for all $a,b\in \mathcal{A}_0$. In this paper we consider only [*unital*]{} Banach algebras, i.e., algebras with a multiplicative unit $I$ with $\|I\|=1$.

An [*involution*]{} on a Banach algebra $\mathcal{A}_0$ is a map $a\mapsto a^*$ of $\mathcal{A}_0$ into itself satisfying

-   $(a^*)^* = a$

-   $(ab)^* = b^*a^*$

-   $(\lambda a + b)^* = \overline{\lambda} a^* + b^*$

for all $a,b\in \mathcal{A}_0$ and $\lambda \in \mathbb{C}$. A [*$C^*$-algebra*]{} is a Banach algebra with an involution such that the $C^*$-[*identity*]{} $$\|a^*a\| = \|a\|^2$$ holds for all $a\in \mathcal{A}_0$. Note that we do not make any assumption on whether $\mathcal{A}_0$ is commutative or not.

Basic examples of $C^*$-algebras are:

1.  The commutative algebra $C(\mathcal{X})$ of all continuous complex-valued functions on a compact Hausdorff space $\mathcal{X}$. Here the addition and multiplication operations are defined pointwise, and the norm is given by $\|f\|_{\infty} = \max_{t\in \mathcal{X}} |f(t)|$. The involution on $C(\mathcal{X})$ maps each function $f$ to its complex conjugate $f^*$, defined by $f^*(t) = \overline{f(t)}$ for all $t\in \mathcal{X}$.

2.  The algebra $\mathcal{B}(\mathcal{H})$ of all bounded linear operators on a complex Hilbert space $\mathcal{H}$, with the operator norm $\|T\|_{op} = \sup \|Tx\|_{\mathcal{H}} /\|x\|_{\mathcal{H}}$, where the supremum is taken over all nonzero $x\in {\mathcal{H}}$. The involution 