Finite von Neumann algebra

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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 {\displaystyle V^{*}V=I} {\displaystyle V^{*}V=I}, then V V ∗ = I {\displaystyle VV^{*}=I} {\displaystyle 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 M {\displaystyle {\mathcal {M}}} {\displaystyle {\mathcal {M}}} denote a finite von Neumann algebra with center Z {\displaystyle {\mathcal {Z}}} {\displaystyle {\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 M {\displaystyle {\mathcal {M}}} {\displaystyle {\mathcal {M}}} is finite if and only if there exists a normal positive bounded map τ : M → Z {\displaystyle \tau :{\mathcal {M}}\to {\mathcal {Z}}} {\displaystyle \tau :{\mathcal {M}}\to {\mathcal {Z}}} with the properties:

  • τ ( A B ) = τ ( B A ) , A , B ∈ M {\displaystyle \tau (AB)=\tau (BA),A,B\in {\mathcal {M}}} {\displaystyle \tau (AB)=\tau (BA),A,B\in {\mathcal {M}}},
  • if A ≥ 0 {\displaystyle A\geq 0} {\displaystyle A\geq 0} and τ ( A ) = 0 {\displaystyle \tau (A)=0} {\displaystyle \tau (A)=0} then A = 0 {\displaystyle A=0} {\displaystyle A=0},
  • τ ( C ) = C {\displaystyle \tau (C)=C} {\displaystyle \tau (C)=C} for C ∈ Z {\displaystyle C\in {\mathcal {Z}}} {\displaystyle C\in {\mathcal {Z}}},
  • τ ( C A ) = C τ ( A ) {\displaystyle \tau (CA)=C\tau (A)} {\displaystyle \tau (CA)=C\tau (A)} for A ∈ M {\displaystyle A\in {\mathcal {M}}} {\displaystyle A\in {\mathcal {M}}} and C ∈ Z {\displaystyle C\in {\mathcal {Z}}} {\displaystyle 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 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 ) = ⨁ i Z ( M ) P i {\displaystyle Z(\mathbf {M} )=\bigoplus _{i}Z(\mathbf {M} )P_{i}} {\displaystyle Z(\mathbf {M} )=\bigoplus _{i}Z(\mathbf {M} )P_{i}}

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 = ⨁ i M P i . {\displaystyle {\mathbf {M} }=\bigoplus _{i}{\mathbf {M} }P_{i}.} {\displaystyle {\mathbf {M} }=\bigoplus _{i}{\mathbf {M} }P_{i}.}

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 ) {\displaystyle L^{\infty }(X)} {\displaystyle L^{\infty }(X)} for some measure space ( X , μ ) {\displaystyle (X,\mu )} {\displaystyle (X,\mu )}. Since an abelian von Neumann algebra M is commutative, for any V ∈ M {\displaystyle V\in M} {\displaystyle V\in M}, V ∗ V = V V ∗ {\displaystyle V^{*}V=VV^{*}} {\displaystyle V^{*}V=VV^{*}}, so clearly any isometry is unitary.

Type I I 1 {\displaystyle II_{1}} {\displaystyle II_{1}} factors


References

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