Complemented copies of 𝑐₀ in 𝐿^{∞}(𝜇,𝐸)

Abstract

Let E E be a Banach space, and let ( Ω , Σ , μ ) (\Omega ,\Sigma ,\mu ) be a measure space. We denote by L ∞ ( μ , E ) {L^\infty }(\mu ,E) the Banach space of all E E -valued μ \mu -measurable essentially bounded functions on Ω \Omega , two functions being identified if they differ only on a locally μ \mu -null set. We prove that if L ∞ ( μ , E ) {L^\infty }(\mu ,E) contains a complemented copy of c 0 {c_0} , then E E contains a copy of c 0 {c_0} .