  A real seminormed involutive algebra is a real associative algebra ${\mathcal
A}$ endowed with an involutive antiautomorphism $*$ and a submultiplicative
seminorm $p$ with $p(a^*) =p(a)$ for $a\in {\mathcal A}$. Then ${\mathop{\tt
ball}\nolimits}({\mathcal A},p) := \{a \in {\mathcal A} \colon p(a) < 1\}$ is
an involutive subsemigroup. For the case where ${\mathcal A}$ is unital, our
main result asserts that a function $\phi \colon{\mathop{\tt
ball}\nolimits}({\mathcal A},p) \to B(V)$, $V$ a Hilbert space, is completely
positive (defined suitably) if and only if it is positive definite and analytic
for any locally convex topology for which ${\mathop{\tt
ball}\nolimits}({\mathcal A},p)$ is open. If $\eta_{\mathcal A} \colon
{\mathcal A} \to C^*({\mathcal A},p)$ is the enveloping $C^*$-algebra of
$({\mathcal A},p)$ and $e^{C^*({\mathcal A},p)}$ is the $c_0$-direct sum of the
symmetric tensor powers $S^n(C^*({\mathcal A},p))$, then the above two
properties are equivalent to the existence of a factorization $\phi = \Phi
\circ \Gamma$, where $\Phi \colon e^{C^*({\mathcal A},p)} \to B(V)$ is linear
completely positive and $\Gamma(a) = \sum_{n = 0}^\infty \eta_{\mathcal
A}(a)^{\otimes n}$. We also obtain a suitable generalization to non-unital
algebras.
  An important consequence of this result is a description of the unitary
representations of ${\rm U}({\mathcal A})$ with bounded analytic extensions to
${\mathop{\tt ball}\nolimits}({\mathcal A},p)$ in terms of representations of
the $C^*$-algebra $e^{C^*({\mathcal A},p)}$.
