On p-approximation properties for p-operator spaces

Abstract

This paper has a two-fold purpose. Let 1 < p < ∞ . We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on L p -spaces. We then apply these properties to the study of the pseudofunction algebra PF p ( G ) , the pseudomeasure algebra PM p ( G ) , and the Figà–Talamanca–Herz algebra A p ( G ) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C ∗-algebra C λ ∗ ( G ) , the group von Neumann algebra VN ( G ) , and the Fourier algebra A ( G ) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PF p ( G ) , PM p ( G ) , and A p ( G ) , respectively. The p-completely bounded multiplier algebra M cb A p ( G ) plays an important role in this work.