Let $A$ be a Banach *-algebra. By a *-representations of $A$, we mean a *-homomorphism $\pi:A\to B(H_\pi)$, where $B(H_\pi)$ is the space of all bounded linear maps on a Hilbert space $H_\pi$. Let $\pi_u=\oplus_{\pi\in Rep(A)}\pi.$
Q. Suppose that $\ker \pi_u=0$. Is $\pi_u$ an order isomorphism? (that is $x\geq0$ iff $\pi_u(x)\geq0$).
Remark. For a given self-adjoint element $x$ in $A$ we write $x\geq0$ if $x=\sum_1^n a^*_ia_i$ where $a_i\in A$ and $n$ is a natural number.
