[210.2.1] In general,
the set of observables 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 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 .
[210.2.5] The notation is used for
the value of the observable in the state .
[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 is a subset of its bidual,
,
its elements
can be considered as functions on the set
of its characters^{d} (This is a footnote:) ^{d}
A character is an algebraic *-homomorphism from a
commutative C-algebra to .,
i.e. for
.
[211.0.2] By virtue of this correspondence, known as the Gelfand isomorphism
[30, 31],
a commutative C-algebra is isomorphic to the
algebra of continuous functions
on the set 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 with consists of all adjoint operators on the dual space [33, 28]. [211.1.3] If denotes the state at time , then equation (8) implies

(9) | ||||

where is arbitrary. [211.1.4] For left translations the adjoint group

(10) |

is the group of right translations with . [211.1.5] The adjoint semigroup is weak continuous, but in general not strongly continuous, unless the Banach space is reflexive [32].