Murray–von Neumann subequivalence of projections

Definition

Let $p$ and $q$ be projections in a $C^*$ -algebra $A$ . The projection $p$ is Murray–von Neumann subequivalent to $q$ , written $p\precsim q$ , if $p$ is Murray–von Neumann equivalent to a subprojection of $q$ . Equivalently, there is a partial isometry $v\in A$ such that

v^*v=p,\qquad vv^*\leq q.

Thus $v$ identifies the range represented by $p$ with a part of the range represented by $q$ . The relation depends on the ambient algebra: enlarging $A$ can introduce additional partial isometries and hence additional subequivalences.

Order properties

Subequivalence is reflexive and transitive. Murray–von Neumann equivalence implies subequivalence in both directions, and for projections in a von Neumann algebra the converse also holds. Passing to equivalence classes therefore turns $\precsim$ into a partial order. Orthogonal sums respect the relation: if $p_i\precsim q_i$ with mutually orthogonal families, then the corresponding sums are subequivalent whenever they exist.

Central obstruction

In a von Neumann algebra, $p\precsim q$ implies $c_M(p)\leq c_M(q)$ , where $c_M$ denotes central support. This condition alone does not compare the sizes of $p$ and $q$ inside each central summand. The comparison theorem supplies the central decomposition on which one or the other subequivalence holds.

Dimension-theoretic role

For finite-dimensional matrix algebras, $p\precsim q$ exactly when $\operatorname{rank}(p)\leq\operatorname{rank}(q)$ . In general von Neumann algebras, subequivalence replaces rank comparison. It is used to define finite, infinite, and properly infinite projections and underlies the Murray–von Neumann classification of factors.

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 equivalence and comparison of projections.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter V on projection comparison.