definition of $C^*$-subalgebra

Question

Every time I come across something with a $C^*$-subalgebra, I am confused. There is no definition in my course notes for this. Could someone please help me with the definition, i.e. the things I have to check to prove something is a $C^*$-subalgebra. Furthermore, when I come across $C^*$-subalgebra there often is some argument about it being closed. The way I see it, it looks like they want to argue that since some subset it closed it must be $C^*$-subalgebra. Could someone please help me?

Answer 1

At this point in your mathematical career, you should be able to spell out the definition yourself.

Indeed, let $A$ be a $C^*$-algebra and $B \subseteq A$ a subset of $A$. We call $B$ a $C^*$-subalgebra of $A$ if $B$ becomes a $C^*$-algebra for the operations induced by $A$.

In particular, we require:

• $x + \lambda y \in B$ for all $x,y \in B$ and all $\lambda \in \mathbb{C}$.
• $xy \in B$ for all $x,y \in B$.

Note that these conditions mean that $B$ is a subalgebra of $A$.

• $x^* \in B$ when $x \in B$.

This condition implies that $B$ is a $*$-subalgebra of $A$.

• $B$ is complete as a normed space, or equivalently it is a closed subset of $A$.