Compactifications of $\omega $ and the Banach space $c_0$

Abstract

Abstract. We investigate for which compactifications γω of the discrete space of natural numbers ω, the natural copy of the Banach space c 0 is complemented in C(γω). We show, in particular, that the separability of the remainder γω \ ω is neither sufficient nor necessary for c 0 being complemented in C(γω) (for the latter our result is proved under the continuum hypothesis). We analyse, in this context, compactifications of ω related to embeddings of the measure algebra into P (ω)/fin.We also prove that a Banach space C(K) contains a rich family of complemented copies of c 0 whenever the compact space K admits only measures of countable Maharam type.

Extension operators and twisted sums of c0 and C(K) spaces

“…The second author is very grateful to the anonymous referee of [13] who pointed out interesting connection of the results presented there with Problem 1.1.…”

“…Following [13] we say that a compactification γω of the discrete space ω is tame if the natural copy of c 0 in C(γω), consisting of all functions from vanishing on the remainder K = γω \ω, is complemented in C(γω). This is equivalent to saying that γω ∈ CDE(K) and there is a corresponding extension operator.…”

“…Note that ult(A/fin) is homeomorphic to the remainder of such a compactification. In this setting, the equivalence of conditions (i) and (iii) from Lemma 2.7 can be stated as follows (see Lemma 3.1 in [13]). Lemma 7.1.…”

On the c_0-extension property

“…It states that if X is a separable Banach space, then every c 0 -valued bounded operator defined on a closed subspace of X admits a c 0 -valued bounded extension defined on X. The search for generalizations of Sobczyk's theorem in the context of nonseparable Banach spaces has attracted a lot of attention in the last decades [5,6,7,8,14,21,22]. In [5], the c 0 -extension property was introduced.…”

On the $c_0$-extension property

“…It states that if X is a separable Banach space, then every c 0 -valued bounded operator defined on a closed subspace of X admits a c 0 -valued bounded extension defined on X. The search for generalizations of Sobczyk's theorem in the context of nonseparable Banach spaces has attracted a lot of attention in the last decades [6,5,7,8,14,21,22]. In [6], the c 0 -extension property was introduced.…”