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.

## Contents

* 1 Properties
* 2 Examples
* 2.1 Finite-dimensional von Neumann algebras
* 2.2 Abelian von Neumann algebras
* 2.3 Type II1 factors
* 3 References

## Properties

Let ℳ denote a finite von Neumann algebra with center 𝒵. One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra ℳ is finite if and only if there exists a normal positive bounded map τ:ℳ->𝒵 with the properties:

* τ(AB)=τ(BA),A,B∈ℳ,
* if A≥0 and τ(A)=0 then A=0,
* τ(C)=C for C∈𝒵,
* τ(CA)=Cτ(A) for A∈ℳ and C∈𝒵.

## Examples

### Finite-dimensional von Neumann algebras

The finite-dimensional von Neumann algebras can be characterized using Wedderburn's theory of semisimple algebras. Let Cn × n be the n × n matrices with complex entries. A von Neumann algebra M is a self adjoint subalgebra in Cn × n such that M contains the identity operator I in Cn × 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) {Pi} such that PiPj = 0 for i ≠ j, Σ Pi = I, and

Z(𝐌)=⨁iZ(𝐌)Pi

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

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

𝐌=⨁i𝐌Pi.

One can see that Z(MPi) = Z(M)Pi. So Z(MPi) is one-dimensional and each MPi 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.

### Abelian von Neumann algebras

Abelian von Neumann algebras are all isomorphic to a multiplication algebra L∞(X) for some measure space (X,μ). Since an abelian von Neumann algebra M is commutative, for any V∈M, V*V=VV*, so clearly any isometry is unitary.

### Type II1 factors

## 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.

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