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In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

Let H be a complex separable Hilbert space with dim ⁡ H ≥ 3 and B ( H ) the Banach algebra of all bounded linear operators on H. Denote by w ( A ) the numerical radius of a bounded linear operator A ∈ B ( H ) and A B − B A ⁎ the skew Lie product of two operators A , B ∈ B ( H ) . In this paper, it is shown that, if a surjective map Φ : B ( H ) → B ( H ) satisfies w ( A B − B A ⁎ ) = w ( Φ ( A ) Φ ( B ) − Φ ( B ) Φ ( A ) ⁎ ) for all A , B ∈ B ( H ) , then there exist a unitary operator U ∈ B ( H ) , a functional h : B ( H ) → { − 1 , 1 } and a subset S ⊆ B ( H ) consisting of some normal operators such that Φ ( A ) = h ( A ) U A U ⁎ if A ∈ B ( H ) ∖ S and Φ ( A ) = h ( A ) U A ⁎ U ⁎ if A ∈ S . Particularly, if dim ⁡ H ∞ , a complete characterization of S can be obtained.

The aim of this paper is to introduce a concept of a cone inner product space over Banach algebras. This is done by replacing the co-domain of the classical inner product space by an ordered Banach algebra. Some properties such as Cauchy-Schwarz inequality, parallelogram identity and Pythagoras theorem are established in this setting. Similarly, the notion of cone normed algebra was introduced. Some illustrative examples are given to support our findings.

In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

We define and study approximate versions of σ-biflatness and σ-biprojectivity of a Banach algebra A where σ ∈Hom(A). We generalize the concepts pseudo amenability and pseudo contractibility via homomorphisms. We investigate their relations.

In this paper, we introduce a new class of algebras satisfying certain conditions which has been named power algebras and we prove some fixed point theorems for a new class of contractions. We give new conditions for the existence and uniqueness of fixed points for this new class of mappings which are not a contraction in the usual algebra. Also, we introduce a new iterative algorithm for fixed-point problems in a Banach algebra. This results improve and extend the corresponding conclusions of the Ishikawa algorithm under weaker conditions and lead to stronger results. In conclusion, we give an application to nonlinear integral equations.

Let $A$ be a Banach algebra. For $f\in A^{\ast}$, we inspect the weak sequential properties of the well-known map $T_f:A\to A^{\ast}$, $T_f(a) = fa$, where $fa\in A^{\ast}$ is defined by $fa(x) = f(ax)$ for all $x\in A$. We provide equivalent conditions for when $T_f$ is completely continuous for every $f\in A^{\ast}$, and for when $T_f$ maps weakly precompact sets onto L-sets for every $f\in A^{\ast}$. Our results have applications to the algebra of compact operators $K(X)$ on a Banach space $X$.

U ovom radu bavimo se lokalno kompaktnim grupama kao najopćenitijim matematičkim objektima koji imaju lijevu (desnu) Haarovu mjeru, tj. Radonovu mjeru invarijantnu s obzirom na lijeve (desne) translacije. Nakon preliminarnih sadržaja sakupljenih u prvom poglavlju, u drugom poglavlju razmatramo osnovna svojstva topoloških i lokalno kompaktnih grupa te kao glavni rezultat dokazujemo egzistenciju i jedinstvenost lijeve (desne) Haarove mjere na proizvoljnoj lokalno kompaktnoj grupi. Iduće poglavlje bavi se algebrom mjera M(G) i prostorom \(L^1(G)\), gdje je G neka lokalno kompaktna grupa, i pokazujemo kako s uvedenom konvolucijom oba prostora posjeduju strukturu Banachove ∗-algebre. Naposljetku, u četvrtom poglavlju predstavljamo dva teorema o egzistenciji i jedinstvenosti invarijatnih i kvazi-invarijantnih mjera na homogenim prostorima, što su prostori snabdjeveni djelovanjem neke lokalno kompaktne grupe. In this paper, we study the locally compact groups as the most general mathematical objects which have left (right) Haar measure, i.e. left (right) translation-invariant Radon measure. After the preliminaries assembled in Chapter 1, in Chapter 2 we examine basic features of topological and locally compact groups and as a main result we prove the existence and uniqueness of left (right) Haar measure on a locally compact group. Chapter 3 deals with the measure algebra M(G) and space \(L^1(G)\), where G is some locally compact group, and following the introduction of convolution, we show that both spaces have the structure of Banach ∗-algebra. Finally, in Chapter 4 we present two theorems about the existence and uniqueness of invariant and quasi-invariant measures on the homogeneous spaces, which are spaces equipped with an action of some locally compact group.

We study the problem of a special factorisation of an orthogonal projector~$P$ acting in the Hilbert space $L_2(\mathbb R)$ with $\dim\ker P

Let $${{\mathcal {A}}}$$ be a Banach algebra and let $$\varphi $$ be a non-zero character on $${{\mathcal {A}}}$$ . Suppose that $${{\mathcal {A}}}_M$$ is the closure of the faithful Banach algebra $${{\mathcal {A}}}$$ in the multiplier norm. In this paper, topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}_M^*$$ are defined and studied. Under some conditions on $${{\mathcal {A}}}$$ , we will show that the set of topologically left invariant $$\varphi $$ -means on $${{\mathcal {A}}}^*$$ and on $${{\mathcal {A}}}_M^*$$ have the same cardinality. The main applications are concerned with the quantum group algebra $$L^1({\mathbb {G}})$$ of a locally compact quantum group $${\mathbb {G}}$$ . In particular, we obtain some characterizations of compactness of $${\mathbb {G}}$$ in terms of the existence of a non-zero (weakly) compact left or right multiplier on $$L^1_M({\mathbb {G}})$$ or on its bidual in some senses.

Let $$\mathcal {A}$$ be a unital algebra and $$\mathcal {M}$$ be a unital $$\mathcal {A}$$ -bimodule. We characterize the linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal {A}$$ into $$\mathcal {M}$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B \in \mathcal {A}$$ with $$AB=0$$ when $$\mathcal {A}$$ contains a separating ideal $$\mathcal {T}$$ of $$\mathcal {M}$$ , which is in the algebra generated by all idempotents in $$\mathcal {A}$$ . We apply the result to $$\mathcal {P}$$ -subspace lattice algebras, completely distributive commutative subspace lattice algebras, and unital standard operator algebras. Furthermore, suppose that $$\mathcal {A}$$ is a unital Banach algebra and $$\mathcal {M}$$ is a unital Banach $$\mathcal {A}$$ -bimodule, we give a complete description of linear mappings $$\delta $$ and $$\tau $$ from $$\mathcal A$$ into $$\mathcal M$$ , satisfying $$\delta (A)B+A\tau (B)=0$$ for every $$A,B\in \mathcal {A}$$ with $$AB=I$$ .

In this research work, we demonstrate the Hyers-Ulam stability for Cauchy-Jensen functional equation in multi-Banach algebras by the fixed point technique. In fact, we prove that for a function which is approximately Cauchy-Jensen in multi Banach algebra, there is a unique involution near it. Next, we show that under some conditions the involution is continuous, the multi-Banach algebra becomes multi-$ C^* $-algebra and the Banach algebra is self-adjoint.

Let IK be a complete ultrametric field and let A be a unital commutative ultrametric Banach IK-algebra. Suppose that the multiplicative spectrum admits a partition in two open closed subsets. Then there exist unique idempotents u, v ϵ A such that ϕ (u) = 1, ϕ (v) = 0 ∀ ϕ ϵ U, ϕ (u) = 0 ϕ (v) = 1 ∀ ϕ ϵ V . Suppose that IK is algebraically closed. If an element x ϵ A has an empty annulus r < |ξ − a| < s in its spectrum sp(x), then there exist unique idempotents u, v such that ϕ (u) = 1; ϕ (v) = 0 whenever ϕ (x − a) ≤ r and ϕ (u) = 0; ϕ (v) = 1 whenever ϕ (x − a) ≥ s.

We give an exposition of Novodvorskii’s theorem in Banach algebra K-theory, asserting that the Gelfand transform for a commutative Banach algebra induces an isomorphism in topological K-theory.

For a Banach algebra $${{\mathcal {A}}}$$ and a norm one character $$\phi $$ on $${{\mathcal {A}}}$$ , we characterize $$\phi $$ -amenability and $$\phi $$ -contractibility of $${{\mathcal {A}}}$$ in terms of metrically homological properties of some Banach $${\mathcal A}$$ -modules. Also, for any Banach $${{\mathcal {A}}}$$ -module, metrically injectivity and projectivity of submodules and quotient modules are investigated. Finally, we give some results on metrically injectivity of $${{\mathcal {A}}}^*$$ and its submodules.

We present an extension, for Banach algebras, of the extension, due to Corach et al, of Jacobson’s lemma on the g-Drazin inverse (Comm. Algebra, 41(2013), 520–531).

Let $${\mathcal {A}}$$ and $${\mathcal {U}}$$ be Banach algebras such that $${\mathcal {U}}$$ is a Banach $${\mathcal {A}}$$ -bimodule with compatible algebra multiplication, module actions and norm. By defining an appropriate action, we turn $$\ell ^1$$ -direct product $${\mathcal {A}}\times {\mathcal {U}}$$ into a Banach algebra such that $${\mathcal {A}}$$ is a closed subalgebra and $${\mathcal {U}}$$ is a closed ideal of $${\mathcal {A}}\times {\mathcal {U}}$$ . This algebra is, in fact, the semidirect product of $${\mathcal {A}}$$ and $${\mathcal {U}}$$ , $${\mathcal {A}}\ltimes {\mathcal {U}}$$ , and as well as, every semidirect product of Banach algebras can be represented as this form. Our aim in this paper is to study the first cohomology group of $${\mathcal {A}}\ltimes {\mathcal {U}}$$ and investigate the relation between the first cohomology group of $${\mathcal {A}}\ltimes {\mathcal {U}}$$ and those of $${\mathcal {A}}$$ and $${\mathcal {U}}$$ . As an application of our results, we show that the earlier results obtained for the first cohomology group of various classes of Banach algebras such as direct products of Banach algebras, module extension Banach algebras and $$\theta$$ -Lau products of Banach algebras can be obtained directly by applying our main results and techniques of proofs. Some examples are also given.

Segal extensions, not necessarily injective, of a Banach algebra into a Banach algebra are discussed resulting into answering when do the Gelfand map of a commutative Banach algebra as well as Gelfand-Naimark construction on a, not necessarily commutative, Banach $$*$$ -algebra give Segal extensions. Analogous to $$C^*$$ -Segal algebras arising from Segal extension into $$C^*$$ - algebras, uB-Segal algebras arising from Segal extensions into uniform Banach algebras are intrinsically characterized.

2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra $$M_{2^n}(A)$$ M 2 n ( A ) ($$n\ge 2$$ n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW$$^*$$ ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property.

Let A be a Banach algebra and X be a (Banach) A-bimodule. A linear map T : A→X is called an n-Jordan multiplier if T(an) = aT(an− 1) for all a ∈ A. In this paper, among other things, we show that under special hypotheses every (n + 1)-Jordan multiplier is an n-Jordan multiplier and vice versa.

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.

The totally-real embeddability of any $2k$-dimensional compact manifold $M$ into $\mathbb C^n$, $n\geq 3k$, has several consequences: the genericity of polynomially convex embeddings of $M$ into $\mathbb C^n$, the existence of $n$ smooth generators for the Banach algebra $\mathcal C(M)$, the existence of nonpolynomially convex embeddings with no analytic disks in their hulls, and the existence of special plurisubharmonic defining functions. We show that these results can be recovered even when $n=3k-1$, $k>1$, despite the presence of complex tangencies, thus lowering the known bound for the optimal $n$ in these (related but inequivalent) questions., 15 pages; paper has been reorganized, Prop.1.3 is new and Thm 1.4 is a substantial expansion of Cor. 1.3 from Version 1

We investigate under what conditions n-Jordan homomorphisms between rings are n- homomorphism, or homomorphism; and under what conditions, n-Jordan homomorphisms are continuous.One of the main goals in this work is to show that every n-Jordan homomorphism f : A → B, from a unital ring A into a ring B with characteristic greater than n, is a multiple of a Jordan homomorphism and hence, it is an n-homomorphism if every Jordan homomorphism from A into B is a homomorphism. In particular, if B is an integral domain whose characteristic is greater than n, then every n-Jordan homomorphism f : A → B is an n-homomorphism.Along with some other results, we show that if A and B are unital rings such that the characteristic of B is greater than n, then every unital n-Jordan homomorphism f : A → B is a Jordan homomorphism and hence, it is an m-Jordan homomorphism for any positive integer m ≥ 2.We also investigate the automatic continuity of n-Jordan homomorphisms from a unital Banach algebra either into a semisimple commutative Banach algebra, onto a semisimple Banach algebra, or into a strongly semisimple Banach algebra whenever the n-Jordan homomorphism has dense range.

Let $\mathcal{A}$ be a commutative complex semisimple Banach algebra. Denote by ${\rm kh}({\rm soc}(\mathcal{A}))$ the kernel of the hull of the socle of $\mathcal{A}$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $\mathcal{A}$. This allows us to show that a "result" from Riesz theory in commutative Banach algebras is not true.

Let $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras and let $\theta \in \Delta (\mathcal B)$ . In this paper, we prove that $\mathcal A\times _{\theta }\mathcal B$ is a type I-BSE algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are so. As a main application of this result, we prove that $\mathcal A\times _{\theta }\mathcal B$ is isomorphic with a $C^*$ -algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are isomorphic with $C^* $ -algebras. Moreover, we derive related results for the case where $\mathcal A$ is unital.

Recently it has been proved that if σ ∈ GL n ( R ) where R is a commutative ring and n ≥ 3 , then each of the matrices t k l ( σ i j ) ( i ≠ j , k ≠ l ) is a product of eight E n ( R ) -conjugates of σ and σ − 1 . In this article we show that similar results hold true if R is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.

The main aim of this paper is to introduce the concept of $\mathcal{N}_{b}$ N b -cone metric spaces over a Banach algebra as a generalization of $\mathcal{N}$ N -cone metric spaces over a Banach algebra and b-metric spaces. Also, we study some coupled common fixed point theorems for generalized Lipschitz mappings in this framework. Finally, we give an example and an application to the existence of solutions of integral equations to illustrate the effectiveness of our generalizations. Some results in the literature are special cases of our results.

Our aim of this paper is to prove some fixed point and common fixed theorems for contractive type maps in a cone metric space over Banach algebra, which unify, extend and generalize most of the existing relevant fixed point theorems from Jiang et al. [1].

In this article we discuss the commutativity of a prime Banach algebra A with the help of its generalized $$(\alpha ,\alpha )$$ -derivation. In particular, we prove that if A is a unital prime Banach algebra and A has a nonzero continuous linear generalized $$(\alpha ,\alpha )$$ -derivation g associated with a nonzero continuous linear $$(\alpha ,\alpha )$$ -derivation d such that $$g((xy)^n)-d(x^n)d(y^n)=0$$ or $$g((xy)^n)-d(y^n)d(x^n)=0$$ for sufficiently many x, y and integer $$n=n(x,y)>1$$ then A must be commutative. In this connection some more results has been found. Further examples are given to show that hypotheses are not superfluous.

UDC 517.98 Essential amenability of Banach algebras have been defined and investigated. Here, this concept will be introduced for Frechet algebras. Then a number of well-known results of essential amenability of Banach algebras are generalized for Fréchet algebras. Moreover, related results about Segal–Fréchet algebras are provided. As the main result, it is provedthat if ( 𝒜 , p ℓ ) is an amenable Fréchet algebra with a uniformly bounded approximate identity, then every symmetric Segal – Fréchet algebra in ( 𝒜 , p ℓ ) is essentially amenable.

Let ℜ {\mathfrak{R}} be a ring with center Z ⁢ ( ℜ ) {Z(\mathfrak{R})} . In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 {\mathfrak{F}} such that ( [ 𝔉 ⁢ ( s m ) ⁢ s n + s n ⁢ 𝔉 ⁢ ( s m ) , s r ] k ) l ∈ Z ⁢ ( ℜ ) {([\mathfrak{F}(s^{m})s^{n}+s^{n}\mathfrak{F}(s^{m}),s^{r}]_{k})^{l}\in Z(% \mathfrak{R})} for every s ∈ ℜ {s\in\mathfrak{R}} , then either 𝔉 ⁢ ( s ) = p ⁢ s {\mathfrak{F}(s)=ps} for every s ∈ ℜ {s\in\mathfrak{R}} or ℜ {\mathfrak{R}} satisfies s 4 {s_{4}} and 𝔉 ⁢ ( s ) = s ⁢ p {\mathfrak{F}(s)=sp} for every s ∈ ℜ {s\in\mathfrak{R}} and p ∈ 𝔘 {p\in\mathfrak{U}} , the Utumi quotient ring of ℜ {\mathfrak{R}} . As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.