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Topologies of compact families on the ideal space of a Banach algebra

Ferdinand Beckhoff (1996)

Studia Mathematica

Let be a family of compact sets in a Banach algebra A such that is stable with respect to finite unions and contains all finite sets. Then the sets U ( K ) : = I ∈ I d ( A ) : I ∩ K = ∅ , K ∈ define a topology τ() on the space Id(A) of closed two-sided ideals of A. is called normal if I i → I in (Id(A),τ()) and x ∈ A╲I imply l i m i n f i ∥ x + I i ∥ > 0 . (1) If the family of finite subsets of A is normal then Id(A) is locally compact in the hull kernel topology and if moreover A is separable then Id(A) is second countable. (2) If the family of countable compact sets...

Topologies on the space of ideals of a Banach algebra

Ferdinand Beckhoff (1995)

Studia Mathematica

Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely τ ∞ , coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra τ ∞ coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if τ ∞ is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A)...

Totally convex algebras

Dieter Pumplün, Helmut Röhrl (1992)

Commentationes Mathematicae Universitatis Carolinae

By definition a totally convex algebra A is a totally convex space | A | equipped with an associative multiplication, i.eȧ morphism μ : | A | ⊗ | A | ⟶ | A | of totally convex spaces. In this paper we introduce, for such algebras, the notions of ideal, tensor product, unitization, inverses, weak inverses, quasi-inverses, weak quasi-inverses and the spectrum of an element and investigate them in detail. This leads to a considerable generalization of the corresponding notions and results in the theory of Banach spaces.

U-ideals of factorable operators

Kamil John (1999)

Czechoslovak Mathematical Journal

We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons’s construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are u -ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.

Uncomplementability of spaces of compact operators in larger spaces of operators

Giovanni Emmanuele, Kamil John (1997)

Czechoslovak Mathematical Journal

In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of c 0 in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results...

Weak invertibility and strong spectrum

Michael Meyer (1993)

Studia Mathematica

A notion of weak invertibility in a unital associative algebra A and a corresponding notion of strong spectrum of an element of A is defined. It is shown that many relationships between the Jacobson radical, the group of invertibles and the spectrum have analogues relating the strong radical, the set of weakly invertible elements and the strong spectrum. The nonunital case is also discussed. A characterization is given of all (submultiplicative) norms on A in which every modular maximal ideal M...

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