Copies of c_0(Γ) in the space of bounded linear operators

Abstract

The space L(𝑋,𝑌) stands for the Banach space of all bounded linear operators from X to Y endowed with the operator norm. It is shown that 𝑐0(Γ) embeds into (𝑋,𝑌) if and only if 𝑙∞(Γ) embeds into (𝑋,𝑌) or 𝑐0(Γ) embeds into Y. As a consequence, we extend a classical Kalton theorem by proving that if 𝑐0(Γ) embeds into (𝑋,𝑌) and X has the |Γ|-Josefson–Nissenzweig property, then 𝑙∞(Γ) also embeds into (𝑋,𝑌). We also show that, for certain Banach spaces X and Y, 𝑐0(Γ) embeds complementably into (𝑋,𝑌) if and only if 𝑐0(Γ) embeds into Y.

Sergio A. Pérez

Instructor