action of $C^*$ algebras

Question

In group theory,given a group $G$,we can define a group action of $G$ on $G$.Can we define a $C^*$ algebra action similarly?To be more precise,suppose we have a $C^*$ algebra $A$,can we define a map from $A\times A$ to $A$ which satisfies some conditions.

Answer 1

You say

To be more precise,suppose we have a $C^∗$ algebra A,can we define a map from $A\times A$ to $A$ which satisfies some conditions.

but this is not terribly precise. What conditions do you want to consider?

Furthermore, there isn't a unique way a group acts on itself. You can consider left (resp. right) multiplication by a group element, for which I suppose the most naturally analogous $C^*$-algebra action would probably be left (resp. right) multiplication (or addition) by an element of the algebra. Additionally, you can consider an action of a group on itself by conjugation, for which there isn't a naturally occuring $C^*$-algebra analogue.