  We provide an unified approach of results of L. Dor on the complementation of
the range, and of D. Alspach on the nearness from isometries, of small into
isomorphisms of L1. We introduce the notion of small subspace of L1 and show
lifting theorems for operators between quotients of L1 by small subspaces. We
construct a subspace of L1 which shows that extension of isometries from
subspaces of L1 to the whole space are no longer true for isomorphisms, and
that nearly isometric isomorphisms from subspaces of L1 into L1 need not be
near from any isometry.
