Isometric isomorphisms in proper \(CQ^*\)-algebras (Q839735)

Let \(A\) be a Banach module over a \(C^*\)-algebra \(A_0\) with involution \(*\) and \(C^*\)-norm \(\|.\|_0\) such that \(A_0\subseteq A\). We say that \((A,A_0)\) is a proper \(CQ^*\)-algebra if (i) \(A_0\) is dense in \(A\) with respect to its norm \(\|.\|;\) (ii) An involution \(*\) which extends the involution of \(A_0\), is defined in \(A\) with the property \((xy)^*=y^*x^*\) for all \(x,y\in A\) whenever the multiplication is defined; (iii)\(\|y\|_0= \sup_{x\in A, \|x\|\leq 1}\|xy\|\) for all \(y\in A_0.\) In this paper the authors investigate the Hyers-Ulam-Rassias stability of isometric homomorphisms in proper \(CQ^*\)-algebras for the Cauchy-Jensen additive mapping \[ 2f(\frac{x_1+x_2}{2}+y)=f(x_1)+f(x_2)+2f(y). \]