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 [math]\displaystyle{ V^*V = I }[/math], then [math]\displaystyle{ VV^* = I }[/math]. 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 [math]\displaystyle{ \mathcal{M} }[/math] denote a finite von Neumann algebra with center [math]\displaystyle{ \mathcal{Z} }[/math]. One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra [math]\displaystyle{ \mathcal{M} }[/math] is finite if and only if there exists a normal positive bounded map [math]\displaystyle{ \tau : \mathcal{M} \to \mathcal{Z} }[/math] with the properties:

• [math]\displaystyle{ \tau(AB) = \tau(BA), A, B \in \mathcal{M} }[/math],
• if [math]\displaystyle{ A \ge 0 }[/math] and [math]\displaystyle{ \tau(A) = 0 }[/math] then [math]\displaystyle{ A = 0 }[/math],
• [math]\displaystyle{ \tau(C) = C }[/math] for [math]\displaystyle{ C \in \mathcal{Z} }[/math],
• [math]\displaystyle{ \tau(CA) = C\tau(A) }[/math] for [math]\displaystyle{ A \in \mathcal{M} }[/math] and [math]\displaystyle{ C \in \mathcal{Z} }[/math].

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

[math]\displaystyle{ Z(\mathbf M) = \bigoplus _i Z(\mathbf M) P_i }[/math]

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,

[math]\displaystyle{ {\mathbf M} = \bigoplus_i {\mathbf M} P_i . }[/math]

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. pp. 676. ISBN 978-0821808207. 

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