Minimal projection - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A minimal projection is a nonzero projection $p\in M$ such that every projection $q\in M$ with $q\leq p$ satisfies $q=0$ or $q=p$ . Here $q\leq p$ means $pq=qp=q$ . Equivalently,

pMp=\mathbb Cp.

Minimality is relative to the ambient algebra $M$ : the same operator may be minimal in one von Neumann subalgebra and not in a larger one. A minimal projection need not be central, and it should not be confused with a minimal central projection.

Equivalent formulations

The corner $pMp$ is itself a von Neumann algebra with identity $p$ . Spectral projections show that it has no projections other than $0$ and $p$ exactly when every self-adjoint element of the corner is scalar, yielding $pMp=\mathbb Cp$ . In a concrete algebra on $H$ , a rank-one orthogonal projection is minimal in $B(H)$ . Higher-rank projections are not, because a one-dimensional subspace of their range gives a proper nonzero subprojection.

Relation to factor type

Every type I factor contains minimal projections: under an isomorphism with $B(K)$ , they correspond to rank-one projections. Type II and type III factors contain no minimal projections Takesaki, Chapter V, §1. For a general type I von Neumann algebra, a diffuse center can prevent the existence of minimal projections, so the factor statement must not be promoted to all type I algebras.

Conventions and scope

“Atomic projection” is commonly synonymous with minimal projection, while an abelian projection only requires $pMp$ to be commutative and can be much weaker. In a general $C^*$ -algebra, $pAp=\mathbb Cp$ is often taken as the definition of a minimal projection. The subprojection condition is equivalent to it for von Neumann algebras, as used here, but not merely from the absence of projections in an arbitrary unital $C^*$ -corner.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §1 on minimal projections and type I structure.
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: Chapter 6 on projections, factors, and type decomposition.