nLab extended von Neumann algebra

Definition

A complex (or real) extended C*-algebra whose bounded part is a von Neumann algebra.

Equivalently (by a theorem due to Dixon and Zakirov–Chilin), an extended von Neumann algebra is a complex (or real) *-algebra $A$ of closed densely defined unbounded operators on a Hilbert space that satisfies the following conditions:

• $A$ is closed under the operations of strong sum, strong multiplication, and passing to adjoints;

• $A$ contains all scalar multiples of the identity operator;

• for every $x\in A$ we have $(1+x^*x)^{-1}\in A$ (equivalently, $x$ is affiliated with $A_b$);

• the *-subalgebra? $A_b$ of bounded operators in $A$ is a von Neumann algebra.

Properties

For any von Neumann algebra there is a unique (up to a unique isomorphism) maximal extended von Neumann algebra that has $A$ as its bounded part, namely, the extended von Neumann algebra of locally measurable operators affiliated with $A$.

The reduction theory? for von Neumann algebras can be extended to the case of extended von Neumann algebras.

Related concepts

• extended C*-algebra

References

• P. G. Dixon, Unbounded operator algebras, Proceedings of the London Mathematical Society s3-23:1 (1971), 53-69. DOI.

  • defines concrete extended von Neumann algebras (EW*-algebras) in Definition 1.2
  • develops reduction theory in Section 3
  • studies measurable operators in Section 4 and 5
  • studies locally measurable operators in Section 6 and 7

• B. S. Zakirov, V. I. Chilin, Abstract characterization of EW*-algebras, Functional Analysis and Its Applications 25:1 (1991), 63-64. DOI.

  • shows that abstract and concrete extended von Neumann algebras are the same
  • proves that extended von Neumann algebras are precisely those extended C*-algebras? whose bounded part is a von Neumann algebra