6
$\begingroup$

This is a cross-post from math.stackexchange.com

What can be said about elements $a$ of a C*-algebra which fulfil the 'symmetric C* property' $$\| a^\ast a+ aa^\ast\|=2\| a\|^2$$

I'd guess that this is not a general property, but I don't have a good idea which elements (except for self-adjoint and normal ones, of course) should satisfy this.

$\endgroup$
1
  • $\begingroup$ That's not much of a wait... $\endgroup$ – David Roberts Mar 19 '18 at 8:11
7
$\begingroup$

This is hardly a complete answer.

As you say, all normal operators $a \in B(\mathcal H)$ satisfy this equality. Moreover, so does any element $c = a\oplus b \in B(\mathcal H\oplus \mathcal H)$ where $a$ is normal and $\|b\| \leq \|a\|$. This is because $$ \|c^*c + cc^*\| = \max\{\|a^*a + aa^*\|, \|b^*b + bb^*\|\} = 2\|a\| = 2\|c\|. $$

On the other hand, this is certainly not a general property. For instance, if $a = \left[\begin{matrix} 0 & 1\\ 0&0\end{matrix}\right]$ then $$\|a^*a + a^*a\| = \left\|\left[\begin{matrix} 1 & 0\\ 0&1\end{matrix}\right]\right\| = 1 \quad \textrm{but} \quad 2\|a\|^2 = 2.$$

And on the third hand, $a = \left[\begin{matrix} 0 & 1 & 0 \\ 0& 0& 1\\ 0&0&0\end{matrix}\right]$ satisfies this equality. This makes me doubt that there will be any nice characterization.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.