Extensions in certain topological algebraic categories

Busby, Robert C.

doi:10.1090/S0002-9947-1971-0282224-6

Extensions in certain topological algebraic categories
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by Robert C. Busby PDF

Trans. Amer. Math. Soc. 159 (1971), 41-56 Request permission

Abstract:

Categories which we call “sufficiently algebraic” are defined, and for certain objects $A$ (called faithful) in such categories, and arbitrary objects $C$, we partially order the sets $\operatorname {Ext} (C,A)$ of extensions of $A$ by $C$. We prove that the maximal elements in $\operatorname {Ext} (C,A)$ (with respect to this ordering) are in bijective correspondence with the morphisms from $C$ to a canonical object $O(A)$. If the short five lemma holds in the category, all extensions are maximal and therefore obtained in this way. As an application we compute extensions in certain categories of topological rings. In particular we investigate the possible extensions of one group algebra (of a locally compact group) by another in the category of Banach algebras with norm decreasing homomorphisms, and using some analytic tools we give conditions for the splitting of such extensions. Previous results of the author on extensions of ${C^ \ast }$-algebras are also included in this theory as a special case.

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Homology

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