A characterization of regular averaging operators and its consequences

Abstract

Abstract. We present a characterization of continuous surjections, between compact metric spaces, admitting a regular averaging operator. Among its consequences, concrete continuous surjections from the Cantor set C to [0, 1] admitting regular averaging operators are exhibited. Moreover we show that the set of this type of continuous surjections from C to [0, 1] is dense in the supremum norm in the set of all continuous surjections. The nonmetrizable case is also investigated. As a consequence, we obtain a new characterization of Eberlein compact sets.

Network characterization of Gul'ko compact spaces and their relatives

“…For an up-to-date account of these classes of compact spaces, as well as their interplay in functional analysis, we recommend the books [6,15,17] together with the survey papers [19,30,33], as well as some very recent papers [2,4,13,16]. We have the following implications:…”

Extension Property and Complementation of Isometric Copies of Continuous Functions Spaces

“…Our result concerns, in particular, the complementation of Banach subalgebras of C(K) (Corollary 2.6). Indeed, recall that a Banach subalgebra (with unity) of C(K) is the range φ * [C(L)] of the linear isometric embedding φ * : Finding conditions under which a continuous surjection admits an averaging operator is a classical problem in Banach space theory [1,5,13].…”

Some remarks on Radon–Nikodym compact spaces

“…We prove that The proof requires techniques introduced in Section 4 and moreover we adopt techniques from [5], related to the well known Ditor Theorem ( [6]). The latter states that for every compact K, there exists a totally disconnected compact space L with the same topological weight as K, mapping continuously onto K by a map that admits a regular averaging operator.…”