Some concepts of $C^*$-algebra generalized from linear algebra. Can anyone help me to check if they are correct, and give some examples?

Question

A Banach algebra is just a Banach space equipped with an operation of multiplication defined such that $\|a b\| \le \|a\|\|b\|$ for all $a,b$ in it. If, in addition, there exists an identity, then it becomes a unital Banach algebra. We may consider the space of all $n$ by $n$ matrices as a unital Banach algebra.

A $C^*$-algebra is a Banach algebra equipped with a so-called conjugate-linear involution, say $*$, such that $(ab)^*=b^*a^*$ and $\|a^*a\|=\|a\|^2$. Again, it is called unital $C^*$-algebra if there is an identity. The space of all $n$ by $n$ matrices also belongs to a unital $C^*$-algebra.

An element $a$ in a unital $C^*$-algebra is called

1. isometry if $a^* a = 1$, e.g., ... any appropriate example?
2. coisometry if $a a^* = 1$, e.g., ... any appropriate example?
3. unitary if $a^* = a^{-1}$, e.g., unitary matrix.
4. normal if $a^*a = a a^*$, e.g., ... any appropriate example?
5. self-adjoint if $a = a^*$, e.g., Hermitian matrix.

Answer 1

In a finite-dimensional C$^*$-algebra, isometry, coisometry, and unitary are the same. Unitary implies normal. Selfadjoint implies normal. One can get all examples in one by taking $a$ to be the identity matrix.

A non-selfadjoint unitary is for instance $$u=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}.$$