Noncommutative Mathematics for Quantum Systems

26 Noncommutative Mathematics for Quantum Systems

Recall that a non-empty subsetK in a real or complex vector
space is called a cone if the following two conditions are satisfied,

(i)x 1 ,x 2 ∈Kimpliesx 1 +x 2 ∈K,

(ii)λ≥0,x∈Kimpliesλx∈K.

A cone is called proper ifK 6 ={ 0 }andK∩(−K) ={ 0 }, that is,
Kdoes not contain a straight line.

Theorem 1.4.9 [Sko91] Let K⊂Rdbe a proper closed cone andμa
probability measure on K. Thenμis infinitely divisible if and only if there
exist b∈K andνa measure on K such that

∫
K^1 ∧‖x‖dν(x)<∞such
that the Fourier transform ofμhas the form

μˆ(y) =

∫

K

ei〈y,x〉dμ(x) =exp

(

i〈y,b〉+

∫

K

(ei〈y,x〉− 1 )dν(x)

)

for y∈Rd.
In this case the Laplace transform is well defined on the dual cone

K′={y∈Rd:〈y,x〉≥ 0 ∀x∈K}

and has the form

ψμ(y) =exp

(

−〈y,b〉−

∫

K

( 1 −e−〈y,x〉)dν(x)

)

for y∈K′.

1.4.6 The Levy–Khintchine formula on ́ (Rd,+)

Theorem 1.4.10 [Sat99, App04] A probability measureμ onRdis
infinitely divisible if and only if its Fourier transform is of the form

μˆ(u) =

∫

Rd

ei〈x,u〉dμ(x) (1.4.1)

=exp

(

i〈b,u〉−

1

2

〈u,Au〉+

∫

Rd−{ 0 }

(ei〈u,y〉− 1 −i〈u,y〉 (^1) ‖y‖< 1 )dν(y)
)
,
for all u∈Rd. The triple(b,A,ν)is uniquely determined byμ.
(b,A,ν)are called thecharacteristicsor thecharacteristic triple ofμ.
Corollary 1.4.11 Any infinitely divisible probability measure onRdis
embeddable into a continuous convolution semigroup.