The study of operator algebras on Hilbert spaces, and $C^{\ast }$-algebras in particular, is one of the most active areas within Functional Analysis. A natural generalization of these is to replace Hilbert spaces (which are $L^2$-spaces) with $L^p$-spaces, for $p\in [1,\infty )$. The study of such algebras of operators is notoriously more challenging, due to the very complicated geometry of $L^p$-spaces by comparison with Hilbert spaces. We give a modern overview of a research area whose beginnings can be traced back to the 50’s, and that has seen renewed attention in the last decade through the infusion of new techniques. The combination of these new ideas with old tools was the key to answer some long standing questions. Among others, we provide a description of all unital contractive homomorphisms between algebras of $p$-pseudofunctions of groups.

希尔伯特空间上的算子代数研究，以及 $C^{＊}$代数尤其是泛函分析中最活跃的领域之一。这些的自然概括是替换希尔伯特空间（它们是${升}^2$-空格）与 ${升}^{磷}$-空格，对于 $磷\in [1,\infty )$. 众所周知，对算子的这种代数的研究更具挑战性，因为非常复杂的几何${升}^{磷}$-spaces 与 Hilbert 空间比较。我们对一个可以追溯到 50 年代的研究领域进行了现代概述，并且在过去十年中通过新技术的注入重新引起了人们的关注。这些新想法与旧工具的结合是回答一些长期存在的问题的关键。其中，我们提供了代数之间的所有单位收缩同态的描述$磷$- 组的伪功能。