We introduce and investigate a new sectional property for topological sequence spaces, which we call the KB property. It turns out that multipliers into l ∞ and multiplication by l 1 play a very similar role for KB as multipliers into bs and multiplication by bv 0 do for the well-known sectional property AB. However, we also encounter significant differences. One of these differences motivates the study of l 1 -invariant sequence spaces, that is, sequence spaces E with l 1 E = E. 2001 Academic Press