Weak module amenability of triangular Banach algebras

Let A and B be unital Banach algebras and let M be a unital Banach A,B-module. Forrest and Marcoux [6] have studied the weak amenability of triangular Banach algebra \(\mathcal{T} = \left[ {_B^{AM} } \right]\) and showed that T is weakly amenable if and only if the corner algebras A and B are weakly amenable. When \(\mathfrak{A}\) is a Banach algebra and A and B are Banach \(\mathfrak{A}\)-module with compatible actions, and M is a commutative left Banach \(\mathfrak{A}\)-A-module and right Banach \(\mathfrak{A}\)-B-module, we show that A and B are weakly \(\mathfrak{A}\)-module amenable if and only if triangular Banach algebra T is weakly \(\mathfrak{T}\)-module amenable, where \(\mathfrak{T}: = \{ [^\alpha _\alpha ]:\alpha \in \mathfrak{A}\} \).     

Synthesis and antimicrobial activity of some netropsin analogues

Nine novel lexitropsins were synthesized by linking two netropsin-like moieties through three different dicarboxylic acids; 9,10-dihydro-2,7-phenanthrenedicarboxylic acid; [(3-[[(carboxymethyl)amino]carbonyl]benzoyl)amino]acetic acid and indole-2,5-dicarboxylic acid. The netropsin residues were modified by the use of N-isopentylpyrrole, 5-methylthiophene or 5-isopropylthiazole heterocyclic building blocks in place of the usual N-methylpyrrole. The compounds were tested against five gram-positive bacteria: Staphylococcus aureus, Streptomyces faecalis, methicillin resistant Staphylococcus aureus, Enterobacter cloacae, Mycobacterium fortuitum, three gram-negative bacteria: Klebsiella aerogenes, Proteus vulgaris, Escherichia coli and three fungi: Aspergillus niger, Candida albicans and Aspergillus nidulans. Some of the compounds showed significant inhibitory effects on the growth of the microorganisms.     

Approximate amenability of certain inverse semigroup algebras

In this article, the approximate amenability of semigroup algebra ?¹(S) is investigated, where (S) is a uniformly locally finite inverse semigroup. Indeed, we show that for a uniformly locally finite inverse semigroup (S), the notions of amenability, approximate amenability and bounded approximate amenability of ?¹(S) are equivalent. We use this to give a direct proof of the approximate amenability of ?¹(S) for a Brandt semigroup (S). Moreover, we characterize the approximate amenability of ?¹(S), where S is a uniformly locally finite band semigroup.