Type I∞ factor - Imported Knowlpedia Content

Definition

A type $\mathrm I_\infty$ factor is a type I factor that is isomorphic, as a von Neumann algebra, to $B(H)$ for an infinite-dimensional complex Hilbert space $H$ . Equivalently, it is a type I factor whose identity is an infinite projection, or whose maximal families of mutually orthogonal equivalent minimal projections are infinite. The subscript $\infty$ distinguishes this case from the finite-dimensional type $\mathrm I_n$ factors $M_n(\mathbb C)$ . No separability hypothesis is part of the definition: the Hilbert-space dimension may be any infinite cardinal.

Classification and projection structure

Every type I factor is $B(H)$ for a Hilbert space $H$ , unique up to dimension. Minimal projections in $B(H)$ are precisely rank-one projections, and a maximal orthogonal family of them corresponds to an orthonormal basis. Thus the cardinality of such a family recovers $\dim H$ , and the infinite cardinals distinguish the nonseparable isomorphism classes Takesaki, Chapter V, §1.

Proper infiniteness and traces

An infinite-dimensional $H$ decomposes as $H\cong H\oplus H$ . The two coordinate embeddings give isometries in $B(H)$ with orthogonal ranges, so every type $\mathrm I_\infty$ factor is a properly infinite [[operator-algebras/von-neumann-algebra|von Neumann algebra]]. The usual operator trace is faithful, normal, and semifinite but takes value $\infty$ at the identity. Hence the algebra is semifinite without being finite.

Examples and boundaries

The algebra $B(\ell^2(\mathbb N))$ is the separable type $\mathrm I_\infty$ factor. More generally, $B(\ell^2(I))$ is type $\mathrm I_\infty$ for every infinite set $I$ . The algebra $\ell^\infty(I)$ is type I but not a factor when $|I|>1$ , because its center is nontrivial. A type $\mathrm{II}_\infty$ factor is properly infinite and semifinite but has no minimal projections, so it is not type I.

Conventions and scope

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §1 on the classification of type I factors.
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: §6.5 on type I factors and properly infinite factors.