A simple and natural example is given of a non-commuting Arens multiplication.

Let G be a locally compact abelian group, M(G) the convolution measure algebra, and X a Banach M(G)-module under the module multiplication μ ∘ x, μ ∈ M(G), x ∈ X. We show that if X is an essential L¹(G)-module, then $\sigma \left(T_{\mu }\right)=\overline{\mu ̂\left(sp\right(X\left)\right)}$ for each measure μ in reg(M(G)), where $T_{\mu }$ denotes the operator in B(X) defined by $T_{\mu }x=\mu \circ x$, σ(·) the usual spectrum in B(X), sp(X) the hull in L¹(G) of the ideal $I_X=f\in L¹\left(G\right)|T_f=0$, μ̂ the Fourier-Stieltjes transform of μ, and reg(M(G)) the largest closed regular subalgebra of M(G); reg(M(G)) contains all...

Denote by any set of cardinality continuum. It is proved that a Banach algebra A with the property that for every collection $a_{\alpha }:\alpha \in \subset A$ there exist α ≠ β ∈ such that $a_{\alpha }\in a_{\beta }{A}^{}$ is isomorphic to ${⨁}_{i=1}^r(ℂ\left[X\right]/X^{d_i}ℂ\left[X\right])\oplus E$, where $d₁,...,d_r\in \mathbb{N}$, and E is either $Xℂ\left[X\right]/X^{d₀}ℂ\left[X\right]$ for some d₀ ∈ ℕ or a 1-dimensional ${⨁}_{i=1}^rℂ\left[X\right]/X^{d_i}ℂ\left[X\right]$-bimodule with trivial right module action. In particular, ℂ is the unique non-zero prime Banach algebra satisfying the above condition.

If X and Y are Banach spaces, then subalgebras ⊂ B(X) and ⊂ B(Y), not necessarily unital nor complete, are called standard operator algebras if they contain all finite rank operators on X and Y respectively. The peripheral spectrum of A ∈ is the set ${\sigma }_{\pi }\left(A\right)=\lambda \in \sigma \left(A\right):\left|\lambda \right|=ma{x}_{z\in \sigma \left(A\right)}\left|z\right|$ of spectral values of A of maximum modulus, and a map φ: → is called peripherally-multiplicative if it satisfies the equation ${\sigma }_{\pi }\left(\phi \left(A\right)\circ \phi \left(B\right)\right)={\sigma }_{\pi }\left(AB\right)$ for all A,B ∈ . We show that any peripherally-multiplicative and surjective map φ: → , neither assumed to be linear nor...

We consider the topological algebra of (Taylor) multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We describe multiplicative functionals and algebra homomorphisms on that algebra as well as idempotents in it. We show that it is never a Q-algebra and never locally m-convex. In particular, we show that Taylor multiplier sequences cease to be so after most permutations.

It is shown that every algebraic isomorphism between standard subalgebras of 𝒥-subspace lattice algebras is quasi-spatial and every Jordan derivation of standard subalgebras of 𝒥-subspace lattice algebras is an additive derivation. Also, it is proved that every finite rank operator in a 𝒥-subspace lattice algebra can be written as a finite sum of rank one operators each belonging to that algebra. As an additional result, a multiplicative bijection of a 𝒥-subspace lattice algebra onto an arbitrary...