# Thread: derivative of operator valued function

1. ## derivative of operator valued function

Consider the following construction

let $f\in L^2(\mathbb{T})$ with $\mathbb{T}$ the unit circle. Define the following mappings

$\varphi_{r}:x\mapsto x+r$ and

$U_{r}:f\mapsto f\circ\varphi_{r}$

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

$\tau_r:a\mapsto U_raU^*_r$
where $a\in A$, a C*-algebra.

Let $A^{\infty}:=\left\{a\in A:a \text{ of class }C^{\infty}\right\}$ where $A$ is a C*-algebra. And by class of $C^{\infty}$ I mean the mapping $r\mapsto \tau_r(a)$ with $a\in A$ is differentiable to any order with $r\in\mathbb{R}$.

Suppose we have a function of the form

$f:\mathbb{R}\rightarrow A$

we can define the derivative of the operator valued function as $f'(x)$ satisfying the equation

$0=\lim_{h\rightarrow\infty}\|\frac{f(x+h)-f(x)}{h}-f'(x) \|$

I define the following mapping

$\delta(a):=\left.\frac{d}{dr}\tau_r(a)\right|_{r=0 }$

Now, what I want to show is that $\delta:A^{\infty}\rightarrow A^{\infty}$.

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

$0=\lim_{h\rightarrow\infty}\|\frac{\tau_{r+h}(a)-\tau_r(a)}{h}-\partial_r\tau_r(a) \|$

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

$0=\lim_{h\rightarrow\infty}\|\frac{\tau_{r+h}(\del ta(a) )-\tau_r(\delta (a))}{h}-\partial_r\tau_r(\delta(a)) \|$

Any ideas?

2. for some reason the $\delta(a)$ did not want to show in the above equation, however, here is what I have done up to now

$\tau_{r+h}(\delta(a))-\tau_r(\delta(a))$

$=U_{r+h}\delta(a)U^*_{r+h}-U_r\delta(a)U^*_{r}$

$=U_r\left[U_h\delta(a)U^*_h-\delta(a)\right]U^*_r$

$=U_r\left[U_h\left(\left.\frac{d}{dr}\tau_r(a) \right|_{r=0}\right) U^*_h-\left.\frac{d}{dr}\tau_r(a)\right|_{r=0}\right]U^*_r$

$=U_r\left[\left.\frac{d}{dr}\tau_{r+h}(a) \right|_{r=0}-\left.\frac{d}{dr}\tau_r(a)\right|_{r=0}\right]U^*_r$

$=U_r\left[\left.\frac{d}{dr}\left(\tau_{r+h}(a)-\tau_r(a) \right) \right|_{r=0} \right]U^*_r$

bringing the $h$ into account I have something similar to what I need, however, the second part does not want to play along..

Any Ideas?