# Projections in the tensor product of von Neumann algebras

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?

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-$$1$$.