A multiplicative Kowalski–Słodkowski theorem for -algebras

We show that a real Banach algebra $A$ such that $\Vert a^{2}\Vert =\Vert a\Vert^{2}$ for $a\in A$ is a subalgebra of the algebra $C_{\mathbb{H}}( X) $ of continuous quaternion-valued functions on a compact set $X$.

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.

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.

Invariants of algebraic derivations and automorphisms in Banach algebras

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.

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.

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

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.

Uniqueness under spectral variation in the socle of a Banach algebra

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

Let E be a complex Banach space and F be a complex Banach algebra. We will be interested in the subspace ℙ_f({}^n E,F) of P({}^n E,F) generated by the collection of functions \varphi^n(n\in ℕ, \varphi \in L(E,F)) where \varphi^n(x) = (\varphi (x))^n for each x\in E .

As a continuation of the paper J. Math. Anal. Appl. 464(2018), 939-954. We will define and examine the $(n,\varepsilon)$-pseudospectra and condition $(n,\varepsilon)$-pseudospectra of an element pencil in a unital complex Banach algebra. We characterize the $(n,\varepsilon)$-pseudospectra of an element pencil in a unital complex Banach algebra. In particular, we demonstrate some properties of $(n,\varepsilon)$-pseudospectra and condition $(n,\varepsilon)$-pseudospectra of an element pencil in unital complex Banach algebras.

In this paper, we introduce bihom derivations in complex Banach algebras. Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of bihom derivations in complex Banach algebras, associated with the bi-additive s-functional inequality $$\Vert f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\Vert \le \Vert s \left( 2f\left( \frac{x+y}{2}, z-w\right) + 2f\left( \frac{x-y}{2}, z+w\right) - 2f(x,z )+ 2 f(y, w)\right) \Vert $$ , where s is a fixed nonzero complex number with $$|s |< 1$$ .

The fundamental equations for the generalized resolvent of an elementary pencil in a unital Banach algebra