AUTOMORPHISMS OF CENTRAL EXTENSIONS OF TYPE III VON NEUMANN ALGEBRAS
Keywords:
Operator, automorphism, algebra, integral, von Neumann algebra.Abstract
We consider the central extension of a Type III von Neumann algebra. For Type III von Neumann algebras, the algebra coincides with itself since its center is trivial. In this case, we show that any arbitrary automorphism of the algebra is inner, meaning the equality holds for some unitary element. Furthermore, we prove that every banded automorphism of the algebra is inner and acts as identity on the center. Our results are based on the Tomita-Takesaki modular theory and Connes' classification of Type III factors.
References
Albeverio S., Ayupov Sh. A., Kudaybergenov K. K., Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra, Siberian Adv. Math. 18 (2008) 86–94.
Albeverio S., Ayupov Sh. A., Kudaybergenov K. K., Structure of derivations on various algebras of measurable operators for type I von Neumann algebras, J. Func. Anal. 256 (2009) 2917–2943.
Ayupov Sh. A., Kudaybergenov K. K., Additive derivations on algebras of measurable operators, ICTP, Preprint, No IC/2009/059, – Trieste, 2009. – 16 p. (accepted in Journal of operator theory).
Ayupov Sh. A., Kudaybergenov K. K., Additive derivations on algebras of measurable operators, J. Operator Theory 67 (2012), 495–510.
Gutman A. E., Kusraev A. G., Kutateladze S. S., The Wickstead problem, Sib. Electron. Math. Reports. 5 (2008) 293–333.
Kadison R.V., Ringrose J.R., Derivations and automorphisms of operator algebras, Comm. Math. Phys. 4 (1967) 32–63.
Kadison R.V., Ringrose J.R., Algebraic automorphisms of operator algebras, J. London Math. Soc. 8 (1974) 329–334.
Kusraev A. G., Automorphisms and derivations in an extended complex f-algebra, Sib. Math. J. 47 (2006) 97–107.
Connes A., Noncommutative Geometry, Academic Press, 1994.
Takesaki M., Theory of Operator Algebras II, Springer, 2003.
