I'm reading Robinson and Bratteli's book for the first time and, in chapter two, there are some definitions about what is a Banach algebra, a $*$-algebra and a Banach and $C^{*}$-algebra. Basically, one starts with an algebra $\mathcal{U}$, that is, a vector space with a product operation $AB$, $A,B \in \mathcal{U}$ which is assumed associative and distributive. Then $\mathcal{U}$ is a normed algebra if it is equipped with a norm $||\cdot ||$ which further satisfy $||AB|| \le ||A||||B||$. This induces a uniform (or metric) topology and if it is complete with respect to this topology, it is called a Banach algebra.
In addition, one defines an involution as a map $\mathcal{U} \ni A \mapsto A^{*} \in \mathcal{U}$ which satisfies: (a) $A^{**} = A$, (b) $(AB)^{*} = B^{*}A^{*}$ and $(\alpha A + \beta B)^{*} = \overline{\alpha}A^{*}+\overline{\beta}B^{*}$, where $\alpha, \beta \in \mathbb{C}$ and the overline denotes complex conjugate. An algebra equipped with an involution is called a $*$-algebra; if it is a Banach algebra and satisfies $||A^{*}|| = ||A||$, then it is called a Banach $*$-algebra. Finally, a $C^{*}$-algebra $\mathcal{U}$ is a Banach $*$-algebra which satisfies, for every $A \in \mathcal{U}$, the condition $||A^{*}A|| = ||A||^{2}$.
It looks quite trivial to me that $\mathbb{C}$, with the involution defined as the complex conjugate itself, is a Banach $C^{*}$-algebra (and, thus, a Banach $*$-algebra). However, this example is not listed in the books. Other lecture notes on the internet do not include such example as well. Also, $\mathbb{R}$ with the involution = identity map (can it be done?) seems also a $C^{*}$-algebra, but it is also not mentioned.
Question: Are these trivial examples of $C^{*}$-algebras?
$\mathbb{C}$ is quite obviously a $C^*$-algebra for the complex conjugation. Probably your book mentions that matrices $M_n(\mathbb{C})$ over $\mathbb{C}$ are a $C^*$-algebra, and this is a special case by taking $n=1$.
On the other hand, $\mathbb{R}$ is not a $C^*$-algebra. A $C^*$-algebra has to be a complex vector space in the first place, and it is unclear how you would define the scalar multiplication. Even if you managed to find a suitable scalar multiplication, the involution $*$ you propose cannot work because it does not satisfy $$(\lambda a)^* = \overline{\lambda} a^*.$$