| W*-Corresponcences, Finite Directed Graphs and Markov Chains Contributor(s): Vega, Victor (Author) |
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ISBN: 3639155246 ISBN-13: 9783639155242 Publisher: VDM Verlag OUR PRICE: $50.27 Product Type: Paperback Published: May 2009 |
| Additional Information |
| BISAC Categories: - Mathematics | Functional Analysis |
| Physical Information: 0.28" H x 6" W x 9" (0.40 lbs) 116 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: Let A be the W*-algebra, L1(E(0), ), where E(0) is a fnite set and is a probability measure with full support. Let P: A->A be a completely positive unital map. In the present context, P is given by a stochastic matrix. We study the properties of P that are refected in the dilation theory developed by Muhly and Solel in Int. J. Math. 13, 2002. Let H be the Hilbert space L2(E(0), ) and let pi: A -> B(H) the representation of A given by multiplication. Form the Stinespring space H1. Then X is a W*-correspondence over P is expressed through a completely contractive representation T of X on H. This representation can be dilated to an isometric representation V of X on a Hilbert space that contains H. We show that X is naturally isomorphic to the correspondence associated to the directed graph E whose vertex space is E(0) and whose edge space is the support of the matrix representing P - a subset of E(0)×E(0). Further, V is shown to be essentially a Cuntz-Krieger representation of E. We also study the simplicity and the ideal structure of the graph C*-algebra associated to the stochastic matrix P. |