Nonlinear completely positive maps and dilation theory for real involutive algebras

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 $e^{C^*({\mathcal A},p)}$.