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

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 $$\alpha \in \mathbb{C}$$.

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.