Wedderburn-type theorems for operator algebras and modules: the traditional and ``quantized'' homological approaches
by
Alexander Ya. Helemskii
Moscow State University

Let H be a Hilbert space, A be a subalgebra of B(H). We call A a Wedderburn algebra if there exists a ``canonical'' decomposition

H= Å { H'\nu\otimesH''\nu :\nu in \Lambda}

such that A consists of all a in B(H) such that, for any \nu in \Lambda, the restriction of a onto H'_\nu\otimesH^''_\nu has the form b_\nu\otimes1 for some b_\nu in B(H'_\nu). Which conditions of homological nature can distinguish Wedderburn algebras?

We suggest to consider the so-called spatial projectivity of A, that is, the projectivity of the A-module H. There are two versions of this property: the traditional one, based on the usual notion of a Banach module, and the ``quantum'' one, based on that of the quantum, or operator, module.

We show that a given von Neumann algebra is Wedderburn if and only if it is quantum spatially projective, whereas traditionally projective von Neumann algebras are exactly Wedderburn algebras with the following additional property: for any \nu in the canonical decomposition of H we have min{dimH_\nu', dimH_\nu^''} < \infty. Departing from this two-fold assertion, we describe at first all spatially projective (in both senses) operator C^*-algebras, and then all projective Hilbert modules over (arbitrary) C^*-algebras. Some parts of these results can be extended to certain classes of non-selfadjoint operator algebras.

(T)

Date received: October 26, 1999