For $x$ a self adjoint element of a $C^*$ algebra it is equivalent:
1. $x$ has non negative spectrum
2. $x$ has a self adjoint square root $x=y^*y$x=y^2$3.$x$is a finite sum of squares$x=\sum {a_i}^*a_i$in this case$x$is indeed called positive. 1 [made Community Wiki] For$x$a self adjoint element of a$C^*$algebra it is equivalent: 1.$x$has non negative spectrum 2.$x$has a square root$x=y^*y$3.$x$is a finite sum of squares$x=\sum {a_i}^*a_i$in this case$x\$ is indeed called positive.