Stone's theorem for 1-parameter groups of unitary multipliers?

Question

Let $A$ be a nonunital C*-algebra and let $M(A)$ denote its multiplier algebra. Let $(u_t)_{t \in \mathbb{R}}$ be a strictly continuous 1-parameter group of unitary multipliers. That is, $u_t x \to x$ as $t \to 0$ for all $x \in A$. I am wondering whether there exists an unbounded operator $H : A \to A$ such that $u_t = e^{itH}$ in some sense? I would want some condititions on $H$:

• $H$ should be densely-defined and closed. That is, the domain of $H$ should be a dense subspace $A_0$ of $A$ and, whenever $x_n \in A_0$ are such that $x_n \to x \in A$ and $H x_n \to y \in A$ we should have $x \in A_0$ and $H x = y$.
• We should have $H$ symmetric with respect to the $A$-valued inner-product $\langle \cdot, \cdot \rangle : A \times A \to A$ given by $\langle x,y \rangle = x^* y$. That is, whenever $x,y \in A_0$, it should hold that $\langle Ax,y \rangle = \langle x,Ay \rangle$.

A good "core" for $H$ should be something like the set of smooth elements $x \in A$, that is elements for which $t \mapsto u_t x$ is a $C^\infty$ path $\mathbb{R} \to A$. There, we can define $H$ by $$ H(x) = \frac{1}{i} \frac{d}{dt} \big|_{t = 0} u_t x .$$

Notice that, for $x$ and $y$ smooth in the above sense, we have $$ \langle u_t x,y \rangle = (u_tx)^* y = x^* u_{-t} y = \langle x, u_{-t} y \rangle. $$ Differentiating both sides and evalutating at $t = 0$ we get gives the desired self-adjointness condition. So one, wants to do things like:

• Prove that smooth elements are a dense subalgebra
• Prove that the operator defined as above on smooth elements is closable, probably by using the symmetry condition
• Find a sense in which $e^{itH} = u_t$, probably strong convergence of the series $\sum_{n=0}^\infty \frac{(itH)^n}{n!}$ on the $\bigcap_{n=1}^\infty \operatorname{dom}(H^n)$ or something.

Hopefully someone can spell out what sorts of results are true in this context? Or point me towards references? Thanks.

Answer 1

There certainly exist references for this topic. Here is the one I found the most helpful.

• Stone's Theorem in C*-algebras, J. HOLLEVOET, J. QUAEGEBEUR and S. VANKEER

I found it in the references of the paper

• On generalized Stone's Theorem, Massoud Amini

which is suggested by user "Argument" in the comments. There is also apparently a book, though I did not look at it.

• Multipliers of Pedersen's Ideal, Aldo Joram Lazar, Donald Curtis Taylor