## Illinois Journal of Mathematics

### Ultraproducts of crossed product von Neumann algebras

Reiji Tomatsu

#### Abstract

We study a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.

#### Article information

Source
Illinois J. Math., Volume 61, Number 3-4 (2017), 275-286.

Dates
Revised: 23 January 2018
First available in Project Euclid: 22 August 2018

https://projecteuclid.org/euclid.ijm/1534924828

Digital Object Identifier
doi:10.1215/ijm/1534924828

Mathematical Reviews number (MathSciNet)
MR3845722

Zentralblatt MATH identifier
06932505

Subjects
Primary: 46L10: General theory of von Neumann algebras
Secondary: 46L40: Automorphisms

#### Citation

Tomatsu, Reiji. Ultraproducts of crossed product von Neumann algebras. Illinois J. Math. 61 (2017), no. 3-4, 275--286. doi:10.1215/ijm/1534924828. https://projecteuclid.org/euclid.ijm/1534924828

#### References

• H. Ando and U. Haagerup, Ultraproducts of von Neumann algebras, J. Funct. Anal. 266 (2014), 6842–6913.
• C. Houdayer, A. Marrakchi and P. Verraedt, Fullness and Connes' $\tau$ invariant of type III tensor product factors, available at \arxivurlarXiv:1611.07914.
• E. Kirchberg, Commutants of unitaries in UHF algebras and functorial properties of exactness, J. Reine Angew. Math. 452 (1994), 39–77.
• A. Marrakchi, Spectral gap characterization of full type III factors, to appear in J. Reine Angew. Math.
• T. Masuda and R. Tomatsu, Rohlin flows on von Neumann algebras, Mem. Amer. Math. Soc. 244 (2016), no. 1153, ix $+$ 111 pp.
• T. Masuda and R. Tomatsu, Classification of actions of discrete Kac algebras on injective factors, Mem. Amer. Math. Soc. 245 (2017), no. 1160, ix $+$ 118 pp.
• A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras, Lecture Notes in Mathematics, vol. 1138, Springer-Verlag, Berlin, 1985.
• Y. Raynaud, On ultrapowers of non commutative $L_p$ spaces, J. Operator Theory 48 (2002), no. 1, 41–68.
• W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987.
• M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9 (1972), 306–321.
• M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Operator Algebras and Non-commutative Geometry, vol. 6, Springer-Verlag, Berlin, 2003.
• R. Tomatsu and Y. Ueda, A characterization of fullness of continuous cores of type III$_1$ free product factors, Kyoto J. Math. 56 (2016), 599–610.