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A real seminormed involutive algebra is a real associative algebra ${\mathcalA}$ endowed with an involutive antiautomorphism $*$ and a submultiplicativeseminorm $p$ with $p(a^*) =p(a)$ for $a\in {\mathcal A}$. Then ${\mathop{\ttball}\nolimits}({\mathcal A},p) := \{a \in {\mathcal A} \colon p(a) < 1\}$ isan involutive subsemigroup. For the case where ${\mathcal A}$ is unital, ourmain result asserts that a function $\phi \colon{\mathop{\ttball}\nolimits}({\mathcal A},p) \to B(V)$, $V$ a Hilbert space, is completelypositive (defined suitably) if and only if it is positive definite and analyticfor any locally convex topology for which ${\mathop{\ttball}\nolimits}({\mathcal A},p)$ 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 thesymmetric tensor powers $S^n(C^*({\mathcal A},p))$, then the above twoproperties are equivalent to the existence of a factorization $\phi = \Phi\circ \Gamma$, where $\Phi \colon e^{C^*({\mathcal A},p)} \to B(V)$ is linearcompletely positive and $\Gamma(a) = \sum_{n = 0}^\infty \eta_{\mathcalA}(a)^{\otimes n}$. We also obtain a suitable generalization to non-unitalalgebras.  An important consequence of this result is a description of the unitaryrepresentations of ${\rm U}({\mathcal A})$ with bounded analytic extensions to${\mathop{\tt ball}\nolimits}({\mathcal A},p)$ in terms of representations ofthe $C^*$-algebra $e^{C^*({\mathcal A},p)}$.

Nonlinear completely positive maps and dilation theory for real involutive algebras