Subspaces of Maximal Operator Spaces

A Banach space E can be embedded into B(H) in a variety of ways. The embeddings giving rise to the minimal and maximal possible operator space norms (introduced in [1], [3] and [4] and further investigated in [21] and [22]) are especially interesting to us. Let A be the set of all functionals on E of norm not exceeding 1 and let B be the set of all linear maps T : E → Mn such that ||T || ≤ 1 (here and below, Mn = B(l n 2 ) and Mnk = B(l n 2 , l k 2)). Consider isometric embeddings JMIN : E → l∞(A) (here l∞(A) is viewed as the set of diagonal operators on l2(A)) and JMAX : E → ( ∑ Mn)l∞(B) →֒ B(H) (here H = ( ∑ l2 )l2(B)), defined by JMINe = ⊕f∈Af(e) and JMAXe = ⊕T∈BT (e). The pairs (E, JMIN) and (E, JMAX) are called the minimal and maximal operator space structures on E and denoted by MIN(E) and MAX(E), respectively. Note that if x ∈ E ⊗B(H), then