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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.

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  • $\begingroup$ Ask Michal Wojciechowski. $\endgroup$ – Piotr Hajlasz Jan 3 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 at 15:35
  • $\begingroup$ I am curious if there exists any homomorphism but a more symmetric one would be better $\endgroup$ – Maciej Ciechowski Jan 3 at 15:47
  • $\begingroup$ Do you want your homomorphism to be linear? $\endgroup$ – Adrián González-Pérez Jan 5 at 17:22
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    $\begingroup$ @Adrian, yes but maybe BV quotients onto some subalgebra that has such property? Homs of Banach algebras are linear and multiplicative. $\endgroup$ – Maciej Ciechowski Jan 7 at 14:38

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