Asymptotic lifts of positive linear maps

Asymptotic Lifts of Positive Linear Maps

We show that the notion of asymptotic lift generalizes naturally to normal positive maps φ : M → M acting on von Neumann algebras M . We focus on cases in which the domain of the asymptotic lift can be embedded as an operator subsystem M∞ ⊆ M , and characterize when M∞ is a Jordan subalgebra of M in terms of the asymptotic multiplicative properties of φ.

Asymptotic Stability I: Completely Positive Maps

We show that for every “locally finite” unit-preserving completely positive map P acting on a C∗-algebra, there is a corresponding ∗-automorphism α of another unital C∗-algebra such that the two sequences P, P , P , . . . and α,α, α, . . . have the same asymptotic behavior. The automorphism α is uniquely determined by P up to conjugacy. Similar results hold for normal completely positive maps o...

Positive Linear Maps and Spreads of Matrices

Irreducible Positive Linear Maps on Operator Algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible...

Hamiltonian flows, cotangent lifts, and momentum maps

Let (M,ω) and (N, η) be symplectic manifolds. A symplectomorphism F : M → N is a diffeomorphism such that ω = F ∗η. Recall that for x ∈ M and v1, v2 ∈ TxM , (F η)x(v1, v2) = ηF (x)((TxF )v1, (TxF )v2); TxF : TxM → TF (x)N . (A tangent vector at x ∈ M is pushed forward to a tangent vector at F (x) ∈ N , while a differential 2-form on N is pulled back to a differential 2-form on M .) In these not...