Subspaces of quotients of $L^p$ spaces

Question

Is the collection of subspaces of quotients of $L^p$ spaces considered to be a large class of Banach spaces?

Answer 1

These are called $SQ_p$ spaces. The size varies: for $p=1,\infty$ the class is perhaps too large (includes all separable spaces); for $p=2$ it consists only of the Hilbert space. In between, these spaces have interesting structure related to factorization of operators. They are often called subquotients of $L^p$. Some references:

1. S. Kwapien, On operators factorizable through $L^p$ space, Mémoires de la Société Mathématique de France 31-32 (1972): 215-225.
2. Y. Gordon and P. Saphar, Ideal norms on $E\otimes L^p$, Illinois J. Math. Volume 21, Issue 2 (1977), 266-285.
3. G. Pisier, Complex Interpolation Between Hilbert, Banach and Operator Spaces, Memoirs of the AMS, 2010. Preprint available from ArXiv.

Kwapien's paper has been influential, but a little hard to read now that we're spoiled by $\LaTeX$. Gordon-Saphar is in the same spirit but LaTeX'd. Pisier's memoir shows how these spaces naturally come up in the theory of interpolation.