Normal conditional expectation - Imported Knowlpedia Content

Definition

Let $N\subseteq M$ be unital von Neumann algebras with the same identity. A normal conditional expectation $E:M\to N$ is a conditional expectation that is normal. Explicitly, $E$ is a positive norm-one linear projection onto $N$ and, for every bounded increasing net $(x_i)$ in $M_+$ ,

E\!\left(\sup_i x_i\right)=\sup_i E(x_i).

The norm-one projection property implies $E(n_1xn_2)=n_1E(x)n_2$ for $n_1,n_2\in N$ . Normality is the additional compatibility with ultraweak limits; faithfulness is not included.

Equivalent formulations

Because $E$ is bounded, normality is equivalent to the existence of a preadjoint

E_*:N_*\longrightarrow M_*, \qquad E_*(\omega)=\omega\circ E.

It is also equivalent to preservation of suprema of bounded increasing nets of projections. Tomiyama's theorem supplies positivity, complete positivity, and $N$ -bimodularity from the norm-one projection hypothesis Takesaki, Chapter IV, §2.

Examples and a near-miss

On $B(\ell^2)$ , taking the diagonal matrix entries defines a normal faithful conditional expectation onto the diagonal von Neumann algebra $\ell^\infty$ . More generally, averaging a normal action of a compact group against Haar measure gives a normal conditional expectation onto the fixed-point algebra.

If $\varphi$ is a singular state on $M$ , then

E_\varphi(x)=\varphi(x)1

is a $C^*$ -algebraic conditional expectation from $M$ onto $\mathbb C1$ , but it is not normal. Thus normality does not follow merely because the domain and range are von Neumann algebras.

Structure and consequences

The range of a normal expectation is automatically ultraweakly closed, and normal positive functionals on $N$ pull back to normal positive functionals on $M$ . If $E$ is faithful, then $E(x^*x)=0$ implies $x=0$ ; this extra condition is often required in modular theory.

Existence is not automatic for an arbitrary inclusion $N\subseteq M$ . Given a faithful normal state $\varphi$ on $M$ , Takesaki's expectation theorem characterizes the existence of a $\varphi$ -preserving normal conditional expectation onto $N$ by invariance of $N$ under the modular automorphism group $\sigma^\varphi$ Takesaki, Chapter IX, §4.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. DOI record. Relevant: Chapter IV, §2 on norm-one projections and conditional expectations.
Masamichi Takesaki, Theory of Operator Algebras II, Springer, 2003. DOI record. Relevant: Chapter IX, §4 on modular invariance and normal conditional expectations.