banach algebra with a closed cone. If the cone contains all squares then the radical only consists of elementsx eA withx 3=0. This characterization yields an elementary proof of a result of Scheffold forff-Banach lattice algebras.

In this article, we introduce a new geometrical structure which is the hybrid of cone metric space over Banach algebra and extended b-metric space. We prove analogues of Banach, Kannan and Reich type fixed point theorems in our introduced space. We also furnish with various concrete examples to establish the validity of our results. The obtained results generalize many well-known results in literature, especially the main results due to Vujakovic et al., Hussain et al., Huang, Radenovic, Xu become special cases of our results.

In the theory of Banach algebras, we use the Schauder fixed-point theorem to obtain some results that concern the existence property for mild solutions of fractional Cauchy problems that involve the Lie bracket operator, the almost sectorial operator, and the φ−Hilfer derivative operator. For any Banach algebra and in two types of non-compact associated semigroups and compact associated semigroups, we prove some properties of the existence of these mild solutions using the Hausdorff measure of a non-compact associated semigroup in the collection of bounded sets. That is, we obtain the existence property of mild solutions when the semigroup associated with an almost sectorial operator is compact as well as non-compact. Some examples are introduced as applications for our results in commutative real Banach algebra R and commutative Banach algebra of the collection of continuous functions in R.

LetA be a matrix over a complex commutative unital Banach algebra. We give necessary and sufficient conditions forA to have a generalized inverse. Moreover, if the Banach algebra has a symmetric involution, these are also necessary and sufficient conditions forA to admit the Moore-Penrose inverse.

We show that for any semisimple, commutative, complex Banach algebra $A$ with unit there are norms on $A$, which we call natural norms, equivalent to the original norm on $A$ with the following property: Let $(A,|| \cdot |{|_A},{e_A})$ and $(B,|| \cdot |{|_B},{e_B})$ are commutative, semisimple Banach algebras with units and natural norms. Assume $T$ is a linear isometry from $(A,|| \cdot |{|_A})$ onto $(B,|| \cdot |{|_B})$ with $T{e_A} = {e_B}$. Then $T$ is an isomorphism in the category of Banach algebras. For a fairly large class of algebras, for example, for uniform algebras, for algebras of the form ${C^k}(X),{\text { Lip}}(X),{\text { AC}}(X)$, the natural norm we have defined coincides with a usual norm.

For a matrix over a complex commutative unital Banach algebra, necessary and sufficient conditions are given for the existence of its group inverse, and more generally, its Drazin inverses. The conditions are easy to check and explicit formulas for the inverses are provided. Some properties of the inverses and an application to operator theory are discussed. This note is a continuation of an earlier work of the author.

In the eighties of the last century Rennison studied and characterized the set of quasi-central elements of some unital Banach algebras. In this paper, we improve one of Rennison's result assert that for given unital Banach algebra \(A\) the set of quasi-central elements \(Q(A)\), need not be equal to centre of \(A\) and need not be closed under addition or multiplication. Further, we prove that if \(A\) is unital ultraprime Banach algebra, then \(Q(A)\) need not be equal to centre of \(\mathrm{A}\) and need not be closed under addition or multiplication.

In this paper, we present some results concerning spectral properties of multipliers on commutative semi-simple Banach algebras. The study concentrates on multipliers whose (local) spectra are in a sense natural. Particular emphasis is given to the determination of the (local) spectra of multipliers in connection with Beurling algebras, Dunford property (C) and Hille-Yosida space.

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.

We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a $C^\star $ -algebra, and if $\phi :A\to \mathbb C $ is a continuous function satisfying $ \phi (x)\phi (y) \in \sigma (xy) $ for all $x,y\in A$ (where $\sigma $ denotes the spectrum), then either $\phi $ is a character of A or $-\phi $ is a character of A.

The relative spectrum of elements in a real Banach algebra