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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.