Support projection of a normal positive functional

For a projection $p\in M$, the following conditions are equivalent: $\varphi(p)=\varphi(1)$, $\varphi(1-p)=0$, and $\varphi(x)=\varphi(pxp)$ for every $x\in M$. Normality is what permits the supremum of all $\varphi$-null projections to remain null. Consequently, $\varphi$ is faithful exactly when $s(\varphi)=1$. These support properties are part of the standard theory of normal positive forms Takesaki, treatment of normal positive functionals.

For $M=B(H)$ and $\varphi(x)=\operatorname{Tr}(\rho x)$, where $\rho$ is positive and trace class, $s(\varphi)$ is the orthogonal projection onto $\overline{\operatorname{ran}\rho}$. In particular, the support of the vector functional $x\mapsto\langle x\xi,\xi\rangle$ is the rank-one projection onto $\mathbb C\xi$. In a commutative model $M=L^\infty(X,\mu)$, the support is multiplication by the essential support of the density representing $\varphi$.

The support projection need not be central. The central support of $\varphi$ is the least central projection dominating $s(\varphi)$, and can be strictly larger. Support also differs from the support of an individual element: $s(\varphi)$ records which corner of $M$ the functional detects.

1. Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: the chapter on normal positive functionals, supports, and polar decomposition.