Abstract
We define a notion of quantum or non-commutative, ergodicity for a class ofC *-dynamical systems (A, G, α) which we callquantized GNS systems. Such a system possesses a natural classical limit state ω, which induces a classical limit system by the GNS construction. The criterion for quantum ergodicity is that the time average 〈A〉 of an observable A ∈A equals the “space average” ω(A)I plus an errorK which is negligible in the classical limit. We prove that ergodicity of ω is a sufficient condition for quantum ergodicity of (A, G, α) if the classical limit system is abelian, give a conditional converse, and discuss a number of applications.
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Communicated by Ya.G. Sinai
Partially supported by NSF grant # DMS-9404637.
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Zelditch, S. Quantum ergodicity ofC * dynamical systems. Commun.Math. Phys. 177, 507–528 (1996). https://doi.org/10.1007/BF02101904
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DOI: https://doi.org/10.1007/BF02101904