Infinite projection - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra and let $p\in M$ be a projection. The projection $p$ is infinite if there exists a projection $q\in M$ with $q<p$ such that $p$ and $q$ are Murray–von Neumann equivalent. Explicitly, some partial isometry $v\in M$ satisfies $v^*v=p$ and $vv^*=q$ , while $q\neq p$ . Thus infiniteness is witnessed inside the ambient algebra by moving all of $p$ onto a proper part of itself. A projection is finite precisely when no such proper equivalent subprojection exists.

Examples and permanence

The identity of $B(H)$ is infinite when $H$ is infinite-dimensional: a unitary from $H$ onto a proper closed infinite-dimensional subspace gives the required partial isometry. Every finite-rank projection in $B(H)$ is finite. If $p$ is infinite and $p\leq r$ , then $r$ is infinite as well. Unitary conjugacy and Murray–von Neumann equivalence also preserve infiniteness.

Proper infiniteness

An infinite projection need not be properly infinite. The latter means that $p$ contains two orthogonal subprojections $p_1,p_2\leq p$ , each Murray–von Neumann equivalent to $p$ . Proper infiniteness is therefore a stronger doubling condition. In a factor, every infinite projection is properly infinite, but this implication can fail in a von Neumann algebra with nontrivial center. The distinction matters in comparison theory and in the classification of nonfactor algebras.

Type-theoretic role

Finite and infinite are properties of projections relative to $M$ , not statements about the cardinality of a set. A von Neumann algebra is type III exactly when every nonzero projection is infinite, while semifinite algebras have enough nonzero finite subprojections to meet every nonzero projection. These projection comparisons underlie the Murray–von Neumann type classification Kadison–Ringrose, §6.3.

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 on finite, infinite, and properly infinite projections.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on projection comparison and factor types.