# Prove the uniqueness of the hereditary $C^*$ algebra generated by a positive element

The following question is from $$C^*$$-Algebras by Example written by Kenneth R. Davidson. The original question is the Problem I.11.

$$\mathit{Definition}:$$ Say $$\mathcal{W}$$ is a $$C^*$$-subalgebra of a $$C^*$$-Algebra $$\mathcal{U}$$ and $$A, B \in \mathcal{U}, 0 \leq A \leq B$$. We call $$\mathcal{W}$$ hereditary iff $$A \in \mathcal{W}$$ whenever $$B \in \mathcal{W}$$

Given a $$C^*$$-Algebra $$\mathcal{U}$$ and a positive element $$A$$, I am asked to show that $$\overline{A\,\mathcal{U}A}$$ is THE hereditary $$C^*$$-subalgebra generated by $$A$$. I showed that it is hereditary but fail to show it is unique. Could anyone provide me some hints? In general if $$W$$ is a element from a $$C^*$$-subalgebra generated by $$A$$, can we know how $$W$$ look like?

For the second part of the question, which asks every separable hereditary $$C^*$$-subalgebra of $$\mathcal{U}$$ has this form. If we let $$\mathcal{W}$$ be a separable hereditary $$C^*$$-subalgebra, I might need to show $$\mathcal{W} = E_n\,\mathcal{U}\,E_n$$ for some $$n \in \mathbb{N}$$ where $$\{E_k\}_{k \in \mathbb{N}}$$ is a increasing sequence of positive elements that forms an approximation identity. I do not know how to show that given a fixed $$k \in \mathbb{N}, E_n \in \overline{E_k\,\mathcal{U}\,E_k}\,\forall n \in \mathbb{N}$$.

• In case you aren't aware, the notation for a $C^*$-algebra in Davidson's book is \mathfrak A, or $\mathfrak A$ Commented May 16, 2020 at 17:22
• Thank you for pointing that out. I never know that is an $A$. Commented May 16, 2020 at 21:36
• No problem, I thought it was a $U$ for the longest time as well, until someone pointed it out to me. Thought I'd pass it on. Commented May 16, 2020 at 22:39

To say that $$\overline{A\mathfrak AA}$$ is the hereditary $$C^*$$-subalgebra of $$\mathfrak A$$ generated by $$A$$, we need to show two things: That $$\overline{A\mathfrak AA}$$ is a hereditary $$C^*$$-subalgebra, and that any hereditary $$C^*$$-subalgebra $$\mathcal W$$ of $$\mathfrak A$$ containing $$A$$ contains $$\overline{A\mathfrak AA}$$. You claim to have shown the first part. To show the second can be shown by proving that if $$X\in \mathfrak A$$, then $$AXA\in\mathcal W$$. A hint to show this: It suffices to assume $$X$$ is positive, in which case you can use the identity $$AXA\leq \|X\|A^2$$.
To show the second part of the question, given your approximate identity $$\{E_n\}$$, let $$A=\sum_{n=1}^\infty E_n/2^n$$, and show that $$\mathcal W=\overline{A\mathfrak A A}$$.
• Thank you for your answers. Could you explain how to show that $\mathcal{W}$ is also contained in $\overline{A\,\mathfrak{A}\,A}$? My question to your hint for the second part is the same. It is clear that $A \in \mathcal{W}$ and hence $\overline{A\,\mathfrak{A}\,A}\,\subset\,\mathcal{W}$ but how about the other direction? Commented May 16, 2020 at 21:46
• Note that $E_n/2^n\leq A$, so $E_n\in\overline{A\mathfrak AA}$ for all $n$. If now $B\in\mathcal W$, then $B=\lim_nE_nBE_n$, and it follows that $B\in\overline{A\mathfrak AA}$. Commented May 17, 2020 at 2:28