Similarity and ergodic theory of positive linear maps

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In this paper we study the operator inequality ϕ(X)≤ X and the operator equation ϕ(X)= X, where ϕis 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 ϕis 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 ₙ+, the complex free semigroup algebra generated by the free semigroup on n generators.