Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 125
contractive). Another theorem due to Gelfand and Naimark states
that each C∗-algebra has a faithful representation — in other
words, eachC∗-algebra is isomorphic to a closed∗-subalgebra of
B(H)for certain Hilbert spaceH.
Applying the last statement one can easily show that ifAis a
C∗-algebra, then the ∗-algebraMn(A) of n by n matrices with
coefficients inApossesses a natural (and unique)C∗-norm. This
turns out to have a very important role in the theory. IfA,B, are
C∗-algebras and T : A → Bis a linear map, then applying T
separately to each entry of the matrix yields a linear map
T(n):Mn(A)→Mn(B). We say thatTiscompletely positive, ifT(n)
is positive for eachn∈N. Sometimes it is convenient to identify
Mn(A)with the algebraic tensor productMn⊗A. IfT :A → A,
then the mapT(n) : Mn(A) → Mn(A) corresponds under this
identification to the map idMn⊗T:Mn⊗A→Mn⊗A.
Remark 2.1.6 Each ∗-homomorphism acting between
C∗-algebras is completely positive. IfAorBis commutative and
T:A→ Bis a positive map, thenTis automatically completely
positive; therefore, in particular states are completely positive.
Further, sums and compositions of completely positive maps are
completely positive and, ifA⊂B(H)andV∈B(H;K), whereH,K
are Hilbert spaces, then the mapa 7→ V∗aV is a completely
positive map fromAtoB(K). All unital completely positive maps
are contractive.
The following theorem is sometimes called the ‘quantum/
noncommutative Hahn–Banach theorem’.
Theorem 2.1.7 [Arveson Extension Theorem]: LetHbe a Hilbert
space,A– aC∗-algebra, andB– aC∗-subalgebra ofA. IfT:B→
B(H)is completely positive, then there exists a completely positive
mapT ̃:A→B(H)such thatT ̃|B=Tand‖T‖=‖T ̃‖.
Finally, note that if we want to study ‘noncommutative
counterparts’ of compact spaces equipped with some further
structure, say for examplecompact groups, we can do it, provided
we find a way of encoding of this extra structure solely in terms of
the algebra of continuous functions defined on the space and
possibly some transformations acting on algebras of that type. In
the particular example listed above such an approach was