Multiplier algebras and local units

Abstract

Let $A$ be an algebra over any field. We do not assume that $A$ has an identity. The \emph{multiplier algebra} $M(A)$ is a unital algebra associated to $A$. If we require the product in $A$ to be non-degenerate (as a bilinear form), the multiplier algebra can be characterized as the largest algebra containing $A$ as an essential ideal. We recall the basic definitions and provide some more information about this notion. We endow the multiplier algebra $M(A)$ with the {\it strict topology}. Then we show that $A$ is dense in $M(A)$ if and only if there exist local units in $A$. We include various examples. In particular, we are interested in the underlying algebras of multiplier Hopf algebras, algebraic quantum groups, algebraic quantum hypergroups, weak multiplier Hopf algebras and algebraic quantum groupoids. In all these cases, one can show that the algebras have local units. We have also included some examples arising from co-Frobenius coalgebras. For most of the material treated in this note, it is only the ring structure of the algebra that plays a role. For this reason, we develop the theory here for rings. But they are not required to have an identity for the multiplicative structure.