Projective boundedness and convolution of Fréchet measures

Fréchet measures of order $n$ ( $\mathcal{F}_n$-measures) are the measure-theoretic analogues of bounded $n$-linear forms on products of $C_0(K)$ spaces. In an LCA setting, convolution of $\mathcal{F}_2$-measures is always defined, while there exist $\mathcal{F}_3$-measures whose convolution cannot be defined. In a three-dimensional setting, we demonstrate the existence of an $\mathcal{F}_2$-measure which cannot be convolved with arbitrary $\mathcal{F}_3$-measures.