Similarity and Ergodic Theory of Positive Linear Maps

In this paper we study the operator inequality φ(X) ≤ X and the operator equation φ(X) = X, where φ 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 φ is completely positive. Canonical decompositions, ergodic type theorems, and lifting theorems are obtained and used to provide a complete description of all solutions, when φ(I) ≤ 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.