Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F\left(u\right){[F\left(u\right),u]}^n=0$ for all $u\in L$, where $n\ge 1$ is a fixed integer. Then one of the following holds: (1) ...

By considering arbitrary mappings $\omega$ from a Banach algebra $A$ into the set of all nonempty, compact subsets of the complex plane such that for all $a\in A$, the set $\omega \left(a\right)$ lies between the boundary and connected hull of the exponential spectrum of $a$, we create a general framework in which to generalize a number of results involving spectra such as the exponential and singular spectra. In particular, we discover a number of new properties of the boundary spectrum.

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over ℂ whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement...

Dans une algèbre de Banach et dans deux cas particuliers, nous montrons la continuité du centre du plus petit disque contenant le spectre. Pour a ∈ , on donne une condition nécessaire et suffisante pour avoir $R_K=d\left(a\right)$ où d(a) est la distance de a aux scalaires et $R_K$ le rayon du plus petit disque contenant K qui représente le spectre ou le domaine numérique algébrique de a. Dans un espace de Hilbert complexe, K peut représenter certains types de spectres ou de domaines numériques de a.

Let $A$ be a unital Banach algebra over $ℂ$, and suppose that the nonzero spectral values of $a$ and $b\in A$ are discrete sets which cluster at $0\in ℂ$, if anywhere. We develop a plane geometric formula for the spectral semidistance of $a$ and $b$ which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that $a$ and $b$ are quasinilpotent equivalent if...

Let A be a linear closed densely defined operator in a complex Banach space X. If A is of type ω (i.e. the spectrum of A is contained in a sector of angle 2ω, symmetric around the real positive axis, and $\parallel \lambda {(\lambda I-A)}^{-1}\parallel$ is bounded outside every larger sector) and has a bounded inverse, then A has a bounded $H^{\infty }$ functional calculus in the real interpolation spaces between X and the domain of the operator itself.

The purpose of this paper is to introduce a harmonic functional calculus in order to generalize some extended versions of theorems of von Neumann, Heinz and Ky Fan.

Given ⊂ ℕ, let ̂ be the set of all positive integers m for which $h^m$ is hermitian whenever h is an element of a complex unital Banach algebra A with hⁿ hermitian for each n ∈ . We attempt to characterize when (i) ̂ = ℕ, or (ii) ̂ = . A key tool is a Müntz-type theorem which gives remarkable conclusions when 1 ∈ and ∑ 1/n: n ∈ diverges. The set ̂ is determined by a single extremal Banach algebra Ea(). We describe this extremal algebra for various .

Bade, Curtis and Dales have introduced the idea of weak amenability. A commutative Banach algebra A is weakly amenable if there are no non-zero continuous derivations from A to A*. We extend this by defining an alternating n-derivation to be an alternating n-linear map from A to A* which is a derivation in each of its variables. Then we say that A is n-dimensionally weakly amenable if there are no non-zero continuous alternating n-derivations on A. Alternating n-derivations are the same as alternating...

The classical identification of the predual of B(H) (the algebra of all bounded operators on a Hilbert space H) with the projective operator space tensor product $\overline{H}\widehat{⨂}H$ is extended to the context of Hilbert modules over commutative von Neumann algebras. Each bounded module homomorphism b between Hilbert modules over a general C*-algebra is shown to be completely bounded with $\parallel b{\parallel }_{cb}=\parallel b\parallel$. The so called projective operator tensor product of two operator modules X and Y over an abelian von Neumann algebra C is introduced...

We compute the algebraic and continuous Hochschild cohomology groups of certain Fréchet algebras of analytic functions on a domain U in ${ℂ}^n$ with coefficients in one-dimensional bimodules. Among the algebras considered, we focus on A=A(U). For this algebra, our results apply if U is smoothly bounded and strictly pseudoconvex, or if U is a product domain.