Tensor Products of Almost $f$-Algebras: A Counterexample

We show that the natural tensor product multiplication of two Archimedean almost $f$-algebras does not, in general, extend to their Fremlin tensor product. The two algebras in the counterexample have the same underlying vector lattice $C[0,1]$. Their multiplications are explicit, positive and nilpotent of index three. The obstruction is the fact that the function $(s,t)\mapsto e^{st}$ does not belong to $C[0,1]\text{ftensor}C[0,1]$. The same construction gives Banach almost $f$-algebras and has consequences for previously published assertions on Riesz and Fremlin projective tensor products of almost $f$-algebras.