A lemma about the pure states

Question

There is a quotation of a book:

Lemma 3.4.5. Assume that both $A$ and $B$ are unital and abelian C*-algebras. Then for every C*-norm $\|\cdot\|_{\alpha}$ on $A\odot B$ and pair of pure states $\phi\in S(A)$, $\psi\in S(B)$ the linear functional $\phi \odot\psi: A\odot B\rightarrow \mathbb{C}$ extends to a state on $A\otimes_{\alpha} B$. (Here, $S(A)$ denote the state space of $A$).

Proof. Let $P(A)$ (resp. $P(B)$) denote the pure states on $A$ (resp. $B$). Let $$U=\{(\phi, \psi)\in P(A) \times P(B): |\phi\odot \psi(x)|\leq\|x\|_{\alpha} ~, \forall x\in A\odot B \}.$$ Then the lemma asserts that $U=P(A)\times P(B)$.

So, ....

I do not know why "the lemma asserts that $U=P(A)\times P(B)$." Does this equation imply the conclusion hold?

Answer 1

They are not saying that the conclusion holds. They are phrasing it in a way that allows them to prove it. If there were $\varphi$, $\psi$ such that $\varphi\odot\psi$ does not extend to a state, then you would have $\mathcal U\subsetneq P(A)\times P(B)$.