Let G be a discrete group. Then the following is equivalent:

(1) G is amenable;

(2) the reduced group C*-algebra is nuclear;

(3) the group von-Neumann algebra L(G) is semidiscrete.

Let p>1. G. An, J.-J. Lee and Z.-J. Ruan introduced the p-nuclearity for Lp-operator algebras. They show that the reduced group Lp-operator algebra is p-nuclear and the p-pseudomeasure algebra is p-semidiscrete if G is amenable. In this talk, we show that the following are equivalent:

(4) G is amenable;

(5) the reduced group Lp-operator algebra is p-nuclear;

(6) the p-pseudomeasure algebra is p-semidiscrete;

This answers an open problem raised by N. C. Phillips.