Reflections on coproducts for non-unital algebras

A coproduct on a vector space $A$ is defined as a linear map $\Delta:A\to A\otimes A$ satisfying coassociativity $(\Delta\otimes\iota)\Delta=(\iota\otimes\Delta)\Delta$. We use $\iota$ for the identity map. If $G$ is a finite group and if $A$ is the space of all complex functions on $G$, a coproduct on $A$ is defined by $\Delta(f)(p,q)=f(pq)$ where $p,q\in G$. We identify $A\otimes A$ with complex functions on the Cartesian product $G\times G$. Coassociativity follows from the associativity of the product in $G$. Unfortunately, sometimes this notion of a coproduct is not the appropriate one. In this note, we consider the case of an algebra $A$, not necessarily unital but with a non-degenerate product. Now a coproduct is a linear map from $A$ to $M(A\otimes A)$, the multiplier algebra of $A\otimes A$. Unfortunately, it is no longer possible to express coassociativity in its usual form as the maps $\Delta\otimes\iota$ and $\iota\otimes \Delta$, defined on $A\otimes A$, may no longer be defined on the multiplier algebra $M(A\otimes A)$. Similar problems occurs when we want to define a useful notion of a coaction in the case of non-unital algebras. We discuss this in another paper. Not all the results we present in this paper are new. We provide a number of references to the original papers where some of this material is treated. However, in the original papers, results are not always found in an organized form and we hope to improve that here. Further a few solutions to some open questions are included as well as some more peculiar examples. Finally, we discuss some open problems and possible further research.