[210.2.1] In general,
the set of observables A of a physical system is
not only a Banach space, but forms an algebra, more specifically,
a C*-algebra [30].
[210.2.2] In classical physics this algebra is commutative.
[210.2.3] The states μ of a physical system are
normalized, positive linear functionals
on its algebra of observables [30].
[210.2.4] As such they are elements of the dual space A*.
[210.2.5] The notation μ,a is used for
the value μa of the observable a in the state μ.
[210.2.6] Convex combinations of states are again states.
[210.2.7] If a state cannot be written as a convex combination
of other states, it is called pure.
[page 211, §0]
[211.0.1] Because the observable algebra A is a subset of its bidual,
A⊂A**,
its elements
can be considered as functions on the set XA
of its characters,
i.e. aχ=χ,a for
a∈A,χ∈XA.
[211.0.2] By virtue of this correspondence, known as the Gelfand isomorphism
[30, 31],
a commutative C*-algebra is isomorphic to the
algebra C0XA of continuous functions
on the set XA of its characters equipped with
the weak* topology.
[211.0.3] Characters are pure states.

[211.1.1] The time evolution of states is obtained from
the time evolution of observables by passing
to adjoints [28, 32].
[211.1.2]
The adjoint time evolution A*Tt˙:A*→A*
with t˙∈R consists of all adjoint operators
ATt˙* on the dual space A* [33, 28].
[211.1.3] If μt˙1 denotes the state at time t˙1∈R,
then equation (8) implies

μt˙1,ATt˙at˙1 | = | μt˙1,at˙1+t˙ | | (9) |

| = | μt˙2-t˙,at˙2=A*Tt˙μt˙2,at˙2 | |

where t˙2=t˙1+t˙∈R is arbitrary.
[211.1.4] For left translations the adjoint group

is the group of right translations with s˙,t˙∈R.
[211.1.5] The adjoint semigroup is weak* continuous,
but in general not strongly continuous, unless
the Banach space A is reflexive [32].