# definition of complete vector lattice

Suppose $M$ is a von-Neumann algebra, $L=M\cap M'$ is the centre of $M$. The last line on page 29, C*-algebras and their automorphism groups, states that the self-adjoint part $L_{sa}$ of $L$ is a complete vector lattice.

Found in wekipeida, a complete lattice is a lattice in which each subset have both a supremum and a infimum.

However, even for $M=\mathbb{C}$, $L_{sa}=\mathbb{R}$ is not complete according to the above definition, as $\mathbb{R}$ itself has neither supremum or infimum.

Well, $(\Bbb R, \min, \max)$ is indeed not a complete lattice, but it is almost complete: $\bar{\Bbb R}:=\Bbb R\cup\{\pm\infty\}$ is complete, or put in other words,