C$^*$-Algebra Beginner's Questions

Question

I am starting with $C^*$ algebras. There are some notations that I don't understand. Please help me.

1. What does the identity representation of $C^*$ algebras mean?
2. Let $A$ be $C^*$ algebra generated by the irreducible operator A and the identity. may be representation is also irreducible?
3. For the operator $T$, then $T^{(n)}$ means that?
4. For the Hilbert space, the notation $H^{(n)}$ means that? Sorry if it is too easy for readers. I'm new in this area.

Answer 1

1. If your C$^*$-algebra is already represented inside some $B(H)$, then the identity map (i.e. "doing nothing") is the identity representation.

2. If a represented C$^*$-algebra contains an irreducible operator, then it is of course irreducible.

3. $H^{(n)}$ is the direct sum of $n$ copies of $H$.

4. $T^{(n)}$ is the operator in $B(H^{(n)})$ that acts by $T^{(n)}(h_1,\ldots,h_n)=(Th_1,\ldots,Th_n)$.