# Existence of conditional expectations map onto subalgebras

Let $$B\subset A$$ be an inclusion of $$C^*$$ - algebras. I am having confusions on the existence of a conditional expectation $$E: A \to B$$. I could see that in general an inclusion need not have any conditional expectation. I couldnt get an example towards this.

1. But if we consider the inclusion of tracial von Neumann algebras, there exists a unique trace preserving faithful normal conditional expectation.

2. Is the same true for inclusion of finite von Neumann algebras ? I knw there exists a unique centre valued trace but can we find a scalar valued trace? Because the construction of the conditional expectation for a tracial von Neumann algebras are via the GNS construction. Is the same true for finite von Neumann algebras also?

3. For a $$C^*$$ algebra $$A$$, with $$B$$ is a Cartan subalgebra, then also there exists a faithful conditional expectation.

Is there any unified theorem which tells the existence of a conditional expectation(faithful) for a $$C^*$$ - algebra inclusion?

• Maybe remark 4.3 page 1177 of this paper is related to your question: bims.iranjournals.ir/… Feb 9 at 15:46
• For $C^*$-algebras, a conditional expectation is a projection of norm 1, e.g. see books.google.ca/books?id=6b_T1j3Ib8oC&pg=PA132. Thus, there exists a conditional expectation $E:A\to B \Leftrightarrow B$ is a 1-complemented subspace of $A$ Mar 27 at 4:42

As you mentioned, there are many concrete examples of $$C^*$$-algebra embeddings that admit a conditional expectation.

In general, for a given embedding $$A\subset B$$ this is a difficult question, it depends a lot on the type of embedding.

But there are situations where we always find conditional expectations: this is the case, for instance, if $$A$$ is an injective $$C^*$$-algebra, as for example $$A=B(H)$$ (the $$C^*$$-algebra of all bounded operators); this is due to Arverson's extension theorem. There are many other examples of injective $$C^*$$-algebras: for instance, the bidual von Neumann algebra of a nuclear $$C^*$$-algebra is always injective, so for instance, $$\ell^\infty(X)$$ is injective for any set $$X$$.

A von Neumann algebra $$M\subset B(H)$$ is injective if and only if there is a conditional expectation $$B(H)\to M$$. From this one can give several examples of embeddings that do not admit a conditional expectation: just take any non-injective von Neumann algebra $$M$$. For instance, if $$G$$ is a discrete group, its von Neumann algebra $$W^*_r(G)$$, generated by the image left regular representation $$\lambda\colon G\to B(\ell^2(G))$$, is injective if and only if $$G$$ is amenable. So, if $$G$$ is not amenable (e.g. a non-commutative free group), then the embedding $$W^*_r(G)\subset B(\ell^2(G))$$ does not admit a conditional expectation.

More generally, we can also ask about existence of weak conditional expectations: for an embedding $$A\subset B$$, this means that the bidual embedding $$A''\subset B''$$ has a (normal) conditional expectation.

If for fixed $$A$$, all the embeddings $$A\subset B$$ admit a weak conditional expectation, then $$A$$ is said to have the WEP (Weak Expectation Property).

All nuclear $$C^*$$-algebras have the WEP. It turns out that a $$C^*$$-algebra $$A$$ has the WEP if and only if the algebraic tensor product $$A\odot C^*(\mathbb{F})$$ has a unique $$C^*$$-norm, see Brown-Ozawa's book $$C^*$$-algebras and finite dimensional approximations'', for instance. Here $$\mathbb{F}$$ denotes a non-commutative free group.

Also, it follows that the celebrated Connes embedding conjecture is equivalent to the fact that $$C^*(\mathbb{F})$$ has the WEP. And this conjecture has been announced to be false recently.

To have a concrete example of a non-WEP $$C^*$$-algebra, one can also look at the reduced $$C^*$$-algebra $$C^*_r(\mathbb{F_2})$$ of the free group $$\mathbb{F}_2$$ on two generators (see again the book of Brown-Ozawa, exercise 13.2.2). In fact, this exercise shows that every exact $$C^*$$-algebra that has the WEP must be nuclear.