Type decomposition of von Neumann algebras - Imported Knowlpedia Content

Statement

For every von Neumann algebra $M$ , there are unique pairwise orthogonal central projections $z_{\mathrm I},z_{\mathrm {II}},z_{\mathrm {III}}\in M$ such that

z_{\mathrm I}+z_{\mathrm {II}}+z_{\mathrm {III}}=1

and the central summands $z_{\mathrm I}M$ , $z_{\mathrm {II}}M$ , and $z_{\mathrm {III}}M$ are respectively a type I, type II, and type III von Neumann algebra. Zero summands are allowed. Hence $M\cong z_{\mathrm I}M\oplus z_{\mathrm {II}}M\oplus z_{\mathrm {III}}M$ , canonically up to the uniquely determined central projections.

Construction and uniqueness

The type I projection is the central support of all abelian projections. After removing that summand, projection comparison separates the remaining algebra into the central part supported by finite projections and the part containing no nonzero finite projection. These are the type II and type III summands. Because each defining class is stable under central summands and central orthogonal sums, the three largest central supports are forced, which proves uniqueness. A detailed proof appears in Kadison–Ringrose, Theorem 6.5.2.

Finer decomposition

The theorem is only the first layer of the classification. The type I part decomposes into homogeneous pieces of type $\mathrm I_\kappa$ . The type II part separates into finite type $\mathrm {II}_1$ and properly infinite type $\mathrm {II}_\infty$ central summands. Type III factors admit the finer Connes classes $\mathrm {III}_0$ , $\mathrm {III}_\lambda$ , and $\mathrm {III}_1$ , obtained from modular theory rather than projection finiteness alone.

Consequences and scope

Any property of von Neumann algebras that respects central direct sums can be studied separately on the three summands. For a factor, central projections are only $0$ and $1$ , so exactly one of the three types occurs.

References

R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. II, American Mathematical Society, 1997. DOI record. Relevant: Theorem 6.5.2 and the surrounding type-decomposition theory.
M. Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V on the type classification of von Neumann algebras.