Strictly positive element in a von Neumann algebra

Question

If $A$ is a $C^*$-algebra and $a\in A^+$. $a$ is said to be strictly positive if $aA$ is dense in $A$.

Let $M$ be a von Neumann algebra. Does there exist another sufficient and necessary condition to ensure that there exists a strictly positive element in $M$?

Answer 1

Note that in unital $C^*$-algebras, a positive element $a$ is strictly positive if and only if it is invertible. Now, since von Neumann algebras are unital, this would be such a condition.

To see a proof of this, confer this question:

strictly positive elements in $C^*$-algebra