Pseudocharacters on free products of groups (Q1267452)

A real function \(f\) from a group \(G\) is called a pseudocharacter on \(G\) if there exists \(\varepsilon>0\) such that \(| f(xy)-f(x)-f(y)|\leq\varepsilon\) for all \(x,y\in G\) and \(f(x^n)=nf(x)\) for all \(x\in G\) and all positive integers \(n\). Let \(PX(G)\) denote the linear space of all pseudocharacters of \(G\). Let \(A\) and \(B\) be groups and let \(G=A*B\) be the free product of \(A\) and \(B\). Then there exists a natural homomorphism from \(G\) into the direct product \(A\times B\). Let \(F\) denote the kernel of this homomorphism and let \(BPX(F,A\cup B)\) denote the subspace of \(PX(F)\) consisting of all pseudocharacters \(f\) of \(F\) with \(f(x^{-1}vx)=f(v)\) for all \(v\in F\) and all \(x\in A\cup B\). It is proved that \(PX(G)\) is isomorphic to \(PX(A)\dotplus PX(B)\dotplus BPX(F,A\cup B)\).