Dilation theory, commutant lifting and semicrossed products

By K. Davidson, Elias G. Katsoulis. Documenta Mathematica, 2011

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We take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a repre- sentation, there is a special property of being fully extremal. This allows a refinement of some of the classical notions which are im- portant when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and Ando's theorem. This is applied to the study of semicrossed products by au- tomorphisms, and endomorphisms which lift to the C*-envelope. In particular, we obtain several general theorems which allow one to conclude that semicrossed products of an operator algebra naturally imbed completely isometrically into the semicrossed product of its C*- envelope, and the C*-envelopes of these two algebras are the same.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

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