Abstract. A morphism / in a category <ß is (von Neumann) regulär if there exists a morphism /' in # with, //'/=/ Iß this paper, we study regularity of morphisms in certain types of categories; in particular, we derive a necessary and sufficient condition for the composite fg of two regulär morphisms to be regulär. We then apply these results to the category g^y of locally convex spaces and show that the subcategory ^ of g^Zf with Fredholm operators äs morphisms is a regulär category, in the sense that every morphism of &* is regulär. Finally we study algebraic and topological properties of the semigroup &(X) of Fredholm operators ön a locally convex Hausdorff space X. We introduce a new integral invariant k(X) for a topological vector space Jf and show that: (i) &(X) is semisimple if and only ifk(X) = 1; (ii) ^(X) is completely semisimple and unit regulär if and only if k(X) = 0 and (iii) ^(X) is simple but not semisimple if and only if k(X) > 1. We also study some important congruences on ^(X) and certain subsemigroups of 1991 Mathematics Subject Classification: 46A03, 47A53, 47D03; 18A32, 20M17, 22A15. Contents 1. Introduction and
Forum Mathematicum – de Gruyter
Published: Jan 1, 1993
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