Consider the following construction
let with the unit circle. Define the following mappings
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