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In a unital C-star algebra $A$, an element is Hermitian when $x=x^{\ast }$, unitary when $x^{\ast }x=x{x}^{\ast }=1$, and normal when $x^{\ast }x=x{x}^{\ast }$. It is a positive element of a C-star algebra when $x=y^{\ast }y$ for some $y$, equivalently when it is Hermitian and its spectrum lies in $[0,\infty )$.

Suppose first that $\Vert \tau \Vert =\tau (1)$, and normalize so that both equal one. If $x=x^{\ast }$, the C-star identity givesWriting $\tau (x)=a+ib$, we obtainfor every real $t$, which forces $b=0$. Decomposing an arbitrary $x$ into its Hermitian real and imaginary parts now gives

Let $x=x^{\ast }$. Every character of an algebra on the commutative unital Banach algebra generated by $x$ has norm and value at one equal to one, so the preceding argument makes its value on $x$ real. The character description of the spectrum of an element therefore gives $\sigma (x)\subset \mathbb{R}$. Iterating the C-star identity,so $\Vert x\Vert =r(x)$. If $c=\Vert x\Vert$, then $x+c1$ has nonnegative spectrum and is positive, as is $c1$; henceis a difference of positive elements.

If $0\le x$ and $\Vert x\Vert \le 1$, then spectral translation givesThus $1-x$ is positive and, by the norm--spectral-radius equality for Hermitian elements, $\Vert 1-x\Vert \le 1$.

Return to a norm-one $\tau$ with $\tau (1)=1$. For $0\le x\le 1$, the preceding paragraph and reality on Hermitian elements giveScaling proves that $\tau$ is a positive functional on a C-star algebra. Every character has norm and value at one equal to one, so every character is positive. On $C[0,1]$, the functionalis positive but is not multiplicative, and hence is not a character.

Conversely, let $\tau$ be positive. Writing a Hermitian element as a difference of positive elements shows that $\tau$ is real on Hermitian elements. Positivity offor every $\lambda \in \mathbb{C}$ says that the associated quadratic polynomial is nonnegative. Minimizing it in $\lambda$ gives the Cauchy--Schwarz inequalityTaking $y=1$ yields $|\tau (x){|}^2\le \tau (x^{\ast }x)\tau (1)$. Since $0\le x^{\ast }x\le \Vert x{\Vert }^{2}1$, positivity givesTogether with $\tau (1)\le \Vert \tau \Vert$, this proves $\Vert \tau \Vert =\tau (1)$.

Convex combinations preserve positivity and value one at the identity, so the state space $S(A)$ is convex. If $x$ is normal, the unital C-star subalgebra $C^{\ast }(1,x)$ is commutative and its Gelfand transform identifies it with $C(\sigma (x))$. Choose $\lambda \in \sigma (x)$ with $|\lambda |=\Vert x\Vert$. Evaluation at $\lambda$ is a state taking $x$ to $\lambda$. Its norm-preserving Hahn--Banach extension to $A$ still takes $1$ to one, so the norm criterion makes the extension a state $\tau$ satisfying $|\tau (x)|=\Vert x\Vert$.

The state space is nonempty, convex, and weak-star compact by the Banach-Alaoglu theorem. The Krein-Milman theorem gives an extreme point, so a pure state on a C-star algebra exists. For positive $x$, the preceding norm-attainment result makesa nonempty weak-star compact convex set. It is a face: if a convex combination has the maximal possible value on $x$, each summand does. An extreme point of $F$ exists by Krein--Milman and, because $F$ is a face, is extreme in $S(A)$. It is the required pure state.

Let $M^{+}(K)$ be the positive linear functionals on $C(K)$: those $\phi$ for which $f\ge 0$ implies $\phi (f)\ge 0$. For real $f$,so positivity gives $|\phi (f)|\le \phi (1_K)\Vert f{\Vert }_{\infty }$; decomposition into real and imaginary parts, or the positive-functional Cauchy--Schwarz inequality, gives the same bound for complex $f$. Hence $\phi$ is continuous andThe Riesz-Markov-Kakutani representation theorem says that there is a unique finite regular positive Borel measure $\mu$ withMore generally, $C(K{)}^{\ast }=M(K)$ is the Banach space of finite regular complex Borel measures with the total-variation norm.

Now let $A\subseteq \mathcal{B}(H)$ be commutative, unital, and C-star, and put $K={\Phi }_A$. The Gelfand transform is an isometric star-isomorphism $A\cong C(K)$. For $\xi ,\eta \in H$, the functionalon $C(K)$ is represented by a regular complex measure ${\mu }_{\xi ,\eta }$. The diagonal measures are positive. Polarization and the Riesz theorem assemble them into a projection-valued measure $E$ characterized byfor Borel sets $S\subseteq K$. The multiplication identities first hold for continuous functions and extend to bounded Borel functions by a monotone-class argument. Thereforedefines a unital star-homomorphism with $\Vert \Psi (f)\Vert \le \Vert f{\Vert }_{\infty }$, andfor every $T\in A$.

For a normal $T\in \mathcal{B}(H)$, apply this construction to the commutative C-star algebra $C^{\ast }(1,T)$. Its character space identifies with $\sigma (T)$, and the Gelfand transform of $T$ is the coordinate function $z(\lambda )=\lambda$. We obtain the Borel functional calculus for a normal operatorIf $\sigma (T)\subseteq \mathbb{T}$, then $\overline{z},z=z\overline{z}=1$ on the spectrum. Since $\Psi$ preserves products and involution,Thus $T$ is a unitary operator.