This is my first overflow question, so let me apologize in advance if this question is too low level.


I was asking the same question in stackexchange, but didn't get an answer; check [here](https://math.stackexchange.com/questions/4465381/why-does-the-fourier-algebra-ag-consits-precisely-with-the-set-of-matrix-coe) for details.


Short: The Fourier algebra $A(G)$ of a locally compact group $G$ is the closed linear span of matrix coefficients of the left regular representation $\lambda$ (w.r.t. some certain norm). As one can show, $A(G)$ is isometric isomorphic the predual (=normal functionals =ultraweakly continuous functionals) of the group vn-algebra $VN(G)$.


Now Thm 2.4.3. in Kanituh-Lau (check link for details) asserts that $A(G)$ consists precisely of the matrix coefficients of $\lambda$. This doesn't seem to clear for me and I don't really get the proof, especially the reference to Dixmier.


EDIT:


For the proof Kanituh-Lau start with a second countable lc $G$, conclude that $L^2(G)$ is separable [clear] and that $VN(G)$ is countably generated [also clear]. Then they assert that every normal positive functional of $VN(G)$ is of the form $T \mapsto\langle T\xi,\xi \rangle$ and conclude that then any normal functional is of the form $T\mapsto \langle T\xi,\eta \rangle$ [why??] with a reference to Dixmier, C\*-algebras, which most certainly is a typo. So you will already help me out with a reliable reference, why


(a) Any normal positive functional on $VN(G)$ is of the form $T \mapsto\langle T\xi,\xi \rangle$


and why


(b) This is enough to show that any normal functional on $VN(G)$ is of the form $T\mapsto \langle T\xi,\eta \rangle$.


I guess (a) implies (b) is not that hard and is based on some polarization/ Jordan decomposition type argument. But proving (a) in the first place is where I'm stuck.


If both (a) and (b) are done, then we have $T\mapsto \langle T\xi,\eta \rangle=\varphi\_T(\overline{\eta}\*\breve{\xi})$, where $\varphi\_T$ is the corresponding functional on $A(G)$ induced by the isomorphism of $A(G)^\*$ and $VN(G)$, and $\overline{\eta}\*\breve{\xi}$ is the desired element of $A(G)$.



Why does the Fourier algebra $A(G)$ consist precisely of the set of matrix coefficients of the LRR?
The key concept you need is of a von Neumann algebra being in "standard form". The classical proof, as given by Eymard, and which Kanituh-Lau seem to follow, is to first use that for a second countable $G$, $VN(G)$ has a separating vector in $L^2(G)$, from which it follows that all normal positive functionals have the form $x\mapsto (x\xi|\xi)$. Then some locally compact group theory can boost this up to the general case. As far as I can tell, the Kanituh-Lau presentation is basically identical, excepting their erroneous reference. In any case, the heavy lifting is performed by von Neumann algebra theory, and most textbooks carry the material you need, if you look hard enough.


A more modern approach is perhaps to follow the 2nd volume of Takesaki, and use the general theory of left Hilbert Algebras. In particular, Chapter VII, Section 3 treats the Fourier algebra and group von Neumann algebra. Lemma 3.7 then gives you what you want. However, even here, the key result is delegated to Chapter V, Theorem 3.15, which follows the "separating vector" ideas. However, I think the idea of Chapter IX, Section 1, about "standard forms" put things in a nice context.


For a more gentle approach, I really like the thesis of Zwarich, [Von Neumann Algebras for Abstract Harmonic Analysis](https://uwspace.uwaterloo.ca/handle/10012/3920). See in particular section 4.3 (but the rest of the thesis develops most of the machinery you need in a self-contained way). Note however that Zwarich uses "quasi-Hilbert algebras" (see Defn 4.3.3) which are not the same a left Hilbert Algebras (I *think*).


