A $d$-algebra whose product does not extend to its Dedekind completion

Abstract

We construct a Banach lattice $d$-algebra with a strong order unit whose product has no positive associative bilinear extension to its Dedekind completion. This gives a negative answer to a question of Wickstead and to the $d$-algebra case of a question of Huijsmans. It also disproves an extension theorem of Chil. The example shows that norm completeness and a strong order unit do not ensure that multiplication extends to the Dedekind completion.