Finite von Neumann algebra - Imported Knowlpedia Content

Definition

A von Neumann algebra $M$ is finite if its identity $1_M$ is a finite projection in $M$ . Explicitly, whenever $v\in M$ satisfies $v^*v=1_M$ , one must also have $vv^*=1_M$ ; thus every isometry belonging to $M$ is unitary. Equivalently, no proper subprojection of $1_M$ is Murray–von Neumann equivalent to $1_M$ . Finiteness is an internal comparison property of projections. It does not require $M$ to be finite-dimensional, a factor, or represented on a finite-dimensional Hilbert space.

Equivalent formulations and structure

Finiteness of the identity forces every projection in $M$ to be finite. Conversely, that condition plainly includes the identity. A major structure theorem supplies every finite von Neumann algebra with a unique faithful normal center-valued trace normalized at $1_M$ ; its scalar specializations organize the ordinary finite traces. See Kadison–Ringrose, §6.5 for the projection-comparison and trace formulations.

Finite factors split into two classes. A finite type I factor is a matrix algebra, whereas a finite factor without nonzero abelian projections is of type $\mathrm{II}_1$ . A finite von Neumann algebra with nontrivial center can have both sorts of central summands.

Examples and non-examples

Every matrix algebra $M_n(\mathbb C)$ is finite. An infinite-dimensional commutative von Neumann algebra is also finite: a partial isometry in an abelian algebra has equal initial and final projections.

By contrast, $B(H)$ for an infinite-dimensional Hilbert space $H$ is not finite. A unilateral shift is an isometry whose range projection is strictly smaller than the identity.

Conventions and scope

References

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 finite von Neumann algebras, comparison, and center-valued traces.
M. Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter V, §2 on finite von Neumann algebras.