Similarity and ergodic theory of positive linear maps

Abstract

Abstract. 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.

Characteristic functions and joint invariant subspaces

“…Extending some results obtained by Sz.-Nagy [25], Sz.-Nagy and Foiaş [29], and the author [10,21], we prove, in particular, that a one-to-one power bounded n-tuple [T 1 , . .…”

Amenability, Completely Bounded projections, Dynamical Systems and Smooth Orbits

“…In the special case when M = B(H), the next corollary shows that the set C = (ϕ) = {X ∈ B(H) | ϕ(X) = X} studied in [Po03] is an injective operator space provided ϕ: B(H) → B(H) is a weak * -continuous, completely positive, completely contractive map. …”

Toeplitz operators associated to commuting row contractions

“…In this case, F(T ) is precisely the space of all Toeplitz operators on H 2 (see [9]) and the symbol map is T φ → φ for φ ∈ L ∞ (T). We also mention that, in the case of non-commuting row contractions, lifting theorems for operators in F(T ) to corresponding Toeplitz operators associated to the minimal row isometric dilation ofT have been proved in [8] and [19]. We will show that to each commutative row contractionT = {T n } n 1 on H for which F(T ) is non-trivial, there exists a triple {K, Γ, {U n } n 1 } where K is a Hilbert space, Γ : K → H is a bounded operator and {U n } n 1 is a spherical unitary on K (i.e.…”