We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if has a brai (blai), then the right (left) module action of on * is Arens regular if and only if is reflexive. We find that Arens regularity is implied by the factorization of * or ** when is a left or a right ideal in **. The Arens regularity and strong irregularity of are related to those of the module actions of on the nth dual ${}^{\left(n\right)}$ of . Banach algebras for which Z( **) = but $⊊Z^t(**)$ are...

Let ϰ be a positive, continuous, submultiplicative function on ${ℝ}^{+}$ such that $li{m}_{t\to \infty }{e}^{-\omega t}{t}^{-\alpha }\varkappa \left(t\right)=a$ for some ω ∈ ℝ, α ∈ $\overline{{ℝ}^{+}}$ and $a\in {ℝ}^{+}$. For every λ ∈ (ω,∞) let ${\varphi }_{\lambda }\left(t\right)=e^{-\lambda t}$ for $t\in {ℝ}^{+}$. Let $L_{\varkappa }^1\left({ℝ}^{+}\right)$ be the space of functions Lebesgue integrable on ${ℝ}^{+}$ with weight $\varkappa$, and let E be a Banach space. Consider the map ${\varphi }_{•}:(\omega ,\infty )\ni \lambda \to {\varphi }_{\lambda }\in L_{\varkappa }^1\left({ℝ}^{+}\right)$. Theorem 5.1 of the present paper characterizes the range of the linear map $T\to T{\varphi }_{•}$ defined on $L(L_{\varkappa }^1\left({ℝ}^{+}\right);E)$, generalizing a result established by B. Hennig and F. Neubrander for $\varkappa \left(t\right)=e^{\omega t}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder’s characterization...

We survey some old and new results in the theory of derivations on Banach algebras. Although our overview is broad ranging, our principal interest is in recent results concerning conditions on a derivation implying that its range is contained in the radical of the algebra.

E. Tutaj has introduced classes of Schauder bases termed "unconditional-like" (UL) and "unconditional-like*" (UL*) whose intersection is the class of unconditional bases. In view of this association with unconditional bases, it is interesting to note that there exist Banach spaces which have no unconditional basis and yet have a basis of one of these two types (e.g., the space 𝓞[0,1]). In the same spirit, we show in this paper that the space of all compact operators on a reflexive Banach space...

*Supported by the Grants AV ˇCR 101-97-02, 101-90-03, GA ˇCR 201-98-1449, and by the Grant of the Faculty of Civil Engineering of the Czech Technical University No. 2003.We further develop the theory of the so called Asplund functions, recently introduced and studied by W. K. Tang. Let f be an Asplund function on a Banach space X. We prove that (i) the subspace Y := sp ∂f(X) has a projectional resolution of the identity, and that (ii) if X is weakly Lindel¨of determined, then X admits a projectional...

When treating spaces of holomorphic functions with growth conditions, one is led to introduce associated weights. In our main theorem we characterize, in terms of the sequence of associated weights, several properties of weighted (LB)-spaces of holomorphic functions on an open subset $G\subset {ℂ}^N$ which play an important role in the projective description problem. A number of relevant examples are provided, and a “new projective description problem” is posed. The proof of our main result can also serve to characterize...