Properly infinite von Neumann algebra - Imported Knowlpedia Content

Definition

A von Neumann algebra $M$ is properly infinite when its identity $1_M$ is a properly infinite projection: there are orthogonal projections $p_1,p_2\leq1_M$ and partial isometries $v_1,v_2\in M$ such that

v_i^*v_i=1_M,\qquad v_iv_i^*=p_i\quad(i=1,2).

Equivalently, $1_M$ contains two orthogonal subprojections, each Murray–von Neumann equivalent to $1_M$ . This property is stronger than the identity merely being infinite in a general $C^*$ -algebra, although the projection theory of von Neumann algebras supplies powerful equivalent central criteria.

Central characterization

A von Neumann algebra is properly infinite if and only if every nonzero central projection is infinite, equivalently if it has no nonzero finite central direct summand. In terms of the type decomposition, precisely the type $\mathrm I_\infty$ , type $\mathrm{II}_\infty$ , and type $\mathrm{III}$ central summands may occur. This characterization is part of the comparison theory of projections Kadison–Ringrose, §6.3.

Isometries and amplification

The partial isometries in the core are isometries with orthogonal range projections. By iterating the construction, one obtains countably many orthogonal subprojections equivalent to the identity when the relevant decomposition is countable. Proper infiniteness is consequently stable under matrix amplification, and $M$ absorbs finite matrix factors up to von Neumann algebra isomorphism under the usual spatial identifications.

Examples and non-examples

For every infinite-dimensional Hilbert space $H$ , $B(H)$ is properly infinite: split an orthonormal basis into two subsets of the same cardinality and use the resulting isometries. Type $\mathrm{II}_\infty$ and type $\mathrm{III}$ factors are also properly infinite. By contrast, $M_n(\mathbb C)$ , finite von Neumann algebras, and type $\mathrm{II}_1$ factors are not, because their identities are finite.

Conventions and scope

References

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: §§6.3 and 6.5 on infinite projections and the type decomposition.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §1 on finite and properly infinite von Neumann algebras.