Projections, traces, and factor types

A projection p satisfies p-star=p=p squared. Projections p and q are Murray–von Neumann equivalent when a partial isometry v inside M has v-star v=p and vv-star=q. In matrices this means equal rank, but the definition works without counting a finite basis. The membership of v in M matters: equivalence in a larger ambient algebra is not automatically the same internal relation. A faithful normal trace takes equal values on equivalent projections by its cyclic property.

A projection is finite if it is not equivalent to any proper subprojection of itself. This is not the same as being finite rank in every representation. A projection is minimal if it is nonzero and its corner pMp contains only scalar multiples of p. Matrix rank-one projections are minimal and finite. In a type II₁ factor every projection is finite, yet no nonzero projection is minimal. Keeping these concepts separate is essential before reading the type classification.

A normal trace is positive, preserves increasing positive suprema, and is invariant under exchanging the order of products in its finite-valued setting. A finite factor has a unique normalized faithful normal tracial state. In M_n(C) this is ordinary matrix trace divided by n. Infinite semifinite algebras can instead have a faithful normal semifinite trace with infinite value on the identity. Semifinite means every nonzero positive element dominates some nonzero positive element of finite trace, after requiring faithfulness.

Type I factors are B(K) for a complex Hilbert space K, with minimal projections present. Finite-dimensional ones are type I_n; the separable infinite-dimensional example is type I-infinity. Type II factors have no nonzero minimal projections and admit a faithful normal semifinite trace. Type II₁ has finite identity and a normalized trace, while type II-infinity has infinite identity. Type III factors have no nonzero finite projections and admit no faithful normal semifinite trace. These are structural assertions, not labels determined just by Hilbert-space dimension.

On ℓ² of the nonnegative integers, the unilateral shift S sends the jth basis vector to the next. It is an isometry, so S-star S is the identity, while SS-star misses the first basis vector. The identity is therefore equivalent to a proper subprojection and is infinite. Nevertheless B(ℓ²) contains rank-one minimal finite projections and has its ordinary semifinite trace. It is type I-infinity, not type III.

A type II₁ factor has a continuous range of projection traces from zero to one; those projections are not obtained by dividing a finite matrix rank by a fixed integer. One important example arises by taking the von Neumann closure of an increasing tensor product of two-by-two matrix algebras in its tracial GNS representation. This example is stated as a construction roadmap, not proved here. The difference between its norm completion and von Neumann closure again matters, as does the representation chosen for that closure.