What deos $\mathbb{C}I$ mean?

Question

I've come across this expression $$ \mathbb{C}I $$ while studying operators algebras, $C^*$-algebras and AF-algebras, concretely.

• In Kenneth R. Davidson's book $\boldsymbol{C^*}$-algebras by example page 76, example III.2.3.

  Consider the $C^*$-algebra $\mathbb{C}I+\mathscr{k}\ldots$

• In Bruce Blackadar's book Operator algebras page 102, proposition II.6.1.8 about representation of $C^*$-algebra $A$

  (ii) $\pi(A)' = \mathbb{C}I$

Can anybody tell me, what $\mathbb{C}I$ in this context means? I could not find the meaning in Davidson's book. Blackadar list only $\mathbb{C}G$ where $G$ is a group and $\mathbb{C}G$ denotes group algebra.

I assume $\mathbb{C}$ means complex numbers as always and $I$ might stand for the identity element. Then $\mathbb{C}I$ might stand for all multiples of the identity with each $\alpha \in \mathbb{C}$.

Answer 1

Yes, $$ \mathbb C\,I=\{\lambda\,I:\ \lambda\in\mathbb C\}. $$ It is coherent with the notation $$ AB=\{ab:\ a\in A,\ b\in B\}, $$ and with $$ A+B=\{a+b:\ a\in A,\ b\in B\}, $$ etc.