I am reading "Noncommutative Geometry and Particle Physics" by van Suijlekom. I have problems to identify one map as homomorphism.
Let $M\times F$ be an almost-commutative manifold. The unital $* $-algebra $A$ of its spectral triple is given by $C^\infty(M,A_F)$. Furthermore $U(A)$ is the set of all unitary elements of the $C^* $-algebra and $A_F=\{u\in U(A) | aJ=Ja^*\}$. The local gauge group is given by $\mathfrak{G}(F)$ and it holds that $\mathfrak{G}(F)\cong U(A_F)/\mathfrak{H}(F)$, where $\mathfrak{H}(F)=U((A_F)_{J_F})$.
The homomorphism $C^\infty(M,U(A_F))\to C^\infty(M,U(A_F)/\mathfrak{H}(F))$ is surjective. I would define the homomorphism by $f\mapsto f+\mathfrak{H}(F)$. This map is obviously surjective. I have problems to see why $(f+\mathfrak{H}(F))(g+\mathfrak{H}(F))=fg+\mathfrak{H}(F)$. The terms $f(x)\mathfrak{H}(F)$ and $\mathfrak{H}(F)g(x)$ are not equal $\mathfrak{H}(F)$.
Thanks for your help.
The map in question is not an algebra homomorphism; instead, it's a group homomorphism, and it sends $f \in C^\infty(M,U(A_F))$ to $[f] \in C^\infty(M,U(A_F)/\mathfrak{H}(F))$ defined by $[f](x) := f(x)\mathfrak{H}(F)$ for all $x \in M$, where the multiplicative coset $f(x)\mathfrak{H}(F)$ is an element of the quotient group $U(A_F)/\mathfrak{H}(F)$. If you want to know what groups are appearing here, you should always think about the unitary group $U(A)$ of a unital $*$-algebra $A$ acting as $\ast$-automorphisms on $A$ by conjugation: for all $u \in U(A)$ and $a \in A$, $\operatorname{Ad}_u(x) := uau^\ast$.