I am looking at Arveson's book, an invitation to $C^*$ algebras. There, it is explained p. 21 ($C^*$ algebras of compact operators) that any representation of a compact $C^*$ algebra can be decomposed into direct sum of multiples of irreducible representations. Then this is applied to the $C^*$ algebra generated by a compact operator. My problem is that I don't understand what this means already for a finite dimensional Hilbert space. That is, to what topic in linear algebra is this (the decomposition of the $C^*$ algebra generated by a compact operator) related to? To compare for instance, I understand the relation between non cyclicity and multiple eigenvalues for self adjoint operators.
Thank you for any help
In my experience, the topic of subalgebras of $M_n(\mathbb C)$ is not part of the usual linear algebra curriculum.
What you need to understand first is the form that finite-dimensional C$^*$-algebras have. A finite-dimensional C$^*$-algebra $A$ is always a finite direct sum $$\bigoplus_{k=1}^m M_{n(k)}(\mathbb C).$$ The "blocks" can be identified via the minimal projections of the centre of $A$.
And because C$^*$-algebras of compact operators always have minimal projections, the same game can be played.