Conditional expectation of C*-algebras - Imported Knowlpedia Content

Definition

Let $B\subseteq A$ be a nonzero $C^*$ -subalgebra. A conditional expectation from $A$ onto $B$ is a completely positive contractive linear map $E:A\to B$ such that

E(b)=b,\qquad E(b_1ab_2)=b_1E(a)b_2

for $a\in A$ and $b_1,b_2\in B$ . The first condition makes $E$ a retraction and hence an idempotent projection: $E^2=E$ . The second says that $E$ is a $B$ -bimodule map. If $A$ and $B$ have the same identity, then $E$ is automatically unital.

Norm-one projection characterization

Tomiyama's theorem states that a bounded linear projection $P:A\to B$ onto a $C^*$ -subalgebra, with $\lVert P\rVert=1$ , is positive, $B$ -bimodular, and completely positive. Conversely, every conditional expectation in the core definition is such a norm-one projection. Thus many sources define a conditional expectation simply as a contractive projection onto $B$ ; the substantial bimodule and positivity properties then follow Takesaki, vol. I, §IV.2.

Basic properties

A conditional expectation fixes $B$ pointwise and has range exactly $B$ . It is positive, so $a\geq0$ implies $E(a)\geq0$ , and it is $*$ -preserving. When the algebras share an identity, Kadison's inequality gives

E(a)^*E(a)\leq E(a^*a).

The expectation is faithful when $E(a^*a)=0$ implies $a=0$ . Faithfulness is an additional condition and is not implied by contractivity or complete positivity.

Conditional expectations let positive functionals on $B$ produce positive functionals on $A$ by composition. They also provide canonical $B$ -valued inner products in operator-algebra and Hilbert-module constructions.

Examples and non-examples

For the diagonal subalgebra $D_n\subseteq M_n(\mathbb C)$ ,

E([a_{ij}])=\operatorname{diag}(a_{11},\ldots,a_{nn})

is a faithful conditional expectation. If a compact group acts continuously on a $C^*$ -algebra $A$ , averaging the action against normalized Haar measure gives a conditional expectation onto the fixed-point algebra.

A positive map $A\to B$ whose image lies in $B$ is not necessarily an expectation: it must fix $B$ and satisfy the projection and bimodule requirements. Similarly, an arbitrary algebraic projection onto $B$ can have norm greater than one and need not be positive.

Von Neumann algebra convention

For an inclusion $N\subseteq M$ of von Neumann algebras, the same definition applies, but authors often require the expectation to be normal, meaning ultraweakly continuous. Normality is extra structure; a $C^*$ -algebraic conditional expectation between von Neumann algebras need not be normal. This distinction is important when expectations are used with preduals, weights, or increasing operator limits.

References

Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV, especially §IV.2, on norm-one projections and conditional expectations.
Nathanial P. Brown and Narutaka Ozawa, C-Algebras and Finite-Dimensional Approximations, Graduate Studies in Mathematics 88, American Mathematical Society, 2008. AMS/DOI record. Relevant: §1.5 on conditional expectations and completely positive maps.