Let $\mathcal{A}$ be a $C^\*$-algebra and $p\in\mathcal{A}^{\*\*}$ be an open projection, that is, $p=p^\*=p^2$ and $p\in\overline{(p\mathcal{A}^{\*\*}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^\*}$, where $\hat{\mathcal{A}}$ is the canonical copy of $\mathcal{A}$ in $\mathcal{A}^{\*\*}$ and the closure is taken in the weak$^\*$ topology on $\mathcal{A}^{\*\*}$.



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> **Question:** Is every orthogonal projection in $\mathcal{A}^{\*\*}$ which is Murray-von Neumann equivalent to $p$ open?
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Open projections and Murray-von Neumann equivalence

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> The answer is **no**.
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*Proof (Thomas Schick)*. The idea of the proof is due to Thomas Schick. I thank him for allowing me to reproduce it here. Let $\mathcal{A}:=C([0,1])\otimes\mathbb{M}\_2$, where $\mathbb{M}\_2$ is the $W^{\star}$-algebra of $2\times2$ matrices with entries in $\mathbb{C}$. Since the set of bounded Borel functions $B^{\infty}([0,1])$ is contained in $C([0,1])^{\*\*}$, we canonically identify $B^{\infty}([0,1])\otimes\mathbb{M}\_2$ with a $C^\*$-subalgebra of $\mathcal{A}^{\*\*}$. Let $F:=\{0\}$ and $F^c:=(0,1]$, then the characteristic functions $\chi\_F$ and $\chi\_{F^c}$ are Borel measurable. Define a partial isometry $u:=\chi\_F\otimes E\_{11}+\chi\_{F^c}\otimes E\_{12}\in\mathcal{A}^{\*\*}$, where $E\_{ij}\in\mathbb{M}\_2$ is the canonical matrix unit, that is, it has $1$ in its $(i.j)$-entry and $0$ elsewhere. Then the orthogonal projection $p:=uu^{\star}=1\otimes E\_{11}$ is open, whereas $q:=u^{\star}u=\chi\_F\otimes E\_{11}+\chi\_{F^c}\otimes E\_{22}$ is not. $\square$



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> Cf. [Here](https://mathoverflow.net/questions/301257/is-the-ideal-property-of-x-inheritable-to-x) is my related question.
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