Positive square root in a C*-algebra

Let $A$ be a $C^*$ -algebra and let $a\in A$ be a positive element. There exists a unique positive element $b\in A$ satisfying $b^2=a$ . This element is the positive square root of $a$ and is denoted $a^{1/2}$ . It belongs to the commutative $C^*$ -subalgebra generated by $a$ and is obtained from the continuous functional calculus applied to $t\mapsto\sqrt t$ on $\sigma(a)\subseteq[0,\infty)$ . The adjective “positive” is essential: an element can have other, nonpositive square roots.

Basic properties

The construction satisfies

\|a^{1/2}\|=\|a\|^{1/2},\qquad (a^{1/2})^*a^{1/2}=a.

It is order preserving: $0\leq a\leq b$ implies $a^{1/2}\leq b^{1/2}$ . Every $*$ -homomorphism $\pi:A\to B$ preserves the construction, since uniqueness gives $\pi(a^{1/2})=\pi(a)^{1/2}$ . The existence, uniqueness, and monotonicity properties follow from functional calculus.

Examples and a near-miss

For a positive matrix $a=u\operatorname{diag}(\lambda_1,\ldots,\lambda_n)u^*$ ,

a^{1/2}=u\operatorname{diag}(\sqrt{\lambda_1},\ldots,\sqrt{\lambda_n})u^*.

For a nonnegative function $f\in C_0(X)$ , the positive square root is the pointwise function $\sqrt f$ . A self-adjoint element with negative spectrum has no positive square root, because the square of a positive element is positive.

Multiplicative caution

If positive $a$ and $b$ commute, then $ab$ is positive and $(ab)^{1/2}=a^{1/2}b^{1/2}$ . Without commutativity, $ab$ need not even be self-adjoint, so this formula is generally meaningless as an identity of positive square roots.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: the chapter on positive elements and continuous functional calculus.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the introductory treatment of positivity and order.