We revisit the characterisation of modules over non-unital C∗-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical inde...

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 τ be a linear map from a unital C∗-algebra A to a von Neumann algebra B and let C be a unital C∗-algebra. A map T from a Hilbert A-module E to a von Neumann C-B module F is called a τ -map if 〈T (x), T (y)〉 = τ(〈x, y〉) for all x, y ∈ E. A Stinespring type theorem for τ -maps and its covariant version are obtained when τ is completely positive. We show that there is a bijective correspondenc...

Let K be a Banach space, B a unital C∗-algebra, and π : B → L(K) an injective, unital homomorphism. Suppose that there exists a function γ : K×K → R+ such that, for all k, k1, k2 ∈ K, and all b ∈ B, (a) γ(k, k) = ‖k‖2, (b) γ(k1, k2) ≤ ‖k1‖ ‖k2‖, (c) γ(πbk1, k2) = γ(k1, πb∗k2). Then for all b ∈ B, the spectrum of b in B equals the spectrum of πb as a bounded linear operator on K. If γ satisfies ...

Let A be an algebra over a commutative unital ring C. We say that A is zero triple product determined if for every C-module X and every trilinear map {·, ·, ·}, the following holds: if {x, y, z} 0 whenever xyz 0, then there exists a C-linear operator T : A3 −→ X such that {x, y, z} T xyz for all x, y, z ∈ A. If the ordinary triple product in the aforementioned definition is replaced by Jordan t...

Let B be a σ–unital C–algebra. We show that every strongly continuous E0–semigroup on the algebra of adjointable operators on a full Hilbert B–module E gives rise to a full continuous product system of correspondences over B. We show that every full continuous product system of correspondences over B arises in that way. If the product system is countably generated, then E can be chosen countabl...

Slender rings and their products are characterized in terms of algebraic and topological properties. Possible limitations on uncountable products of such rings are discussed. Ring, module, and topological terminology and notation are standard. R denotes a unital ring with identity 1, and R ω denotes its countable product, that is, the set of all sequences of elements of R. Of course ω {0, 1,. ....

‎Let $mathcal{A}$ be a unital Banach algebra‎, ‎$mathcal{M}$ be a left $mathcal{A}$-module‎, ‎and $W$ in $mathcal{Z}(mathcal{A})$ be a left separating point of $mathcal{M}$‎. ‎We show that if $mathcal{M}$ is a unital left $mathcal{A}$-module and $delta$ is a linear mapping from $mathcal{A}$ into $mathcal{M}$‎, ‎then the following four conditions are equivalent‎: ‎(i) $delta$ is a Jordan left de...

We show that every infinite-dimensional commutative unital C∗-algebra has a Hilbert C∗-module admitting no frames. In particular, this shows that Kasparov’s stabilization theorem for countably generated Hilbert C∗-modules can not be extended to arbitrary Hilbert C∗-modules.

In this note all rings are commutat ive rings with identity and all modules are unital. Let R be a commutat ive ring with identity. An R-module M is called a mult ipl icat ion module provided for each submodule N of M there exists an ideal I of R such that N = I M. Various propert ies of mult ipl icat ion modules are considered. If there is a common theme it is that the methods used generalise ...