Ideals of Completely Bounded Operators

The ideal structures of the Haagerup and weak*Haagerup tensor products of B(H) with itself are explored: the closed ideals of B(H) ®h B(H) are completely determined and some of the closed ideals of B(H) ®*h B(H) are identified. It is shown that if A is the Calkin algebra then A ®h A is simple. An example of a compact completely bounded operator on K(H) which is not approximable in completely bounded operator norm by finite rank operators is constructed. It is shown that a completely bounded operator W on a C*algebra A is p-summing for 1 2 then we can choose V, W E Ci,. It is shown that a version of the Grothendieck inequality using the usual nonsymmetric modulus holds for completely bounded operators on a 2-concave subspace of a C*algebra, and that if such an inequality is satisfied by all completely bounded operators on a subspace of a C*algebra then that subspace is, 2-concave.