Standard form of a von Neumann algebra - Imported Knowlpedia Content

Definition

A standard form of a von Neumann algebra $M$ is a quadruple $(M,H,J,P)$ , with $M\subseteq B(H)$ faithful and nondegenerate, $J$ a conjugate-linear isometric involution, and $P\subseteq H$ a closed self-dual cone, such that

JMJ=M',\qquad J\xi=\xi\quad(\xi\in P),

xJxJ(P)\subseteq P\quad(x\in M), \qquad JzJ=z^*\quad(z\in Z(M)).

The cone $P$ is the natural positive cone, and $J$ plays the role of modular conjugation. All four compatibility conditions belong to the standard-form package; merely realizing $M$ faithfully on a Hilbert space does not give a standard form.

Existence and uniqueness

Every von Neumann algebra has a standard form. Moreover, if $(M,H,J,P)$ and $(N,K,J_N,P_N)$ are standard forms and $\alpha\colon M\to N$ is a normal $*$ -isomorphism, there is a unique unitary $U\colon H\to K$ implementing $\alpha$ and satisfying

UJ=J_NU,\qquad U(P)=P_N.

This is Haagerup's standard-form uniqueness theorem Haagerup, Theorem 2.3.

Positive functionals as vectors

For every normal positive functional $\varphi$ on $M$ , there is a unique vector $\xi_\varphi\in P$ such that

\varphi(x)=\langle x\xi_\varphi,\xi_\varphi\rangle \qquad(x\in M).

Thus the cone removes the nonuniqueness that normally occurs when a positive functional is represented by a vector in an arbitrary representation.

Conventions and scope

Some authors call a representation “standard” when only a conjugation $J_0$ with $J_0MJ_0=M'$ is specified. That weaker usage omits the canonical cone and does not, by itself, specify Haagerup's standard form. Here “standard form” always means the full quadruple and the displayed axioms.

References

U. Haagerup, “The Standard Form of von Neumann Algebras,” Mathematica Scandinavica 37 (1975), 271–283. DOI record. Relevant: Definition 2.1 and Theorem 2.3 on the standard-form axioms and uniqueness.
H. Araki, “Some Properties of Modular Conjugation Operator of von Neumann Algebras and a Non-commutative Radon–Nikodym Theorem with a Chain Rule,” Pacific Journal of Mathematics 50 (1974), 309–354. DOI record. Relevant: the natural cone and its representation of normal positive functionals.