Spaces of pseudocharacters of free products of semigroups (Q1324939)

A mapping \(\varphi : S \to R^ +\) (\(R^ +\) the additive group of reals) is called a (real) pseudocharacter of the semigroup \(S\) if the set \(\{\varphi(xy)- \varphi(x) - \varphi(y)\mid x,y \in S\}\) is bounded and \(\varphi(x^ n) = n\varphi(x)\) for every \(n\) and \(x \in S\); \(PX(S)\) is the real vector space of all pseudocharacters of the semigroup \(S\). Let \(A*B\) be the free product of semigroups \(A\) and \(B\), \(D\) the free subsemigroup of \(A*B\) generated by the set of free generators \(M = \{ab \mid a\in A,\;b\in B\}\) and \(BPX(D)\) denote the subspace of \(PX(D)\) consisting of pseudocharacters bounded on \(M\). Theorem. \(PX(ABB) = PX(A) + PX(B) + BPX(D)\).