Conditional expectation onto maximal abelian subalgebras

Question

If you take a von Neumann algebra $M$ and its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take the Cuntz algebra $O_2$ and its canonical masa $C(\Delta)$. Is this true in general?

Let $A$ be an arbitrary C*-algebra. Is there a masa $D$ of $A$ onto which there is a conditional expectation (norm-one projection from $A$ onto $D$)?

Answer 1

This is not a proper answer, but maybe you find this interesting.

Definition Let $D\subset A$ be an inclusion of $C^*$-algebras. We say that $D$ is a Cartan subalgebra of $A$ when the following conditions are satisfied:

1. $D$ is a MASA in $A$
2. $D$ contains an approximate unit of $A$ (non-degeneracy)
3. The set of normalizers of $D$ in $A$, namely $N_A(D):=\{a\in A: a\cdot D\cdot a^*+a^*\cdot D\cdot a\subset D\}$ generates $A$ as a $C^*$-algebra (regularity)
4. There exists a faithful conditional expectation $E:A\to D$.

At first this might seem as overly complicated, but soon you realize this is a very good and accurate definition. Examples include: the diagonal matrices inside a matrix algebra, or $C(X)$ inside $C(X)\rtimes_r G$ for a topologically free action $G\curvearrowright X$ on a compact Hausdorff space. The most general examples come from the groupoid $C^*$-algebra construction, but I won't get into that; the point is that many, many $C^*$-algebras have a Cartan subalgebra, which is a (much) stronger form of what you're asking.

In classification theory of $C^*$-algebras, it is known that all simple, separable, unital, nuclear, $\mathcal{Z}$-stable $C^*$-algebras that satisfy the UCT are classified by the Elliott invariant, whatever the last two conditions mean. The last condition however, namely satisfying the UCT is a bit mysterious and it is a long standing problem whether the UCT is automatic for $C^*$-algebras satisfying the rest of the adjectives. Recently Xin Li showed that under these assumptions (or maybe a few more or less, I'm not sure), satisfying the UCT is equivalent to the existence of a Cartan subalgebra. So, if the answer to the long standing question is positive, i.e. satisfying the UCT is indeed automatic for these $C^*$-algebras, then your question gets a very strong positive answer for a large class of $C^*$-algebras (the well-behaved ones: unital, simple, separable, nuclear, $\mathcal{Z}$-stable).

On the other hand, it is known that not all $C^*$-algebras (not even all vN algebras) have a Cartan subalgebra: counter examples come from group algebras of free groups.

Now for your question, my guess is that in complete generality this will fail, some counter-example probably can be cooked up without too much trouble (I've been trying, but no luck yet). But, if you were to add some adjectives between the words "a" and "$C^*$-algebra" in the phrase "Let $A$ be a $C^*$-algebra", then this might be a hard question.