Finite von Neumann algebra

In mathematics, a finite von Neumann algebra is a von Neumann algebra in which every isometry is a unitary. In other words, for an operator V in a finite von Neumann algebra if $V^{*}V=I$, then $VV^{*}=I$. In terms of the comparison theory of projections, the identity operator is not (Murray-von Neumann) equivalent to any proper subprojection in the von Neumann algebra.

Properties

Let ${\mathcal {M}}$ denote a finite von Neumann algebra with center ${\mathcal {Z}}$. One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra ${\mathcal {M}}$ is finite if and only if there exists a normal positive bounded map $\tau :{\mathcal {M}}\to {\mathcal {Z}}$ with the properties:

• $\tau (AB)=\tau (BA),A,B\in {\mathcal {M}}$,
• if $A\geq 0$ and $\tau (A)=0$ then $A=0$,
• $\tau (C)=C$ for $C\in {\mathcal {Z}}$,
• $\tau (CA)=C\tau (A)$ for $A\in {\mathcal {M}}$ and $C\in {\mathcal {Z}}$.

Examples

Finite-dimensional von Neumann algebras

The finite-dimensional von Neumann algebras can be characterized using Wedderburn's theory of semisimple algebras. Let C^{n × n} be the n × n matrices with complex entries. A von Neumann algebra M is a self adjoint subalgebra in C^{n × n} such that M contains the identity operator I in C^{n × n}.

Every such M as defined above is a semisimple algebra, i.e. it contains no nilpotent ideals. Suppose M ≠ 0 lies in a nilpotent ideal of M. Since M* ∈ M by assumption, we have M*M, a positive semidefinite matrix, lies in that nilpotent ideal. This implies (M*M)^k = 0 for some k. So M*M = 0, i.e. M = 0.

The center of a von Neumann algebra M will be denoted by Z(M). Since M is self-adjoint, Z(M) is itself a (commutative) von Neumann algebra. A von Neumann algebra N is called a factor if Z(N) is one-dimensional, that is, Z(N) consists of multiples of the identity I.

Theorem Every finite-dimensional von Neumann algebra M is a direct sum of m factors, where m is the dimension of Z(M).

Proof: By Wedderburn's theory of semisimple algebras, Z(M) contains a finite orthogonal set of idempotents (projections) {P_i} such that P_iP_j = 0 for i ≠ j, Σ P_i = I, and

$Z(\mathbf {M} )=\bigoplus _{i}Z(\mathbf {M} )P_{i}$

where each Z(M)P_i is a commutative simple algebra. Every complex simple algebras is isomorphic to the full matrix algebra C^{k × k} for some k. But Z(M)P_i is commutative, therefore one-dimensional.

The projections P_i "diagonalizes" M in a natural way. For M ∈ M, M can be uniquely decomposed into M = Σ MP_i. Therefore,

${\mathbf {M} }=\bigoplus _{i}{\mathbf {M} }P_{i}.$

One can see that Z(MP_i) = Z(M)P_i. So Z(MP_i) is one-dimensional and each MP_i is a factor. This proves the claim.

For general von Neumann algebras, the direct sum is replaced by the direct integral. The above is a special case of the central decomposition of von Neumann algebras.

References

• Kadison, R. V.; Ringrose, J. R. (1997). Fundamentals of the Theory of Operator Algebras, Vol. II : Advanced Theory. AMS. p. 676. ISBN 978-0821808207.

• Sinclair, A. M.; Smith, R. R. (2008). Finite von Neumann Algebras and Masas. Cambridge University Press. p. 410. ISBN 978-0521719193.