0
$\begingroup$

Assume $\mathcal{A}$ is a unital $C^*$ algebra and consider some positive-definite element $\Psi\in M_n(\mathcal{A})$. Can we say something about $C^*(\langle \Psi^{-\frac{1}{2}}E_{i,i}\Psi^{\frac{1}{2}}:1\leq i\leq n\rangle)\subset M_n(\mathcal{A})$?

This question came up during my research. It may be very specific, but more generally, if $\mathcal{A}$ is a $C^*$ algebra and $x\in\mathcal{A}$ some positive element, are there any known results about the $C^*$ subalgebra $\mathcal{B}$ generated by $\{xa_1x^{-1},\ldots,xa_nx^{-1}\}$ with $a_1,\ldots,a_n\in\mathcal{A}$?

I tried looking at some combinations of the elements (including taking $*$, products and sums) but couldn't come up with something that will help me understand the $C^*$ algebra I get.

Any help would be appreciated.

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ What would you like to say? $\endgroup$ Commented May 13, 2024 at 23:46
  • 2
    $\begingroup$ "Are there any known results" is too broad. You need to make the question more specific; as it stands I have no idea what kind of result you're looking for. $\endgroup$ Commented May 14, 2024 at 0:26

0

Reset to default

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.