Some properties of a class of diagonalizable states of von Neumann algebras

N. N. Ganikhodzhaev; F. M. Mukhamedov

doi:10.1023/B:MATN.0000043460.76177.5d

Some properties of a class of diagonalizable states of von Neumann algebras

N. N. Ganikhodzhaev, F. M. Mukhamedov

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

Abstract

In this paper, a class of representations of uniformly hyperfinite algebras is constructed and the corresponding von Neumann algebras are studied. It is proved that, under certain conditions, the Markov states generate factors of type III _λ, where λ ε (0,1), in the GNS representation; this gives a negative answer to the conjecture that the factors corresponding to Hamiltonians with nontrivial interactions have type III ₁. It is shown that, for a certain class of Hamiltonians, there exists a unique translation-invariant ground state.

Original language	English
Pages (from-to)	329-338
Number of pages	10
Journal	Mathematical Notes
Volume	76
Issue number	3-4
DOIs	https://doi.org/10.1023/B:MATN.0000043460.76177.5d
Publication status	Published - Sept 2004
Externally published	Yes

Keywords

Connes spectrum
Markov state
associated representation
diagonalizable state
factor of type iii
translation-invariant ground state
von Neumann algebra

ASJC Scopus subject areas

Mathematics(all)

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10.1023/B:MATN.0000043460.76177.5d

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T1 - Some properties of a class of diagonalizable states of von Neumann algebras

AU - Ganikhodzhaev, N. N.

AU - Mukhamedov, F. M.

PY - 2004/9

Y1 - 2004/9

N2 - In this paper, a class of representations of uniformly hyperfinite algebras is constructed and the corresponding von Neumann algebras are studied. It is proved that, under certain conditions, the Markov states generate factors of type III λ, where λ ε (0,1), in the GNS representation; this gives a negative answer to the conjecture that the factors corresponding to Hamiltonians with nontrivial interactions have type III 1. It is shown that, for a certain class of Hamiltonians, there exists a unique translation-invariant ground state.

AB - In this paper, a class of representations of uniformly hyperfinite algebras is constructed and the corresponding von Neumann algebras are studied. It is proved that, under certain conditions, the Markov states generate factors of type III λ, where λ ε (0,1), in the GNS representation; this gives a negative answer to the conjecture that the factors corresponding to Hamiltonians with nontrivial interactions have type III 1. It is shown that, for a certain class of Hamiltonians, there exists a unique translation-invariant ground state.

KW - Connes spectrum

KW - Markov state

KW - associated representation

KW - diagonalizable state

KW - factor of type iii

KW - translation-invariant ground state

KW - von Neumann algebra

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M3 - Article

AN - SCOPUS:5044233148

SN - 1757-7489

VL - 76

SP - 329

EP - 338

JO - Mathematical Notes of the Academy of Sciences of the USSR

JF - Mathematical Notes of the Academy of Sciences of the USSR

IS - 3-4

ER -

Some properties of a class of diagonalizable states of von Neumann algebras

Abstract

Keywords

ASJC Scopus subject areas

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