Topological groups and Dugundji compacta

V. V. Uspenskii

Sbornik. Mathematics, Tome 67 (1990) no. 2, pp. 555-580

Résumé

A compact space $X$ is called a Dugundji compactum if for every compact $Y$ containing $X$, there exists a linear extension operator $$\Lambda\colon C(X)\to C(Y),$$ which preserves nonnegativity and maps constants into constants. It is known that every compact group is a Dugundji compactum. In this paper we show that compacta connected in a natural way with topological groups enjoy the same property. For example, in each of the following cases, the compact space $X$ is a Dugundji compactum: 1) $X$ is a retract of an arbitrary topological group; 2) $X=\beta P$, where $P$ is a pseudocompact space on which some $\aleph_0$-bounded topological group acts transitively and continuously. Bibliography: 57 titles.