Definition 6.4.2 (Real and Imaginary Parts).label Let $A$ be an involutive Banach algebra then each $x\in A$ is represented uniquely as
\begin{align*}x=x_{1}+ix_{2}\end{align*}
for some self-adjoint elements $x_{1},x_{2}\in A$ where
\begin{align*}x_{1}=\frac{1}{2}\parens{x+x^*}&&x_{2}=\frac{1}{2i}\parens{x-x^*}\end{align*}
called the real part and imaginary part of $x$ respecitvely. In this form $x$ is normal if and only if $x_{1}$ and $x_{2}$ commute. We denote
\begin{align*}A_{h}=\curl{x\in A:x\text{ is self-adjoint}}&&A_{u}=U(A)=\curl{x\in A:x\text{ is unitary}}\end{align*}
Then $A_{h}$ is a real Banach space such that
\begin{align*}A=A_{h}+iA_{h}\end{align*}
We call the closed subgroup $U(A)$ of the general linear group $G(A)$ of $A$ the unitary group of A