Journal article
Extensions in certain topological algebraic categories
Transactions of the American Mathematical Society, v 159, pp 41-56
01 Jan 1971
Abstract
Categories which we call “sufficiently algebraic” are defined, and for certain objects
A
A
(called faithful) in such categories, and arbitrary objects
C
C
, we partially order the sets
Ext
(
C
,
A
)
\operatorname {Ext} (C,A)
of extensions of
A
A
by
C
C
. We prove that the maximal elements in
Ext
(
C
,
A
)
\operatorname {Ext} (C,A)
(with respect to this ordering) are in bijective correspondence with the morphisms from
C
C
to a canonical object
O
(
A
)
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
∗
{C^ \ast }
-algebras are also included in this theory as a special case.
Metrics
3 Record Views
1 citations in Scopus
Details
- Title
- Extensions in certain topological algebraic categories
- Creators
- Robert C. Busby
- Publication Details
- Transactions of the American Mathematical Society, v 159, pp 41-56
- Publisher
- American Mathematical Society
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- [Retired Faculty]
- Scopus ID
- 2-s2.0-84966204385
- Other Identifier
- 991019173879104721