Derivations on Operator Algebras

The infinitesimal generator of a continuous observable-algebra evolution is generally a closed, densely defined *-derivation. The continuity here is norm continuity of the orbit of each algebra element. We will prove density and closedness and identify the generator’s domain. Dynamics as Automorphisms explains the algebraic formulation, the corresponding state evolution, and the connection to equilibrium.

Helpful background. Operator Algebra Minimum introduces C*-algebras and their norm. Heisenberg Dynamics explains why conjugation by an unbounded Hamiltonian need not satisfy the point-norm condition on all of $\mathcal B(\mathcal H)$.

The generator is an operator on the observable algebra

Let $\mathcal A$ be a C*-algebra and $\alpha:\mathbb R\to\operatorname{Aut}(\mathcal A)$ a group satisfying

$$\|\alpha_t(A)-A\|\longrightarrow0 \quad(t\to0),\qquad A\in\mathcal A.$$

Each automorphism is an isometry. Regard $\alpha$ as a strongly continuous group on the Banach space $\mathcal A$ and define

$$\begin{aligned} D(\delta)&=\left\{A\in\mathcal A: \lim_{t\to0}\frac{\alpha_t(A)-A}{t} \text{ exists in norm}\right\},\\ \delta(A)&=\lim_{t\to0}\frac{\alpha_t(A)-A}{t}. \end{aligned}$$

The generator domain is a *-subalgebra and

$$\delta(AB)=\delta(A)B+A\delta(B),\qquad \delta(A^*)=\delta(A)^*.$$

These product and adjoint identities follow from the norm product rule, whose calculation is given in Dynamics as Automorphisms. They are the algebraic derivation property. The analytic assertions that $D(\delta)$ is dense and $\delta$ is closed require additional arguments.

The group preserves the generator domain and satisfies

$$\delta(\alpha_t(A))=\alpha_t(\delta(A)), \qquad A\in D(\delta),$$

by applying the bounded map $\alpha_t$ to the defining difference quotient.

Time averaging creates dense smooth elements

For $f\in C_c^\infty(\mathbb R)$ and $A\in\mathcal A$, define the norm-convergent Bochner integral

$$A_f=\int_{\mathbb R}f(t)\alpha_t(A)\,dt.$$

Changing variables gives

$$\alpha_s(A_f)=\int_{\mathbb R}f(t-s)\alpha_t(A)\,dt.$$

Difference quotients of $f(t-s)$ converge in $L^1$, so the norm derivative may be passed under the integral. Consequently

$$\delta(A_f)=-\int f'(t)\alpha_t(A)\,dt.$$

Repeated differentiation shows $A_f\in D(\delta^n)$ for every $n$, with

$$\delta^n(A_f)=(-1)^n\int f^{(n)}(t)\alpha_t(A)\,dt, \qquad \|\delta^n(A_f)\|\leq\|f^{(n)}\|_1\|A\|.$$

Now take a nonnegative smooth approximate identity $f_\varepsilon$ with integral one and support shrinking to zero. Then

$$\|A_{f_\varepsilon}-A\| \leq\int f_\varepsilon(t)\|\alpha_t(A)-A\|\,dt \longrightarrow0.$$

Thus infinitely differentiable elements are dense, and in particular $D(\delta)$ is dense. Smoothness should not be confused with analyticity: these estimates do not automatically make the formal series $\sum t^n\delta^n(A)/n!$ converge.

The integral identity proves closedness

For $A\in D(\delta)$, integration of the norm derivative yields

$$\alpha_t(A)-A=\int_0^t\alpha_s(\delta(A))\,ds.$$

Suppose $A_n\in D(\delta)$, $A_n\to A$, and $\delta(A_n)\to B$ in norm. Isometry of the group permits passage to the limit on both sides:

$$\alpha_t(A)-A=\int_0^t\alpha_s(B)\,ds.$$

Dividing by $t$ and taking $t\to0$ gives $B$, by point-norm continuity of the orbit of $B$. Therefore $A\in D(\delta)$ and $\delta(A)=B$. This is precisely closedness of the graph.

The proof also shows why topology is part of the theorem. Replacing norm convergence by strong operator convergence without changing the domain and conclusion would be a different assertion.

Inner and outer generators

If $\mathcal A$ is unital and $K=K^*\in\mathcal A$, then

$$\alpha_t(A)=e^{itK/\hbar}Ae^{-itK/\hbar}$$

has the everywhere-defined bounded generator $\delta(A)=(i/\hbar)[K,A]$, with norm at most $2\|K\|/\hbar$. The derivation is inner because its implementing element belongs to the algebra.

An instructive contrasting example is rotation of continuous functions on the circle:

$$\mathcal A=C(\mathbb T),\qquad (\alpha_t f)(\theta)=f(\theta+t).$$

Compactness of the circle makes each continuous function uniformly continuous, so this is point-norm continuous. Its generator has domain $C^1(\mathbb T)$ and is $\delta f=f'$. The functions $f_n(\theta)=e^{in\theta}$ have norm one and derivative norm $|n|$, proving unboundedness.

This derivation cannot be an internal commutator: the algebra is commutative, so every such commutator vanishes. It can nevertheless be implemented in a Hilbert-space representation by a unitary translation group with an unbounded generator. Innerness in a chosen algebra and unitary implementability in a representation are distinct questions.

Conversely, writing a formal Leibniz rule does not prove that a proposed derivation generates an automorphism group. A generation theorem requires analytic hypotheses, including the appropriate domain and resolvent control. The present argument begins with an existing group and identifies its generator.

Exercises

1. Prove that the generator domain for rotations of $C(\mathbb T)$ is exactly $C^1(\mathbb T)$.

Solution

For a continuously differentiable periodic function, uniform continuity of its derivative gives uniform convergence of translation difference quotients to $f'$. Conversely, if the quotients converge uniformly to $g$, then $g$ is continuous and the integrated group identity gives $f(\theta+t)-f(\theta)=\int_0^t g(\theta+s)\,ds$. Hence $f$ is continuously differentiable with derivative $g$.

2. For $\lambda>0$, show that the operator $R_\lambda A=\int_0^\infty e^{-\lambda t}\alpha_t(A)\,dt$ has norm at most $1/\lambda$ and solves $(\lambda-\delta)R_\lambda A=A$.

Solution

Isometry gives the norm bound by integrating $e^{-\lambda t}$. Changing variables in $\alpha_h(R_\lambda A)$ and taking the norm difference quotient gives $\delta R_\lambda A=\lambda R_\lambda A-A$. Thus the integral takes values in $D(\delta)$ and is the resolvent of the Banach-space generator. Integration by parts gives the reverse inverse identity on $D(\delta)$.

References

• O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987, Chapter 3, publisher edition — automorphism groups, smooth elements, and derivations.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer, 1983, Chapter 1, publisher edition — closed densely defined generators and Laplace resolvents.