Things to remember
- Every C* homomorphism is a bounded linear map which preserve multiplication and involution.
- Every C* algebra is a Banach algebra, and therefore also a Banach space under the norm.
- Every * homomorphism from C* algebra to the set of complex numbers is a linear functional of this Banach space.
- Every * homomorphism between C* algebras are norm decreasing. (Thm 2.1.7, Murphy)
- Hence every multiplicative * linear functional of this C* algebra is in the closed unit ball of the dual space of the Banach space.
- A hermitian linear functional are those linear functional (need not be a * homomorphism) which satisfy $f(a) = \overline{f(a^*)}$ for all $a$ in the C* algebra.
- Positive linear functionals of a C* algebra are those linear functionals (need not be a * homomorphism) where the image of the positive elements of the C* algebra lie the non-negative real line. Clearly positive linear functionals are hermitian.
- Every multiplicative * linear functional is positive.
- A bounded linear functional $\phi$ in a unital C* algebra is positive if and only if $|\phi| = \phi(1)$. (Cor. 3.3.4, Murphy)
- Every hermitian linear functional on a C* algebra can be written as the difference of two positive linear functionals. (Jordan decomposition, Thm 3.3.10, Murphy)
- Every linear functional $f$ on a C* algebra can be written as $f = g+ih$ where $g, h$ are hermitian linear funcional.
- States are defined to be those positive linear functionals with norm 1.
- Convex combinations of states are again states.
- Pure states of a C* algebra are those states which cannot be written as a convex combination of any other state.
So essentially every multiplicative * linear functional on a C* algebra can be written as a linear combination of pure states.