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.

Let $𝔐$ be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$,   $a,b\in 𝔐$(ii) $\parallel a^2\parallel =\parallel a{\parallel }^2$(iii) $\parallel a^2\parallel \le \parallel a^2+b^2\parallel$.$𝔐$ is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set ${𝔐}^{+}$ of squares in $𝔐$ is a closed convex cone. $(𝔐,{𝔐}^{+},\mathbf{1})$ is a complete ordered vector space with $\mathbf{1}$ as a order unit. In addition, we assume $𝔐$ to be monotone complete (i.e. $𝔐$ coincides with the bidual ${𝔐}^{**}$), and that there exists a finite normal faithful trace $\phi$ on $𝔐$.Then the completion $\{{𝔐}^{+}{\}}_{\phi }$ of ${𝔐}^{+}$ with respect to the Hilbert structure...

We give a survey of our recent results on homological properties of Köthe algebras, with an emphasis on biprojectivity, biflatness, and homological dimension. Some new results on the approximate contractibility of Köthe algebras are also presented.

Let S be a Rees semigroup, and let ℓ¹(S) be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of ℓ¹(S) are isomorphic to those of the underlying discrete group algebra.

For every closed subset C in the dual space ${Ĥ}_n$ of the Heisenberg group $H_n$ we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra $S\left(H_n\right)$ and we show that in general for two closed subsets $C_1,C_2$ of ${Ĥ}_n$ the product of $j\left(C_1\right)$ and $j\left(C_2\right)$ is different from $j\left(C_1\cap C_2\right)$.