COMPLEMENTEDLY DENSE IDEALS: DECOMPOSABLE ALGEBRAS

An ideal I of a (non-associative) algebra A is dense if the multiplication algebra of A acts faithfully on I, and is complementedly dense if it is a direct summand of a dense ideal. We prove that every complementedly dense ideal of a semiprime algebra is a semiprime algebra, and determine its central closure and its extended centroid. We also prove that a semiprime algebra is an essential subdirect product of prime algebras if and only if, its extended centroid is a direct product of fields. This result is applied to discuss decomposable algebras with respect to some familiar closures for ideals.

Avoiding families and Tukey functions on the nowhere-dense ideal

We investigate Tukey functions from the ideal of all closed nowhere-dense subsets of 2ℕ. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of ℕ. We also prove non-existence of various special types of Tukey reductions from the nowhere-dense ideal to analytic P-ideals. In connection with these results, we study families$\mathcal{F}$of clopen subsets of 2ℕwith the property that for each nowhere-dense subset of 2ℕthere is a set in$\mathcal{F}$not intersecting it. We call such families avoiding.

Similarité entre l'algèbre de Volterra et un quotient d'algèbre uniforme

Two commutative Banach algebras A and B are said to be similar if there exists a Banach algebra D such that [xD]− = D for some x in D, and two one-to-one continuous homomorphisms φ:D→A and ψ:D→B such that φ(D) is a dense ideal of A and ψ(D) a dense ideal of B.We prove in this paper that the Volterra algebra is similar to A0/e-z A0 where A0 is the commutative uniform, separable Banach algebra of all continuous functions on the closed right-hand half plane , analytic on H and vanishing at infinity. We deduce from this result that multiplication by an element of A0/e-z A0 is a compact mapping.

Contracted primes of the complete ring of quotients

The generic closure of the set of primes contracted from the complete ring of quotients of a reduced commutative ring is shown to be just the set of those primes not containing a finitely generated dense ideal. It is also the smallest generically closed, quasi-compact set containing the minimal primes.

Extension and inversion of extended orthomorphisms on Riesz spaces

Let E be an Archimedean Riesz space and let Orth∞(E) be the f-algebra consisting of all extended orthomorphisms on E, that is, of all order bounded linear operators T:D→E, with D an order dense ideal in E, such that T(B∩D) ⊆ B for every band B in E. We give conditions on E and on a Riesz subspace F of E insuring that every T ∈ Orth∞(F) can be extended to some ∈ Orth∞(E), and we also consider the problem of inversing an extended orthomorphism on its support. The same problems are also studied in the case of σ-orthomorphisms, that is, extended orthomorphisms with a super order dense domain. Furthermore, some applications are given.

Orthomorphisms of a commutative W*-algebra

If M is a commutative W*-algebra of operators and if ReM is the Dedekind complete Riesz space of self-adjoint elements of M, then it is shown that the set of densely defined self-adjoint transformations affiliated with ReM is a Dedekind complete, laterally complete Riesz algebra containing ReM as an order dense ideal. The Riesz algebra of densely defined orthomorphisms on ReM is shown to coincide with , and via the vector lattice Randon-Nikodym theorem of Luxemburg and Schep, it is shown that the lateral completion of ReM may be identified with the extended order dual of ReM.