Definition 1

Let A be a normed algebra. For $a\in A$, denote by $\boldsymbol{\gamma }_a$ and $\boldsymbol{\delta }_a$ the mappings $x\mapsto ax$, and $x\mapsto xa$ from A into A. The mapping $\boldsymbol{\gamma }:a\mapsto \boldsymbol{\gamma }_a$ is a representation of A in A, called the left regular representation of A. The mapping $\boldsymbol{\delta }:a\mapsto \boldsymbol{\delta }_a$ is a representation of the opposite algebra of A in A, called the right regular representation of A.