A band is a semigroup whose elements are idempotents. The paper is motivated by some open problems concerning linear representations of bands. A counter example to a conjecture stated in  is given. On the other hand, it is proved that there exist linear bands with two components such that the associated commutative K-algebra constructed in  cannot be embedded into any Noetherian ring, even in the case where this algebra is reduced. © 2009 Departament de Matemàtiques Universitat Autònoma de Barcelona.