Ring homomorphisms on real Banach 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 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.

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.

The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

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 $$C(X(\mathcal {R}), \tau )$$.

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.

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.

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

For commutative Banach algebras B and I and a continuous representation π: B → M(I), where M(I) are those bounded linear operators T: I → I that satisfy T(ij) = iT(j) for all i,jeI, the direct sum B I can be made into a commutative Banach algebra which contains B and I as respectively a closed subalgebra and a closed ideal. This algebra is called the semidirect product of B and I. Some topological aspects of the character space of a semidirect product are described. Furthermore, decompositions of commutative Banach algebras into the direct sum of a subalgebra and a principal ideal are investigated.

Some problems, concerning non-removable ideals in commutative Banach algebras, are discussed.

Recent research on Dunford-Pettis operators associated with the group algebra L 1 (G) and the Fourier algebra A(G) of a locally compact group has motivated the study of the structure of the Dunford-Pettis space for a Banach module ϵ over a Banach algebra . In this paper, we study conditions under which (ϵ) coincides with the entire Banach right -module ϵ, and investigate its connections with some classical geometric properties. We analyze the structure of Dunford-Pettis elements for various classical Banach spaces and study their behavior under continuous homomorphisms and quotient maps. Moreover, we explore the geometric properties of certain ideals in commutative Banach algebras.

In order to summarize the contents of this paper it is necessary to use some of the definitions in [5]. These definitions are repeated below at the beginning of Section 1. Although this paper adds to the material in [5] we have attempted to reduce the number of references to [o] to a minimum. An approximate diagonal in a Banach algebra 31 is a bounded net {ma} in 31 a for all a in 31 in the norm topologies where 31? 31 is the usual projective tensor product [2] and -k is the continuous linear map 31?31->31 defined by ir(a?b) =ab. The net {m 31) ** with aM = Ma and ?r** (M) a = a for all a ? 31. In the algebraic cohomology of algebras the existence of an element m of A A with ma = am and ir(m) the identity of A is equivalent to the statement that H1(A, X) =0 for all 31 modules X almost never holds?in fact, in the commutative case for all A bimodules X. Among Banach algebras the statement jEP^S^X) =0 (see [3; Theorem 1] and [5; Proposition 8.1]) we can always find a Banach 31 module ? with S^1(3I,36)=t^O unless 31 zz?n. The existence of an approxi? mate diagonal is not unusual and is equivalent to the statement !M* (31,36*) = 0 for all Banach 31 modules X; thatis 31 is an emenable Banach algebra. The element M plays the same role in the theory of amenable tlgebras that the invariant mean does for amenable groups. The definitions used from [5] and the theorem connecting amenability with approximate diagonals are given in ? 1. In ? 2 we show that if 31 has a virtual diagonal with Ma? 31? St for all a in 3T then 913 (31, 3E) ? .0 for all Banach 31 modules 3?. Algebras satisfying this condition are grdup algebras of compact topological groups and the algebra c0 of convergent sequences of complex numbers. Although we do not prove it the result also applies to any annihilator B* algebra with only finite dimensional minimal ideals. In ? 3 we show that if 31 is a semi simple commutative amenable Banach algebra then there is a Banach 31 module X with 3/2(3I,3E) ^0, unless 31 is finite

Topological invariants of the maximal ideal space of a Banach algebra

Embedding Nilpotent Finite Dimensional Banach Algebras into Amenable Banach Algebras