This is related to my recent question in MO. I am sure this is trivial, but I have no intuition here, so my apologies from the very beginning.
Let $G$ be a discrete group, $A$ a $C^*$-algebra, and $\varphi_0:G\to A$ an "involutive homomorphism", i.e. a map with the following properties: $$ \varphi_0(1)=1,\qquad \varphi_0(a\cdot b)=\varphi_0(a)\cdot\varphi_0(b),\qquad \varphi_0(a^{-1})=\varphi_0(a)^*,\qquad a,b\in G $$ Let $\varphi_1:G\to A$ be a "tangent vector" with respect to $\varphi_0$, i.e. a map with the properties $$ \varphi_1(1)=0,\qquad \varphi_1(a\cdot b)=\varphi_0(a)\cdot\varphi_1(b)+\varphi_1(a)\cdot\varphi_0(b),\qquad \varphi_1(a^{-1})=\varphi_1(a)^*,\qquad a,b\in G $$ Consider the $C^*$-algebra $C^*(G)$ of $G$ (i.e. the enveloping $C^*$-algebra of $\ell_1(G)$). From the definition of $C^*(G)$ it follows that $\varphi_0:G\to A$ can be extended to a homomorphism of $C^*$-algebras $\psi_0:C^*(G)\to A$ $$ \psi_0(1)=1,\qquad \psi_0(x\cdot y)=\psi_0(x)\cdot\psi_0(y),\qquad \psi_0(x^*)=\psi_0(x)^*,\qquad x,y\in C^*(G) $$ Question:
Can $\varphi_1:G\to A$ be extended to a "tangent vector of $C^*(G)$ with respect to $\psi_0$", i.e. to a linear continuous map $\psi_1:C^*(G)\to A$ with the following properties: $$ \psi_1(1)=0,\qquad \psi_1(x\cdot y)=\psi_0(x)\cdot\psi_1(y)+\psi_1(x)\cdot\psi_0(y),\qquad \psi_1(x^*)=\psi_1(x)^*,\qquad x,y\in C^*(G) $$ ?