Comparison theorem for projections - Imported Knowlpedia Content

Statement

Let $p$ and $q$ be projections in a von Neumann algebra $M$ . The comparison theorem for projections states that there is a central projection $z\in Z(M)$ such that

zp\precsim zq,\qquad (1-z)q\precsim(1-z)p,

where $\precsim$ denotes Murray–von Neumann subequivalence. Because $z$ is central, $zp,zq,(1-z)p$ , and $(1-z)q$ are projections in the corresponding central summands. Thus any failure of global comparability is resolved by splitting the algebra along its center.

Factor case

If $M$ is a factor, its only central projections are $0$ and $1$ . The theorem then says that any two projections are comparable:

p\precsim q\quad\text{or}\quad q\precsim p.

This total comparability of projection classes is one reason that numerical or extended numerical dimension functions can classify projections in factors.

Interpretation of the central split

A general von Neumann algebra behaves like a family of factors over its center. On some central components $p$ is no larger than $q$ ; on the remaining components the reverse holds. The central projection $z$ records these first components. It need not be unique without an additional maximality convention, but the theorem guarantees that at least one such decomposition exists.

Consequences

Projection comparison yields the Schröder–Bernstein property for Murray–von Neumann equivalence and supports the decomposition of projections into finite and properly infinite parts. Together with central support, it is a principal tool in the type classification and in the construction of center-valued dimension functions.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, American Mathematical Society, 1997. Publisher record. Relevant: Theorem 6.2.7 and the surrounding comparison theory.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter V on comparison of projections.