Noncommutative Mathematics for Quantum Systems

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.