Noncommutative Banach Algebras Research Articles

On the spectrum of a class of non-commutative Banach algebras

On the spectrum of a class of non-commutative Banach algebras

A pair of generalized derivations in prime, semiprime rings and in Banach algebras

Let R be a prime ring with extended centroid C, I a non-zero ideal of R and n ≥ 1 a fixed integer. If R admits the generalized derivations H and G such that (H(xy)+G(yx))n= (xy ±yx) for all x,y ∈ I, then one ofthe following holds:(1) R is commutative;(2) n = 1 and H(x) = x and G(x) = ±x for all x ∈ R.Moreover, we examine the case where R is a semiprime ring. Finally, we apply the above result to non-commutative Banach algebras.

Some differential identities on prime and semiprime rings and Banach algebras

In this manuscript, we discuss the behaviour and nature of generalized derivation \({\mathscr {G}}\) on a (semi-) prime ring \({\mathscr {R}}\) satisfying certain differential identities over \({\mathscr {I}}\), a nonzero ideal of \({\mathscr {R}}\). Moreover, we extend our purely ring theoretic result to a non-commutative Banach algebras and obtained some range inclusion results of continuous generalized derivations.

Generalized derivations on some convolution algebras

Let G be a locally compact abelian group, $$\omega $$ be a weighted function on $${\mathbb {R}}^+$$ , and let $$\mathfrak {D}$$ be the Banach algebra $$L_0^\infty (G)^*$$ or $$L_0^\infty (\omega )^*$$ . In this paper, we investigate generalized derivations on the noncommutative Banach algebra $$\mathfrak {D}$$ . We characterize $$\textsf {k}$$ -(skew) centralizing generalized derivations of $$\mathfrak {D}$$ and show that the zero map is the only $$\textsf {k}$$ -skew commuting generalized derivation of $$\mathfrak {D}$$ . We also investigate the Singer–Wermer conjecture for generalized derivations of $$\mathfrak {D}$$ and prove that the Singer–Wermer conjecture holds for a generalized derivation of $$\mathfrak {D}$$ if and only if it is a derivation; or equivalently, it is nilpotent. Finally, we investigate the orthogonality of generalized derivations of $$L_0^\infty (\omega )^*$$ and give several necessary and sufficient conditions for orthogonal generalized derivations of $$L_0^\infty (\omega )^*$$ .

On Lie Ideals with Generalized Derivations and Non-commutative Banach Algebras

Let R be a prime ring of characteristic different from 2, L be a non-central Lie ideal of R and m, n be the fixed positive integers. If R admits a generalized derivation F associated with a deviation d such that $$F(u^2)^m -d(u)^{2n}\in Z(R)$$ for all $$u \in L$$ , then R satisfies $$s_4$$ , the standard identity in four variables. Moreover, we also examine the case when R is semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

On generalized derivation in rings and Banach algebras

Let R be a prime ring, F be a generalized derivation associated with a derivation d of R and m, n be the fixed positive integers. In this paper we study the case when one of the following holds: (i) F(x) ◦m F(y) = (x ◦ y)n, (ii) F(x)◦md(y) = d(x◦y)n for all x, y in some appropriate subset of R. We also examine the case where R is a semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.

A non-commutative Banach algebra whose maximal commutative subalgebras are all mutually isomorphic

A non-commutative Banach algebra whose maximal commutative subalgebras are all mutually isomorphic

On prime and semiprime rings with generalized derivations and non-commutative Banach algebras

Let R be a prime ring of characteristic different from 2 and m a fixed positive integer. If R admits a generalized derivation associated with a nonzero deviation d such that [F(x),d(y)] m =[x,y] for all x,y in some appropriate subset of R, then R is commutative. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer–Werner theorem.

Derivations as a generalization of Jordan homomorphisms on Lie ideals and non-commutative Banach algebras

Let R be a prime ring with center Z(R), d a nonzero derivation of R, L a noncentral Lie ideal of R and \(n\ge 1\) a fixed integer. If \(d(u^2)^n-d(u)^{2n}\in Z(R)\) for all \(u \in L\), then R satisfies \(s_4\), the standard identity in four variables. We also obtain some related result in case A is a non-commutative Banach algebra and d is continuous.

DERIVATIONS ON CONVOLUTION ALGEBRAS

In this paper, we investigate derivations on the noncommutative Banach algebra <TEX>$L^{\infty}_0({\omega})^*$</TEX> equipped with an Arens product. As a main result, we prove the Singer-Wermer conjecture for the noncommutative Banach algebra <TEX>$L^{\infty}_0({\omega})^*$</TEX>. We then show that a derivation on <TEX>$L^{\infty}_0({\omega})^*$</TEX> is continuous if and only if its restriction to rad(<TEX>$L^{\infty}_0({\omega})^*$</TEX>) is continuous. We also prove that there is no nonzero centralizing derivation on <TEX>$L^{\infty}_0({\omega})^*$</TEX>. Finally, we prove that the space of all inner derivations of <TEX>$L^{\infty}_0({\omega})^*$</TEX> is continuously homomorphic to the space <TEX>$L^{\infty}_0({\omega})^*/L^1({\omega})$</TEX>.

Approximate Derivations with the Radical Ranges of Noncommutative Banach Algebras

We consider the derivations on noncommutative Banach algebras, and we will first study the conditions for a derivation on noncommutative Banach algebra. Then, we examine the stability of functional inequalities with a derivation. Finally, we take the derivations with the radical ranges on noncommutative Banach algebras.

JORDAN DERIVATIONS ON PRIME RINGS AND THEIR APPLICATIONS IN BANACH ALGEBRAS, II

The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. We show that if there exists a continuous linear Jordan derivation D : A <TEX>${\rightarrow}$</TEX> A such that [D(x), x]<TEX>$D(x)^3{\in}$</TEX> rad(A) for all <TEX>$x{\in}A$</TEX>, then D(A) <TEX>${\subseteq}$</TEX> rad(A).

Fourier transform and the ideal of group algebra on the nilpotent Engel-Lie group

We study noncommutative Fourier transform on the nilpotent Engel-Lie group G4 to solve some interesting problems of noncommutative analysis. In fact the finest structure of G4 can be shown as a semi-direct product of two real vector groups. This helps us to form our idea by constructing a new larger group in order to define the Fourier transform and to obtain the Plancherel formula on G4. Moreover we show that our methods lead us to construct several existence theorems for the invariant differential operators on G4 and on G4 × ℝ. Since the heat equation is invariant on G4 × ℝ, so a fundametal solution of this equation will be obtained. Finally we establish a theorem that gives a classification of all left ideals of the noncommutative Banach algebra L1(G4) of G4.

Some Banach algebra properties in the Cartesian product of Banach algebras

For semisimple, commutative Banach algebras $\mathcal A$ and $\mathcal B$, some Banach algebra properties of the Cartesin product $\mathcal A \times \mathcal B$ are characterized in terms of $\mathcal A$ and $\mathcal B$. A couple of results are also proved for non-commutative Banach algebras.

JORDAN DERIVATIONS ON PRIME RINGS AND THEIR APPLICATIONS IN BANACH ALGEBRAS, I

The purpose of this paper is to prove that the noncommutative version of the Singer-Wermer Conjecture is affirmative under certain conditions. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation <TEX>$D:A{\rightarrow}A$</TEX> such that <TEX>$D(x)^3[D(x),x</TEX><TEX>]</TEX><TEX>{\in}rad(A)$</TEX> for all <TEX>$x{\in}A$</TEX>. In this case, we show that <TEX>$D(A){\subseteq}rad(A)$</TEX>.

Corrections to “Riccati Equations on Noncommutative Banach Algebras"

This paper contains corrections to [R. Curtain, SIAM J. Control Optim., 49 (2011), pp. 2542--2557]. While all claims remain valid, and the proof for the linear quadratic Riccati equations is correct, this proof does not cover the positive-real and bounded-real Riccati equations. Here a correct proof is given for these cases.

Riccati Equations on Noncommutative Banach Algebras

Conditions for the existence of a solution of a Riccati equation to be in some prescribed noncommutative involutive Banach algebras are given. The Banach algebras are inverse-closed subalgebras of the space of bounded linear operators on some Hilbert space, and the Riccati equation has an exponentially stabilizing solution on this larger space. Applications to spatially distributed systems are given.

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

It is shown that any generalized Jordan (triple-)derivation on a 2–torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2–torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.

Generalized Derivations on (Semi-)Prime Rings and Noncommutative Banach Algebras

We first give several polynomial identities of semiprime rings which make the additive mappings appearing in the identities to be generalized derivations. Then we study pairs of generalized Jordan derivations on prime rings. Let m,n be fixed positive integers, \mathcal R be a noncommutative 2(m+n)! -torsion free prime ring with the center \mathcal Z and μ, ν be a pair of generalized Jordan derivations on \mathcal R . If μ(x^m)x^n+x^nν(x^m) \in \mathcal Z for all x \in \mathcal R , then μ and ν are left (or right) multipliers. In particular, if μ , ν are a pair of derivations on \mathcal R satisfying the same assumption, then μ = ν = 0 . Then applying these purely algebraic result we obtain several range inclusion results of pair of derivations on Banach algebras.

The canonical test case for the non-commutative Singer–Wermer conjecture

It is a famous conjecture that every derivation on each Banach algebra leaves every primitive ideal of the algebra invariant. This conjecture is known to be true if, in addition, the derivation is assumed to be continuous. It is also known to be true if the algebra is commutative, in which case the derivation necessarily maps into the (Jacobson) radical. Because I. M. Singer and J. Wermer originally raised the question in 1955 for the case of commutative Banach algebras, the conjecture is now usually referred to as the non-commutative Singer-Wermer conjecture (the non-commutative situation being the unresolved case). In a previous paper we demonstrated that if the conjecture fails for some non-commutative Banach algebra with discontinuous derivation, then it fails for at most finitely many primitive ideals, and each of these primitive ideals must be of finite codimension. In this paper we first show that one can make an additional reduction of any counter-example to the simplest case of a non-commutative radical Banach algebra with identity adjoined and discontinuous derivation D such that D does not leave the (Jacobson) radical (which is of codimension one) invariant. Second, we show that this radical Banach algebra with identity adjoined has a formal power series quotient of the form A0[[t]] based at an element t in the radical which is mapped to an invertible element by the discontinuous derivation. Finally, we specialize to the case of a separable Banach algebra and show that the pre-image of the algebra A0 is a unital subalgebra which is not an analytic set. In particular, this shows that A0 cannot be countably generated.