Weak expectation property - Imported Knowlpedia Content

Definition

A unital $C^*$ -algebra $A$ has the weak expectation property (WEP) if, for every faithful unital representation $\pi:A\to B(H)$ , there is a unital completely positive map

\Phi:B(H)\longrightarrow \pi(A)''

such that $\Phi(\pi(a))=\pi(a)$ for every $a\in A$ . Here $\pi(A)''$ is the von Neumann algebra generated by $\pi(A)$ . Equivalently, it suffices to use the universal representation, where the double commutant is $A^{**}$ . The map $\Phi$ is called a weak expectation Brown–Ozawa, §3.3.

Interpretation and distinctions

WEP says that the inclusion $\pi(A)\subseteq B(H)$ can be retracted after the codomain is enlarged from $\pi(A)$ to its weak closure. It is weaker than asking for a conditional expectation $B(H)\to\pi(A)$ : a weak expectation may land only in $\pi(A)''$ , and its range need not lie in the original norm-closed algebra.

The adjective “weak” refers to this bidual target, not to positivity. Complete positivity and preservation of the unit are indispensable parts of the definition. WEP is also distinct from the local lifting property and from exactness, although tensor-product characterizations relate these properties.

Tensor-product characterization

Lance's formulation identifies WEP with a maximal-tensor extension property. In particular, if $A\subseteq B(H)$ is faithful and unital, then for every $C^*$ -algebra $C$ , the canonical map

A\otimes_{\max}C\longrightarrow B(H)\otimes_{\max}C

is isometric. This makes WEP a regularity condition governing how $A$ sits inside larger operator algebras Lance, pp. 157–176.

Every nuclear $C^*$ -algebra has WEP. Also $B(H)$ has WEP, witnessed by the identity map. These examples do not collapse the notions: WEP only asks for a bidual-valued retraction, whereas nuclearity requires agreement of minimal and maximal tensor products against every $C^*$ -algebra.

Conventions and scope

The core uses the standard unital formulation. For a nonunital $C^*$ -algebra, WEP is defined through a unitization or an equivalent nonunital formulation; the convention must be stated because the phrase “unital completely positive” otherwise has no literal meaning.

References

E. Christopher Lance, “On Nuclear $C^*$ -Algebras,” Journal of Functional Analysis 12 (1973), 157–176. Elsevier DOI record. Relevant: weak expectations and the tensor extension property.
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: Chapter 3, especially §3.3 on WEP.