TOPICS
Search

AW^*-Algebra

An AW^*-algebra is a C*-algebra A in which the right annihilator of every subset S subset= A has the form pA for a projection p (Blackadar 2006, p. 243). Thus {a in A:sa=0 for every s in S}=pA. Every von Neumann algebra is an AW^*-algebra. The projections form a complete lattice, but the monotone completeness conjecture asks the stronger order-completeness question for all self-adjoint elements.

An AW^*-factor has ring center C1. Arulseelan and Hanson (2026) construct a finite AW^*-factor with canonical normalized quasitrace tau and elements y_1, y_2, y_3 satisfying

sum_(i=1)^3y_i^*y_i=1,

and

sum_(i=1)^3y_iy_i^*<=1/21.

Taking a quasitracial ultrapower produces a type II_1 AW^*-factor that is not even monotone sequentially complete. Here finite means that every isometry is unitary. This factor is not a von Neumann algebra, although every maximal abelian subalgebra is a W*-algebra.

See also

C*-Algebra, Monotone Completeness Conjecture, Quasitrace, von Neumann Algebra

Explore with Wolfram|Alpha

References

Arulseelan, J. and Hanson, J. E. "Counterexamples to the Monotone Completeness Conjecture." 19 Oct 2026. https://arxiv.org/abs/2610.27184.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. "AW^*-Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AW-Star-Algebra.html

Subject classifications