6
$\begingroup$

Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\mathsf{BV}(\mathbb T)$ onto $\ell_1(\mathbb Z)$ (with convolution)?

My motivation comes from the fact that Fourier coefficients of functions in $\mathsf{BV}(\mathbb T)$ vanish at infinity however I am not sure if this is directedly related.

$\endgroup$
7
  • $\begingroup$ Ask Michal Wojciechowski. $\endgroup$ Jan 3, 2019 at 13:36
  • $\begingroup$ Do you require the homomorphism to respect the symmetry of the circle in any way? (viewing $\ell_1({\bf Z})$ as ${\rm A}({\bf T})$)? $\endgroup$
    – Yemon Choi
    Jan 3, 2019 at 15:35
  • $\begingroup$ I am curious if there exists any homomorphism but a more symmetric one would be better $\endgroup$ Jan 3, 2019 at 15:47
  • $\begingroup$ Do you want your homomorphism to be linear? $\endgroup$ Jan 5, 2019 at 17:22
  • 1
    $\begingroup$ @Adrian, yes but maybe BV quotients onto some subalgebra that has such property? Homs of Banach algebras are linear and multiplicative. $\endgroup$ Jan 7, 2019 at 14:38

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.