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Dilation theory, commutant lifting and semicrossed products | | Kenneth R. Davidson
; Elias G. Katsoulis
; | | Date: |
1 Sep 2011 | | Abstract: | We take a new look at dilation theory for nonself-adjoint operator algebras.
Among the extremal (co)extensions of a representation, there is a special
property of being fully extremal. This allows a refinement of some of the
classical notions which are important 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 automorphisms, 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. | | Source: | arXiv, 1109.0231 | | Services: | Forum | Review | PDF | Favorites | |
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