21 problems

Runde's equality conjecture.

Multiplier-algebra replacement conjecture. In the corresponding result for $A(G)$, the algebra $C_b(G)$ should be replaced by $MC^*_{\lambda}(G)$.

Extension conjecture. The theorem should hold for general closed non-open subgroups of $G$.

Let $G$ be a locally compact group, and let $A(G)$ denote its Fourier algebra. Amenability equivalence conjecture. The group $G$ is amenable if and only if the Banach algebra…

Amenability conjecture. The Fourier algebra $A(G)$ is amenable if and only if $G$ is amenable.

Amenable-group complete contraction conjecture. The identity map from $A(G)$ to $OA(G)$ is a complete contraction for each amenable $G$.

Let $F$ be a group, let $G$ be the group under consideration, let $H$ be the relevant group, and let $A(G)$ denote the Fourier algebra of $G$. A mixed piecewise-affine map is a map…

Figà-Talamanca–Picardello conjecture. $A_r(G)$ is dense in $B_0(G)$, equivalently

Let $G$ be a locally compact group and let $A(G)$ denote its Fourier algebra. In the co-commutative setting, relative 1-biflatness of $A(G)$ is known to be equivalent to the existe…

Let $G$ be a locally compact group. Write $\widetilde{\mathrm{A}}(G)_1$ for the unit ball of the unitization of the Fourier algebra $\mathrm{A}(G)$, let $\mathcal{C}_0(G)$ denote t…

Forrest–Runde conjecture. The Fourier algebra $\operatorname{A}(G)$ is weakly amenable if and only if the connected component of the identity in $G$ is abelian.

Johnson's conjecture. The Fourier algebra $A(G)$ should reflect the amenability of $G$ in the same way that $L^1(G)$ does.

Let $G$ be a locally compact group, and let $\tau_\rho$ denote the representation associated with a representation $\rho$. A representation $\rho$ is spectrally natural when…

Let $G$ be a locally compact group, let $A(G)$ denote its Fourier algebra, and let $\pi:A(G)\rightarrow\mathcal B(H)$ be a completely bounded representation. The Fourier-algebra si…

Let $G$ be a compact group, let $N$ be a normal subgroup such that $G/N$ is a Lie quotient, and let $q:G\rightarrow G/N$ be the quotient map. Let $G_{\omega}$ and…

Let $G$ and $H$ be locally compact groups. Suppose that $G$ is amenable, and let $\Phi:{\mathrm{A}(G)}\to{\mathrm{B}(H)}$ be a bounded homomorphism. A continuous map…

Let $G$ be a locally compact group and let ${\mathrm{A}(G)}$ denote its Fourier algebra. Operator biflatness conjecture. ${\mathrm{A}(G)}$ is always operator biflat. This would ext…

Let $G$ be a locally compact group, and let $G_e$ denote the connected component of the identity. Weak amenability necessity conjecture. The Fourier algebra $\mathrm{A}(G)$ can be…

Let $G$ be a compact group, and let $G_e$ denote the connected component of the identity. Forrest's conjecture. The Fourier algebra $\mathrm{A}(G)$ is weakly amenable precisely whe…

Let $G$ be a compact group. An almost abelian group is one that admits an abelian subgroup of finite index. Losert's conjecture. The Fourier algebra $\mathrm{A}(G)$ is amenable as…

Pointwise parametrization conjecture. Every element of $\Phi^{r}_{A(G)}$ is of the form $\alpha_{\gamma}$ for some $\gamma\in\Delta^{r}$, and $\Delta^{r}$ with the topology of poin…