A band is a semigroup whose elements are idempotents. The
paper is motivated by some open problems concerning linear representations
of bands. A counterexample to a conjecture stated
in F. Cedó and J. Okniński, "Faithful linear representations of bands,"
Publ. Mat. 53 (1) (2009), 119–140 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 Cedó and Okniński cannot be embedded
into any Noetherian ring, even in the case where this algebra is
reduced.