On ergodic properties of operator nets on the predual of von neumann algebras

In this paper, we proved theorems which give the conditions that special operator nets on a predual of von Neumann algebras are strongly convergent under the Markov case. Moreover, we investigate asymptotic stability and existence of a lower-bound function for such nets.

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ASYMPTOTIC STABILITY I: COMPLETELY POSITIVE MAPS

International Journal of Mathematics ◽

10.1142/s0129167x04002284 ◽

2004 ◽

Vol 15 (03) ◽

pp. 289-312 ◽

Cited By ~ 3

Author(s):

WILLIAM ARVESON

Keyword(s):

Asymptotic Behavior ◽

Asymptotic Stability ◽

Von Neumann Algebras ◽

Completely Positive Map ◽

Completely Positive Maps ◽

Completely Positive ◽

Locally Finite ◽

Von Neumann ◽

C Algebra ◽

Positive Maps

We show that for every "locally finite" unit-preserving completely positive map P acting on a C*, there is a corresponding *-automorphism α of another unital C*-algebra such that the two sequences P, P2, P3, … and α, α2, α3, … have the same asymptotic behavior. The automorphism α is uniquely determined by P up to conjugacy. Similar results hold for normal completely positive maps on von Neumann algebras, as well as for one-parameter semigroups. These results are operator algebraic counterparts of the classical theory of Perron and Frobenius on the structure of square matrices with nonnegative entries.

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Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

Journal of Topology and Analysis ◽

10.1142/s1793525321500461 ◽

2021 ◽

pp. 1-54 ◽

Cited By ~ 1

Author(s):

Michael Brannan ◽

Li Gao ◽

Marius Junge

Keyword(s):

Lower Bound ◽

Ricci Curvature ◽

Von Neumann Algebras ◽

Logarithmic Sobolev Inequality ◽

Beltrami Operator ◽

Free Products ◽

Von Neumann ◽

Amalgamated Free Products ◽

Quantum Tori ◽

Bounded Below

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

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