*Wissenschaftliches Programm* *Liste der Vortragenden*

**Sektion 4**

Dienstag, 19.09.2000,
17.00–17.20 Uhr, POT 251

#### Ralf Meyer, Universität Münster

Given an operator algebra A and a representation of A on a Hilbert space, it is usually quite difficult to
decide (numerically) whether is completely contractive, even if A is a commutative algeba of matrices
and is finite dimensional. We consider a situation where there is an easy-to-check criterion for a
representation to be completely contractive.

A d-tuple of commuting elements (T _{j}) of A is called a d-contraction iff the row matrix (T _{1} . . . T _{d}) is a
contraction. We say that a commutative operator algebra A is determined by (T _{j}) iff the following holds:
A representation of A is completely contractive iff (T _{j}) is a d-contraction.

We will consider the case where A is a commutative, d-dimensional subalgebra of the algebra M_{d} of
d × d-matrices. Using Arveson’s dilation theory for d-contractions, we obtain an algorithm to
decide whether a given subalgebra A is determined by a d - 1-contraction (T _{j}) and to compute
(T _{j}).

Endow the set of all commutative d-dimensional subalgebras of M_{d} with the subspace topology
from the Grassmannian manifold. It turns out that the set of A that are determined by some
d - 1-contraction has non-empty interior. Hence the set of “well-behaved” subalgebras is not too
small.

It turns out that a commutative, d-dimensional subalgebra of the algebra of d × d-matrices has a good
chance to be determined by some d - 1-contraction.

If we topologize the set of d-dimensional, commutative subalgebras with the topology of the
Grassmannian manifold, the set of algebras that are determined by a d - 1-contraction has non-empty
interior.