Let $A$ be a C*-algebra. We identify $A$ with its canonical image in the bidual $A^{**}$. Consider the following conditions:

(1) $A$ is a von Neumann algebra.

(2) There is a multiplicative conditional expectation from $A^{**}$ onto $A$, that is, a map $\pi\colon A^{**}\to A$ that is a *-homomorphism and such that $\pi(a)=a$ for all $a\in A$.

Then (1) implies (2): Consider the predual $A_*$ of $A$, and the canonical embedding $\kappa\colon A_*\to (A_*)^{**}$. Then the dual of $\kappa$ has the desired properties.

Note that $A^{**}$ is always a von Neumann algebra. One can show that (2) implies that $A$ is a monotone complete AW*-algebra. However, is it also a von Neumann algebra?

Q1: Does (2) imply (1) ?

Assuming (2), let $J$ denote the kernel of $\pi$. Then $A$ is (*-isomorphic to) the quotient $A^{**}/J$. However, a quotient of a von Neumann algebra by a closed, two-sided ideal need not be a von Neumann algebra. (For example, the Calkin algebra is such a quotient and not a von Neumann algebra.)

In the commutative case, we might also consider the following more general question:

Q2: Is the quotient of a commutative von Neumann algebra by a closed, two-sided ideal again a von Neumann algebra?

Equivalently, is every closed subset of a hyperstonean space again hyperstonean?

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