Consider the following construction

let with the unit circle. Define the following mappings

and

it is easy to show that is unitary and we can then define an anology of time evolution by the mapping

where , a C*-algebra.

Let where is a C*-algebra. And by class of I mean the mapping with is differentiable to any order with .

Suppose we have a function of the form

we can define the derivative of the operator valued function as satisfying the equation

I define the following mapping

Now, what I want to show is that .

I am not sure how to get this results, I think that I am going to have to use

but I cannot manipulate the expression in such a way that the above equation pops out. So I want to show that

Any ideas?