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The Shannon-McMillan Theorem for AF $C^*$-systems (Applications of Renormalization Group Methods in Mathematical Sciences)

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The

Shannon-McMillan

Theorem for

AF

$C^{*}$-systems

Yoshiko Ogata Graduate School of Mathematics, University ofTokyo

数理解析研究所講究録

(2)

References

[BKSS] I. Bjelakovi\’{c}, T. Kr\"uger, R. Siegmund-Schultze, A. Szkola: The Shannon-McMillan theorem

for

ergodic quantum lattice systems, In-vent. Math. 155, 203-222, (2004).

[BS] I. Bjelakovi\’{c},

R.

Siegmund-Schultze:

An

ergodic

theorem

for

the

quan-tum relative entropy, Commun. Math. Phys. 247, 697-712, (2004). [GN] V. Y. Golodets, S. Neshveyev: Gibbs states

for

$AF$ algebras; J. Math.

Phys. 39, 6329-6344, (1998).

[HP1] F. Hiai, D. Petz: Quantum Mechanics in $AFC^{*}$-systems,

Rev. Math.

Phys. 8, 819-859, (1996).

[HP2] F. Hiai, D. Petz: The proper

formula for

relative entropy and its

asymptotics in quantum probability, Commun. Math. Phys.143, 99-114, (1991).

[NS1]

S.

Neshveyev, E.

Strmer:

The McMillan theorem

for

a class

of

asymp-totically abelian$C^{*}$-algebras, Ergodic Theory Dynam. Systems.

22889-897, (2002).

[S] C. Shannon: A

mathematical

theory

of

communication, Bell System Tech. J. 27379-423, 623-656, (1948).

参照

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