The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C∗-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.

We study representations of Banach algebras on reflexive Banach spaces. Algebras which admit such representations which are bounded below seem to be a good generalisation of Arens regular Banach algebras; this class includes dual Banach algebras as defined by Runde, but also all group algebras, and all discrete (weakly cancellative) semigroup algebras. Such algebras also behave in a similar way...

We give a new Banach module characterization of W *-modules, also known as selfdual Hilbert C *-modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of W *-modules, to the setting where the operator algebras are σ-weakly closed algebras of operators on a Hilbert space. That is, we find the appropriate weak* topology variant of our earlier notion of r...

We use a fermionic extension of the bosonic module to obtain a class of B(0, N)-graded Lie superalgebras with nontrivial central extensions. 0 Introduction B(M − 1, N)-graded Lie superalgebras were first investigated and classified up to central extension by Benkart-Elduque (see also Garcia-Neher’s work in [GN]). Those root graded Lie superalgebras are a super-analog of root graded Lie algebras...

assume that $a$‎, ‎$b$ are banach algebras and that $m:atimes brightarrow b$‎, ‎$m^prime:atimes arightarrow b$ are bounded bilinear mappings‎. ‎we study the relationships between arens regularity of $m$‎, ‎$m^prime$ and the banach algebras $a$‎, ‎$b$‎. ‎for a banach $a‎$‎-bimodule $b$‎, ‎we show that $b$ factors with respect to $a$ if and only if $b^{**}$ is unital as an $a^{**}‎$‎-module‎. ‎le...

Let M be a Banach C*-module over a C*-algebra A carrying two A-valued inner products 〈., .〉1, 〈., .〉2 which induce equivalent to the given one norms on M. Then the appropriate unital C*-algebras of adjointable bounded A-linear operators on the Hilbert A-modules {M, 〈., .〉1} and {M, 〈., .〉2} are shown to be ∗-isomorphic if and only if there exists a bounded A-linear isomorphism S of these two Hi...

We define a Banach algebra A to be dual if A = (A∗) ∗ for a closed submodule A∗ of A∗. The class of dual Banach algebras includes all W ∗-algebras, but also all algebras M(G) for locally compact groups G, all algebras L(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception...

We study Banach C̃-algebras, i.e., complete ultra-pseudo-normed algebras over the ring C̃ of Colombeau generalized complex numbers. We develop a spectral theory in such algebras. We show by explicit examples that important parts of classical Banach algebra theory do not hold for general Banach C̃algebras and indicate a particular class of Banach C̃-algebras that overcomes these limitations to a lar...