TOPICS
Search

Continuous Core

The continuous core of a von Neumann algebra M with a faithful normal state phi is the crossed-product von Neumann algebra

c(M)=M×AdjustmentBox[│, BoxMargins -> {{-0.27, 0.13913}, {-0.5, 0.5}}]_(sigma^phi)R,

where sigma^phi is the modular automorphism group. It contains a copy of M and unitaries implementing the modular action. Different faithful normal states give canonically related descriptions of the core. If phi is tracial, the action is trivial and the core is M tensor ^_L(R).

Amenability means injectivity in the von Neumann algebra sense.

See also

Amenable von Neumann Algebra, Bicentralizer, Commutant, Modular Automorphism Group, Operator Algebra Conditional Expectation, von Neumann Algebra

Explore with Wolfram|Alpha

References

Blackadar, B. Operator Algebras: Theory of C*-Algebras and von Neumann Algebras. Berlin, Germany: Springer, 2006. https://doi.org/10.1007/3-540-28517-2.

Cite this as:

Weisstein, Eric W. "Continuous Core." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ContinuousCore.html

Subject classifications