标题： 乘子代数和局部单位 显示英文标题

标题： Multiplier algebras and local units

摘要： 设$A$为任意域上的代数。 我们不假设$A$具有单位元。 \emph{乘子代数} $M(A)$ 是与$A$相关的含单位元的代数。 如果我们要求$A$中的乘积是非退化的（作为双线性形式），则乘子代数可以表征为包含$A$作为本质理想的最大的代数。 我们回顾基本定义，并提供关于这一概念的更多信息。 我们将乘子代数$M(A)$赋予{\it 紧拓扑}。 然后我们证明$A$在$M(A)$中稠密当且仅当$A$存在局部单位。 我们包括了各种例子。 特别是，我们对乘子霍普夫代数、代数量子群、代数量子超群、弱乘子霍普夫代数和代数量子群胚的基本代数感兴趣。 在所有这些情况下，可以证明这些代数具有局部单位。 我们还包括了一些来自余弗罗贝尼乌斯余代数的例子。 对于本文中讨论的大部分内容，只有代数的环结构起作用。 因此，我们在这里为环发展了该理论。 但它们不需要具有乘法结构的单位元。 显示英文摘要

摘要： 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.