Definition 44.4.7 (Simple).label Let $(M, \tau)$ be a tracial von Neumann algebra and $x \in M$, then $x$ is simple if there exists $\seq{\lambda_j}\subset \complex$ and mutually orthogonal projections $\seqf{p_j}\subset \text{Proj}(M)$ such that $x = \sum_{j = 1}^{n} \lambda_{j} p_{j}$.