Similarity and ergodic theory of positive linear maps

Details Similarity and ergodic theory of positive linear maps Similarity and ergodic theory of positive linear maps

2003-10-08

arxiv.org/abs/math/0310113v1

J.reine angew. Math. 561 (2003), 87-129

Mathematics

Operator Algebras

37 pages, Section 6 slightly improved

Scientific paper

In this paper we study the operator inequality \phi(X)\leq X and the operator equation \phi(X)= X, where \phi is a w^*-continuous positive (resp. completely positive) linear map on B(H). We show that their solutions are in one-to-one correspondence with a class of Poisson transforms on Cuntz-Toeplitz C^*-algebras, if \phi is completely positive. Canonical decompositions, ergodic type theorems, and lifting theorems are obtained and used to provide a complete description of all solutions, when \phi(I)\leq I. We show that the above-mentioned inequality (resp. equation) and the structure of its solutions have strong implications in connection with representations of Cuntz-Toeplitz C^*-algebras, common invariant subspaces for n-tuples of operators, similarity of positive linear maps, and numerical invariants associated with Hilbert modules over \CF_n^+, the complex free semigroup algebra generated by the free semigroup on n generators.

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