Center of a von Neumann algebra - Imported Knowlpedia Content

Definition

Let $M\subseteq B(H)$ be a von Neumann algebra. Its center is

Z(M)=\{z\in M:zx=xz\ \text{for every }x\in M\} =M\cap M'.

Here $M'$ is the commutant of $M$ . The center is a commutative von Neumann algebra containing the scalar multiples of $1_M$ . The algebra $M$ is a factor precisely when $Z(M)=\mathbb C1_M$ . Thus factoriality means trivial center, not commutativity: a nontrivial commutative von Neumann algebra is a factor only in the one-dimensional scalar case.

Central projections and decomposition

A projection in $Z(M)$ is a central projection. Each central projection $z$ splits the algebra as

M\cong zM\oplus(1-z)M.

Conversely, direct-sum decompositions of $M$ arise from central projections. More generally, the center supports the central, or factorial, decomposition of a von Neumann algebra into factors; this is formulated as a direct integral under standard separability hypotheses Kadison–Ringrose, vol. II, §6.5.

Examples

For $M=B(H)$ with $H\ne\{0\}$ , the center is $\mathbb C I_H$ , so $B(H)$ is a type I factor. If $M=L^\infty(X,\mu)$ acts by multiplication on $L^2(X,\mu)$ , then $M$ is commutative and $Z(M)=M$ . For a direct sum $M_1\oplus M_2$ , its center is $Z(M_1)\oplus Z(M_2)$ ; even when both summands are factors, their direct sum is not a factor.

Conventions and scope

The formula $M\cap M'$ uses the commutant inside the specified ambient $B(H)$ , but the resulting center is intrinsic to the abstract von Neumann algebra. “Central algebra” sometimes refers to an algebra equipped with a chosen map from another commutative algebra; that broader usage should not be confused with $Z(M)$ . A factor may be of type I, II, or III, so “factor” is not a synonym for “type I factor.”

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: §6.5 on centers and factorial decomposition.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapters IV–V on factors and central decomposition.