Semifinite von Neumann algebra - Imported Knowlpedia Content

Definition

A von Neumann algebra $M$ is semifinite if every nonzero projection $p\in M$ dominates a nonzero finite projection: there is a projection $0\neq q\leq p$ that is finite relative to $M$ . Equivalently, the supremum of the finite projections of $M$ is $1$ , or every nonzero central summand of $M$ contains a nonzero finite projection. This is a property of the algebra’s projection structure. It includes all type I and type II von Neumann algebras and excludes every nonzero type III direct summand.

Tracial characterization

Semifiniteness is equivalent to the existence of a faithful normal semifinite trace on $M$ . Here “semifinite” for the trace means that every nonzero positive element majorizes a nonzero positive element of finite trace; it does not mean that the identity has finite trace. A semifinite von Neumann algebra can also be equipped with a faithful normal semifinite center-valued trace Takesaki, Chapter V.

Examples and boundary cases

Every finite von Neumann algebra is semifinite. The algebra $B(H)$ is semifinite even when $H$ is infinite-dimensional: finite-rank projections provide finite subprojections, and the canonical operator trace is faithful, normal, and semifinite although it sends $1$ to $+\infty$ . Type $\mathrm{II}_\infty$ factors are semifinite but not finite. A nonzero type III algebra is not semifinite because it has no nonzero finite projection.

Classification convention

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, American Mathematical Society, 1997. AMS record. Relevant: §§6.3 and 6.5 on finite projections and semifinite type.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on semifinite von Neumann algebras and traces.