# Homomorphisms from BV

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.

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