Clarification regarding a Lemma on C* algebra of compact operators from Davidson

Question

I am having problem understanding the following lemma from "C*-Algebra by Examples", the book by Kenneth R. Davidson

I am reading this on my own. I am stuck here for a long time. Here are my confusion:

1. What is a irreducible C*- subalgebra? (So far in the book "irreducible" has been used in the context of representations)
2. In the proof, what does it mean by "appropriate dimension" of $\mathcal H$ ? Isn't it already determined by the given statement.
3. In the proof, unit vector "$e$" is chosen which I assume it is an element of $\mathfrak A$. I am not sure if that is okay, as $Ae=x\in \mathcal H$ and
   $Be=y\in \mathcal H$ , I couldn't understand how that is possible.
4. We choose $x,y\in \mathcal H$ but what is the meaning of $y^*$ then?

I am not sure if there is any typo in the proof. Please help me understand this. If you can suggest some reference where this topic is discussed that would be beneficial as well.

Thank you for you time.

Answer 1

1. $\mathcal{A} \subseteq B(H)$, so just take the natural inclusion $\mathcal{A} \to B(H)$ to be the representation.

2. "Appropriate dimension," in the proof, he is handling irreducible subalgebras of $K(H)$ for any possible dimension of $H$ (separable). I think he just wants to emphasize that he is not, for example, restricting to $H$ being infinite-dimensional.

3. $e \notin \mathcal{A}$. $E \in \mathcal{A} \subseteq B(H)$ is a projection of rank-1, so $e \in H$ is some unit vector in the range of $E$ (which is uniquely determined up to a modulus 1 complex number).

4. $y^*$ is the linear functional on $H$ defined by $y^*(h) = \langle h,y\rangle$, i.e., the inner product.