This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.


Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be their von Neumann algebra tensor product.



> 
> **Question:** Can every projection in $M\bar{\otimes}N$ be expressed as
>  the supremum (join) of projections of the form $p\otimes q$, where $p$
>  and $q$ are projections in $M$ and $N$, respectively?
> 
> 
> 



Projections in the tensor product of von Neumann algebras

> 
> The answer is **no**.
> 
> 
> 


*Proof.* Let $\mathcal{H}$ and $\mathcal{K}$ be any infinite-dimensional Hilbert spaces, and let $\{\xi\_n\}\_{n=1}^\infty$ and $\{\eta\_n\}\_{n=1}^\infty$ be sequences of any orthogonal vectors of norm $1$ in $\mathcal{H}$ and $\mathcal{K}$, respectively. Then $\zeta:=\sum\_{n=1}^\infty\frac{1}{2^n}\xi\_n\otimes\eta\_n$ is an element of $\mathcal{H}\otimes\mathcal{K}$ (the completion has been taken), but not an element of $\mathcal{H}\odot\mathcal{K}$ (the algebraic tensor product before taking completion). The rank-$1$ projection onto $\mathbb{C}\zeta$ is in $\mathbb{B}(\mathcal{H})\bar{\otimes}\mathbb{B}(\mathcal{K})$, but not in $\mathbb{B}(\mathcal{H})\otimes\mathbb{B}(\mathcal{K})$, and it cannnot be the supremum of projections of the form described in the question since it is of rank-$1$.


