Représentations d'opérateurs à valeurs dans L1 (X,..,..).
Separately σ-additive and separately finitely additive complex functions on the Cartesian product of two algebras of sets are represented in terms of spectral measures and their finitely additive counterparts. Applications of the techniques include a bounded joint convergence theorem for bimeasure integration, characterizations of positive-definite bimeasures, and a theorem on decomposing a bimeasure into a linear combination of positive-definite ones.
Representations of a module over a *-algebra are considered and some related seminorms are constructed and studied, with the aim of finding bounded *-representations of .
Suppose B is a unital algebra which is an algebraic product of full matrix algebras over an index set X. A bijection is set up between the equivalence classes of irreducible representations of B as operators on a Banach space and the σ-complete ultrafilters on X (Theorem 2.6). Therefore, if X has less than measurable cardinality (e.g. accessible), the equivalence classes of the irreducible representations of B are labeled by points of X, and all representations of B are described (Theorem 3.3).
We give a representation of the spaces as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that is isomorphic to the sequence space , thereby showing that the isomorphy class does not depend on the dimension N if p=2.