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# 4 Time Evolution of States

[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 . [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 . [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, AA**, its elements can be considered as functions on the set XA of its charactersd (This is a footnote:) d A character is an algebraic *-homomorphism from a commutative C*-algebra to C ., i.e. aχ=χ,a for aA,χXA. [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˙1R, then equation (8) implies

 μ⁢t˙1,A⁢Tt˙⁢a⁢t˙1 = μ⁢t˙1,a⁢t˙1+t˙ (9) = μ⁢t˙2-t˙,a⁢t˙2=A*⁢Tt˙⁢μ⁢t˙2,a⁢t˙2

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

 A*⁢Tt˙⁢μ⁢s˙=μ⁢s˙-t˙ (10)

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 .