ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

2. Algebra

1991 MS Classification: 17, 46 Rakhimov, Abdugafur Abdumadjidovich, Uzbekistan Subfactors of a real factor of type III_{λ}, 0 < λ < 1 Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H. A weakly closed real *-subalgebra R in B(H) is called a real W^{∗}-algebra if it contains the identity operator and if R ∩ iR = {0}. Let R be a real factor of type III_{λ}, 0 < λ < 1, ψ be a faithful normal state whose modular automorphism group satisfies σ_{T_{0}}^{ψ} = id or σ_{2T_{0}}^{ψ} = id, where T_{0} = −2π/lnλ. We proved that any real W^{∗}-subalgebra of R, which contains the centralizer R_{ψ} of ψ, generated by R_{ψ} and unitary u^{k}, for some k. Hence, as corollary it follows that if R is a real W^{∗}-subalgebra of R which contains R_{ψ} then modα_{R} = modα_{R}, and conversely, if modα_{R} ≠ modα_{Re}, then there isn’t a faithful normal state with R_{ψ} ⊂ R, where α_{R} is an involutive *-antiautomorphism, which generates R, and mod is the fundamental homomorphism of Connes–Giordano. References: Haagerup U., Stormer E. Subfactors of a factor of type III_{λ} which contain a maximal centralizer. Internat. J. Math. 6, (1995) , 273-