Positive and negative parts of a self-adjoint element

Definition

Let $a$ be a self-adjoint element of a $C^*$ -algebra $A$ , and put $|a|=(a^2)^{1/2}$ . The positive part and negative part of $a$ are

a_+=\frac{|a|+a}{2}, \qquad a_-=\frac{|a|-a}{2}.

Both belong to the positive cone of $A$ , and they satisfy

a=a_+-a_-, \qquad |a|=a_++a_-, \qquad a_+a_-=0.

They are the unique positive elements $b,c\in C^*(a)$ with $a=b-c$ and $bc=0$ . This decomposition is also called the Jordan decomposition of $a$ .

Functional-calculus description

Under the continuous functional calculus for $a$ ,

a_+=f_+(a),\qquad a_-=f_-(a),

where $f_+(t)=\max(t,0)$ and $f_-(t)=\max(-t,0)$ . Hence $a_-$ is the positive part of $-a$ , and both parts commute with every element that commutes with $a$ . Their orthogonality follows pointwise from $f_+(t)f_-(t)=0$ Murphy, §2.2.

Order and norm consequences

The formulas give $a\leq a_+$ , $-a\leq a_-$ , and

\|a\|=\max\{\|a_+\|,\|a_-\|\}.

Moreover, $a$ is positive exactly when $a_-=0$ , and $a$ is negative exactly when $a_+=0$ . Applying a $*$ -homomorphism commutes with taking positive and negative parts because $*$ -homomorphisms commute with continuous functional calculus.

Examples and cautions

For a Hermitian matrix, $a_+$ keeps the positive eigenvalues and replaces the negative ones by zero; $a_-$ replaces each negative eigenvalue $\lambda$ by $-\lambda$ . For a real-valued function, the construction is pointwise. The notation $a_-$ denotes a positive element, not the nonpositive function $\min(a,0)$ . The decomposition also should not be confused with the Jordan decomposition of a functional or measure, which requires a separate uniqueness theorem.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. Elsevier DOI record. Relevant: §2.2 on positive elements and continuous functional calculus.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. Elsevier DOI record. Relevant: §§1.4–1.5 on positivity, order, and functional calculus.