For a Banach algebras $A$ satisfying certain properties,
we construct an operator space $X$ such that the space of
completely bounded maps $CB(X)$ consists of elements of
$A$ (or, at least, $\pi(A)$, where $\pi$ is a faithful
representation), and their "small" perturbations.
As properties of an operator space are reflected in its
space of completely bounded maps, we construct spaces
with various "pathological" properties. The prime example
here is the space $X$, isometric to a separable Hilbert
space, such that any c.b. maps on any subspace of $X$ is a
sum of a scalar and a Hilbert-Schmidt operator. Other
"strange" spaces include $Y$, completely isomorphic to
$Y \oplus Y$, and such that $CB(Y)$ admits a non-trivial
trace.
Part of this work was done jointly with Eric Ricard.
