Hi everyone,

I currently encountered some difficulties on my research related to operator algebras and I wondered if by any chance someone could find my question quite trivial. Here is the context: given a von Neumann algebra $A$, I know that we can construct the so-called von Neumann tensor product $A\otimes A$ of $A$, providing us with a natural completion of the algebraic tensor product of $A$. My question is then the following: can we continuously extend the multiplication map $u\otimes v \to uv$ (defined on the algebraic tensor product) to a linear map defined on the whole product $A\otimes A$? In other words, is this multiplication map continuous with respect to the topology of $A\otimes A$? If not, what if $A$ is endowed with a faithful trace in addition? These questions may be obvious for some operator algebras specialists, but I can’t find any paper/book on this specific issue…

Thank you for any help.