Self-adjoint positive unitary and normal elements

Let $A$ be a complex C*-algebra (C star algebra) and let $a\in A$.

• $a$ is self-adjoint when $a^{\ast }=a$;
• $a$ is normal when $a^{\ast }a=a{a}^{\ast }$;
• $a$ is positive when $a=b^{\ast }b$ for some $b\in A$. (At this stage positivity is a purely algebraic condition; its pointwise description as nonnegativity of the transformed function is proved later, from Nonunital commutative Gelfand Naimark.)

If in addition $A$ is unital, with unit $1$ (Unital Banach algebra), then:

• $a$ is unitary when $a^{\ast }a=a{a}^{\ast }=1$.

No unitary notion is claimed here for a genuinely nonunital C*-algebra, where the equation $u^{\ast }u=u{u}^{\ast }=1$ has no solution since a noninvertible element cannot satisfy it and a left identity in a C*-algebra is an identity, forcing unitality.

• Self-adjoint elements are normal, since $a^{\ast }a=aa=a^2=a{a}^{\ast }$ when $a^{\ast }=a$; and $b^{\ast }b$ is self-adjoint for every $b$, because $(b^{\ast }b{)}^{\ast }=b^{\ast }b$.
• The real and imaginary parts. Every $a$ can be written $a=s+it$ with $s:=\frac{1}{2}(a+a^{\ast })$ and $t:=\frac{1}{2i}(a-a^{\ast })$ self-adjoint; both identities are algebraic, and they are the decomposition used in Characters on a unital commutative C star algebra preserve star.
• Positivity is preserved by star-homomorphisms, since $\phi (b^{\ast }b)=\phi (b{)}^{\ast }\phi (b)$; no positivity notion outside the given algebra is imported.