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Approximate innerness of positive linear maps of finite von Neumann algebras. (English) Zbl 0579.46044

The completely positive linear maps give an important role in the theory of operator algebras. We have the famous Stinespring’s theorem for the completely positive linear maps and the Sakai’s theorem that the inner automorphism group is dense in the automorphism group for the hyperfinite \(II_ 1\)-factors.
From the above mentioned two theorems, this report gives the connection between the completely positive linear maps and the approximate innerness for the finite von Neumann algebras.

MSC:

46L10 General theory of von Neumann algebras
46L30 States of selfadjoint operator algebras
46L40 Automorphisms of selfadjoint operator algebras
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References:

[1] Man Duen Choi, A Schwarz inequality for positive linear maps on \?*-algebras, Illinois J. Math. 18 (1974), 565 – 574. · Zbl 0293.46043
[2] Uffe Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975), no. 2, 271 – 283. · Zbl 0304.46044 · doi:10.7146/math.scand.a-11606
[3] -, A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space (preprint). · Zbl 0586.46047
[4] Shôichirô Sakai, On automorphism groups of \?\?\(_{1}\)-factors, Tôhoku Math. J. (2) 26 (1974), 423 – 430. · Zbl 0303.46060 · doi:10.2748/tmj/1178241136
[5] W. Forrest Stinespring, Positive functions on \?*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211 – 216. · Zbl 0064.36703
[6] Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. · Zbl 0436.46043
[7] J. Tomiyama, Complete positivity in operator algebras, Lecture Note No. 4, RIMS, Kyoto Univ., 1978. (Japanese)
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