Central support of a projection - Imported Knowlpedia Content

Definition

Let $p$ be a projection in a von Neumann algebra $M$ . The central support of $p$ , denoted $c_M(p)$ , is the least projection $z$ in the center $Z(M)$ such that $p\leq z$ . It exists because arbitrary infima of projections exist in $M$ , and it is given by

c_M(p)=\bigwedge\{z\in Z(M):z\text{ is a projection and }p\leq z\}.

Thus $c_M(p)=1$ means that no nonzero central summand of $M$ is orthogonal to $p$ ; one then says that $p$ has full central support. It depends on the ambient algebra, which is recorded by the subscript.

Equivalent descriptions

The central support is also the join of the unitary conjugates of $p$ :

c_M(p)=\bigvee_{u\in\mathcal U(M)}upu^*.

Equivalently, the ultraweakly closed two-sided ideal generated by $p$ is $Mc_M(p)$ . These descriptions express that central support records every central summand reached by $p$ , while forgetting its size inside each summand.

Basic properties

Central support is monotone: $p\leq q$ implies $c_M(p)\leq c_M(q)$ . It is invariant under unitary conjugacy and, more generally, under Murray–von Neumann equivalence. For a family of projections,

c_M\left(\bigvee_i p_i\right)=\bigvee_i c_M(p_i).

If $z$ is already central, then $c_M(z)=z$ .

Factors and direct sums

In a factor, every nonzero projection has central support $1$ , because the only central projections are $0$ and $1$ . In a direct sum $M=\bigoplus_\alpha M_\alpha$ , the central support of $p=(p_\alpha)$ is the central projection whose $\alpha$ -component is $1$ exactly when $p_\alpha\neq0$ . Central support therefore detects which factor summands occur, not the ranks of the components.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II, American Mathematical Society, 1997. Publisher record. Relevant: §6.3 on comparison and central carriers of projections.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter V on projections, central support, and factors.