Bounds on Projection in Unital C*-Algebra

Theorem

Let $\struct {A, \ast, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.

Let $\le_A$ be the canonical preordering of $A$.

Let $p$ be a projection on $A$.


Then we have:

${\mathbf 0}_A \le_A p \le_A {\mathbf 1}_A$


Proof

Since $p$ is a projection, it is Hermitian.

From Spectrum of Projection in *-Algebra: Corollary, we have:

$\map {\sigma_A} p \subseteq \set {0, 1}$

From Bounds on Hermitian Element of Unital C*-Algebra in terms of Bounds on Spectrum, we have:

${\mathbf 0}_A \le_A p \le_A {\mathbf 1}_A$

$\blacksquare$

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