Summary: COMPLETELY BOUNDED ISOMORPHISMS
OF OPERATOR ALGEBRAS
Abstract. In this paper the author proves that any two elements from one of the
following classes of operators are completely isomorphic to each other.
1. fVN(Fn) : n 2g. The II1 factors generated by the left regular representation
of the free group on n-generators.
2. fC (Fn) : n 2g. The reduced C -algebras of the free group on n-generators.
3. Some \non-commutative" analytic spaces introduced by G. Popescu Po].
The paper ends with some applications to Popescu's version of Von Neumann's in-
1. Introduction and preliminaries
E. Christensen and A. M. Sinclair CS] showed that any non-elementary injective
von Neumann algebra on a separable Hilbert space is completely isomorphic to
B(H), and A. G. Robertson and S. Wassemann RW] generalized the work on CS]
and proved that an in nite dimensional injective operator system on a separable
Hilbert space is completely isomorphic to either B(H) or `1.
The techniques on those papers depend on the injectivity of the spaces and do
not extend to interesting non-injective von Neumann algebras or operator algebras.
In the present note we address some of these examples. For instance, we prove