Properly infinite projection - Imported Knowlpedia Content

Definition

Let $p$ be a projection in a $C^*$ -algebra $A$ . The projection $p$ is properly infinite if there are orthogonal projections $p_1,p_2\leq p$ such that

p_1\sim p\sim p_2,

where $\sim$ is Murray–von Neumann equivalence. Equivalently, two orthogonal copies of $p$ can be embedded under $p$ , often written $p\oplus p\precsim p$ after passage to a matrix algebra. Every properly infinite projection is infinite, since either $p_i$ is a proper subprojection equivalent to $p$ . The converse need not hold in a general algebra.

Examples

The identity of $B(H)$ is properly infinite when $H$ is infinite-dimensional: split $H$ into two orthogonal infinite-dimensional subspaces, each unitarily isomorphic to $H$ . More generally, every infinite-rank projection in $B(H)$ is properly infinite. By contrast, no nonzero projection in a finite von Neumann algebra is properly infinite.

Relation to factors

In a von Neumann factor, every infinite projection is properly infinite. For von Neumann algebras with nontrivial center, an infinite projection can have finite behavior on some central summands, so infiniteness need not imply proper infiniteness. Projection comparison and central support isolate the summands on which the doubling condition holds.

Permanence and type theory

Proper infiniteness is invariant under Murray–von Neumann equivalence and unitary conjugacy, and orthogonal sums of properly infinite projections are properly infinite. Merely dominating a properly infinite projection does not force proper infiniteness when an additional finite central summand is present. The identity is properly infinite in every type $\mathrm I_\infty$ , type $\mathrm{II}_\infty$ , or type $\mathrm{III}$ factor, making the condition central to the infinite side of von Neumann algebra classification.

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: §6.3 on infinite and properly infinite projections.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter V on projection comparison and factor types.