I'm having trouble understanding Thm 2.4.3 (for second countable locally compact groups G" of "Fourier and FourierStieltjes Algebras on Locally Compact Groups" by Kaniuth and Lau.
Context: The Fourier Algebra $A(G)$ of a (say second countable) lc group $G$ is defined to be the closed linear span of functions of the coefficients of the left regular representation, i.e. functions of the form $u(x)=\langle \pi(x)\xi,\eta\rangle=\xi*\breve{\eta}$ with respect to a certain norm. They then show that $A(G)$ is isometrically isomorphic to the predual (=space of ultraweak continuous functionals) of the group von Neumann algebra $VN(G)$. So far so good.
Now 2.4.3 asserts (see image 1): $A(G)$ consits precisely of the coefficients of the LRR (hence, we may drop both, span and closure in the definition). And here comes the part which I don't understand at all: Neither do I understand why Proposition 14.5.1, Dixmier (see image 2) implies that every normal positive functional on $VN(G)$ is of the form $T\mapsto \langle T\xi,\xi\rangle$, nor do I understand why this implies that every uwc-functional on $VN(G)$ is of the form $T\mapsto \langle T \xi,\eta \rangle$.
Thank you very much in advance for clarifying!
