Let $V$ be a tracial von Neumann algebra i.e a von Neumann algebra with a tracial faithful normal state $\tau$. It is known that the trace induces a norm on the von Neumann algebra defined by $||a||_2^{2} = \tau(a^*a)$. And this norm metricizes the strong operator topology on bounded sets (bounded with respect to the operator norm). My question is the unit ball (with the respect to the operator norm) of the von Neumann algebra complete with respect to this norm ?
Is the unit ball complete in the 2-norm of tracial von Neumann algebra
operator-algebrasoperator-theoryvon-neumann-algebras
Best Answer
It is. This is Proposition 2.6.4 in https://www.math.ucla.edu/~popa/Books/IIunV15.pdf