Fixed points of normal completely positive maps on B(H)

2011/05/10 by Bojan Magajna, Magajna, Bojan
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.1105.1914

Abstract

Given a sequence of bounded operators a_j on a Hilbert space H with ∑ a_j^*a_j=1=∑ a_ja_j^*, we study the map Ψ defined on B(H) by Ψ(x)=∑ a_j^*xa_j and its restriction Φ to the Hilbert-Schmidt class C²(H). In the case when the sum ∑ a_j^*a_j is norm-convergent we show in particular that the operator Φ-1 is not invertible if and only if the C^*-algebra A generated by (a_j) has an amenable trace. This is used to show that Ψ may have fixed points in B(H) which are not in the commutant A' of A even in the case when the weak* closure of A is injective. However, if A is abelian, then all fixed points of Ψ are in A' even if the operators a_j are not positive.

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