Murray–von Neumann equivalence of projections

Definition

Let $M$ be a von Neumann algebra and let $p,q\in M$ be projections. They are Murray–von Neumann equivalent, written $p\sim q$ , if there is a partial isometry $v\in M$ such that

v^*v=p \qquad\text{and}\qquad vv^*=q.

Thus $v$ restricts to an isometric identification of the initial subspace of $p$ with the final subspace of $q$ , but the implementing operator must belong to $M$ . The same definition applies to projections in any $C^*$ -algebra. Reflexivity, symmetry, and transitivity follow respectively from $p$ , $v^*$ , and products of compatible implementing partial isometries.

Sub-equivalence and finiteness

One writes $p\precsim q$ if $p\sim r$ for some projection $r\leq q$ . This compares the sizes of projections without assigning a numerical dimension. A projection $p$ is finite when $p\sim r\leq p$ forces $r=p$ ; otherwise it is infinite. These notions are internal to the algebra: the same concrete projections can have different comparison behavior when viewed inside different von Neumann algebras.

Concrete interpretation

In $B(H)$ , two projections are Murray–von Neumann equivalent exactly when their ranges have the same Hilbert-space dimension. In a matrix algebra this reduces to equality of ranks. Equivalence need not mean conjugacy by a unitary in $M$ : extending an implementing partial isometry $v$ to a unitary also requires the complementary projections $1-p$ and $1-q$ to be equivalent.

Role in classification

Murray–von Neumann equivalence is the comparison relation behind the dimension theory of von Neumann algebras. Passing to equivalence classes and using orthogonal sums of projections records how pieces of the identity can be decomposed. The distinction between finite, semifinite, and properly infinite behavior—and ultimately the type I, II, and III classification—is formulated through this comparison theory.

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