On a Conjecture of Wood

Abstract

Abstract. We show that there exists a locally compact separable metrizable space L such that C 0 (L), the Banach space of all continuous complex-valued functions vanishing at infinity with the supremum norm, is almost transitive. Due to a result of Greim and Rajagopalan [3], this implies the existence of a locally compact Hausdorff spaceL such that C 0 (L) is transitive, disproving a conjecture of Wood [9]. We totally owe our construction to a topological characterization due to Sánches [8].2000 Mathematics Subject Classification. 54C35, 46B04, 54G99.1. Introduction, main theorem and preliminaries. For a locally compact Hausdorff space L, C 0 (L) denotes the Banach space of all continuous complex-valued functions on L vanishing at infinity, equipped with the supremum norm. A Banach space X is said to be transitive (resp. almost transitive) if the isometry group G(X) acts transitively on the unit sphere S(X) = {x ∈ X| x = 1} (resp. the orbit G(X) · x is dense in S(X) for each x ∈ S(X)). In [9], Wood conjectured that C 0 (L) is not transitive for any locally compact Hausdorff space L unless L is a singleton. In [3], the conjecture was verified for the Banach space C 0 (L : R) of all real-valued continuous functions on L vanishing at infinity. Greim and Rajagopalan proved in [3] that the existence of a locally compact Hausdorff space L with C 0 (L) being almost transitive implies the existence of a locally compact Hausdorff spaceL such that C 0 (L) is transitive. In [7], the verification of the conjecture was reduced to the case in which L has the metrizable onepoint compactification αL. Furthermore, Sánches in [8] gave an explicit topological characterization of the space L with the almost transitive C 0 (L). Theorem 2 and 3 of [8] are restated below. Following [8], we say that a locally compact Hausdorff space L is an Wood space (resp. an almost Wood space) if C 0 (L) is transitive (resp. almost transitive).

ALMOST TRANSITIVITY IN $\mathcal{C}_0$ SPACES OF VECTOR-VALUED FUNCTIONS

“…In [18] and [10] it is proved that if L is a locally compact, non-compact space andL is the topological space known as a pseudoarc, then L is an L 0 -space. However, such an L cannot be an L 1 -space because the pseudoarc is metrizable.…”

A counterexample to Wood's conjecture

“…The author has recently known that the same counterexample to Wood's conjecture has been independently given by Kawamura [9], however his proof and the path leading to the results are substantially different to the ones followed here. As a consequence, the necessary conditions stated in Theorem 2.3 do not appear in [9].…”

Convex-transitivity and function spaces

“…This problem was answered by Greim and Rajagopalan in [13] in the case K = R in the positive and was recently refuted independently by Kawamura in [18] and by Rambla in [26] in the case K = C. Theorem 2.4 below extends the work in [13] (and here K = R).…”

“…In this article we will investigate the abovementioned transitivity conditions, mainly convex-transitivity, in different settings. For some important results related to this investigation, see [12,13,18,26].…”