Type I factor - Imported Knowlpedia Content

Definition

A type I factor is a von Neumann factor $M$ that is also a type I von Neumann algebra. Equivalently, $M$ contains a nonzero abelian projection $p$ , meaning that the corner $pMp$ is abelian. Factoriality makes the central support of every nonzero projection equal to $1_M$ , so one such projection detects type I. This definition is representation-independent. The classification theorem states that there is a Hilbert space $K$ , unique up to Hilbert-space dimension, for which $M$ is isomorphic as a von Neumann algebra to $B(K)$ .

Classification by dimension

Choose a maximal family of mutually orthogonal equivalent minimal projections. Matrix units constructed between them identify the factor with all bounded operators on a Hilbert space. Consequently the isomorphism class of a type I factor is determined by the cardinal dimension of that Hilbert space Takesaki, Chapter V, §1.

When $\dim K=n<\infty$ , the factor is $M_n(\mathbb C)$ and is called type $\mathrm{I}_n$ . When $K$ is infinite-dimensional, $B(K)$ is often denoted type $\mathrm{I}_\infty$ .

Examples and consequences

The algebra $\mathbb C=B(\mathbb C)$ is the type $\mathrm{I}_1$ factor. Every full matrix algebra is a type I factor. The algebra $B(\ell^2)$ is an infinite type I factor: it contains rank-one abelian projections, although its identity is infinite.

An abelian von Neumann algebra with more than one point in its spectrum is type I but is not a factor, because its center is larger than the scalars. A type $\mathrm{II}$ or type $\mathrm{III}$ factor has no nonzero abelian projection and hence is not type I.

Conventions and scope

References

M. Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §1 on the classification of type I factors.
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, American Mathematical Society, 1997. DOI record. Relevant: §6.5 on factor types.