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 Md 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 Md 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.