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# 4 Time Evolution of States

[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 . [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 . [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 charactersd (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 .