Hermitian Element of C*-Algebra Decomposes into Positive Elements/Corollary

Corollary

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

Let $b \in A$.

Then there exists positive elements $b_1, b_2, b_3, b_4 \in A$ such that:

$b = b_1 - b_2 + i \paren {b_3 - b_4}$

Proof

From Element of *-Algebra Uniquely Decomposes into Hermitian Elements, there exists Hermitian elements $\alpha, \beta \in A$ such that:

$b = \alpha + i \beta$

From Hermitian Element of C*-Algebra Decomposes into Positive Elements, there exists positive elements $b_1, b_2, b_3, b_4 \in A$ such that:

$\alpha = b_1 - b_2$

and:

$\beta = b_3 - b_4$

Hence:

$b = b_1 - b_2 + i \paren {b_3 - b_4}$

$\blacksquare$