28 Noncommutative Mathematics for Quantum Systems
Fix a basis(Xj, 1≤j≤n)ofgand define the dense subspace
C 2 L(G)by
C 2 L(G) =
{
f∈C 0 (G);
XLi(f)∈C 0 (G)and
XiLXLj(f)∈C 0 (G)for all 1≤i,j≤n
}
,
whereXLj denotes the action ofXjon appropriate functions onGas
a left invariant vector field.C 2 L(G)is a Banach space with respect to
the norm
‖f‖2,L=‖f‖+
n
∑
i= 1
‖XLif‖+
n
∑
j,k= 1
‖XLjXLkf‖,
where
‖f‖=sup
g∈G
|f(g)|
denotes the supremum norm. The spaceCR 2 (G)and the norms
‖·‖2,Rare defined similarly. Note that the smooth functions of
compact support are included in these spaces, that is,Cc∞(G) ⊆
C 2 L(G)∩C 2 R(G).
There exist functionsxi∈C∞c(G), 1≤i≤nso that(x 1 ,... ,xn)
are a system of canonical co-ordinates forGate.
Theorem 1.4.12 (Hunt’s theorem)[Hun56] Let X be a L ́evy process
in G with infinitesimal generator L. Then
(i) C 2 L(G)⊆Dom(L).
(ii) For each g∈G,f∈C 2 L(G),
L f(g) = ∑
i
biXiLf(g) +∑
i,j
aijXiLXLjf(g)
+
∫
G\{e}
(f(gh)−f(g)−yi(g)XLif(g))ν(dh),
where b= (b^1 ,.. .bn)∈Rn,a= (aij)is a non-negative-definite,
symmetric n×n real-valued matrix andνis a L ́evy measure on
G\{e}.
Conversely, for any linear operator with such a representation there exists
a L ́evy process (unique up to stochastic equivalence) with generator L.