Non-commuting n-tuples of operators and dilation theory

Abstract

Every contractive n-tuple of operators has a minimal joint isometric dilation to isometries acting on a larger space. Each of these dilations decomposes into a Cuntz part and a pure part. The Cuntz part determines a representation of the Cuntz C*-algebra. When an n-tuple acts on finite dimensional space, its dilation is completely described in terms of the original n-tuple. This is accomplished by classifying the associated Cuntz representations. In fact, simple complete unitary invariants for the representations are obtained. The pure part of a dilation is determined by copies of the left regular representation of the free semigroup on n letters. The number of copies can be computed directly in terms of the original n-tuple. Davidson and Pitts have shown that the non-selfadjoint WOT-closed algebras generated by the pure isometries or 'left creation operators' are the appropriate non-communatative analytic Toeplitz algebras. Factorization problems in these algebras are investigated. Positive results are obtained when norm conditions are placed on possible factors; however, over the full algebra deep factorization pathologies are exposed. This leads to information on the left ideals in these algebras. Finally, non-commutative versions of Arveson's curvature invariant and Euler characteristic for a commuting n-tuple of operators are developed. They are sensitive enough to detect when an n-tuple of operators are developed. They are sensitive enough to detect when an n-tuple is free. The curvature invariant is shown to be upper semi-continuous. A new class of examples is introduced and is used to obtain information on the ranges of the invariants.