Nonlinear completely positive maps and dilation theory for real involutive algebras

A real seminormed involutive algebra is a real associative algebra ${\mathcal A}$endowedwithaninvolutiveantiautomorphism$$ and a submultiplicative seminorm $p$ with $p(a^{\ast })=p(a)$ 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 ${\mathop{\tt ball}\nolimits}({\mathcal A},p)$isopen\mathrm{.}If$\eta_{\mathcal A} \colon {\mathcal A} \to C^({\mathcal A},p)$istheenveloping$C^$-algebraof$({\mathcal A},p)$and$e^{C^({\mathcal A},p)}$isthe$c_0$-direct sum of the symmetric tensor powers $S^n(C^{\ast }(\mathcal{A},p))$, 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 $\mathrm{U}(\mathcal{A})$ with bounded analytic extensions to $(\mathcal{A},p)$ in terms of representations of the $C^{\ast }$-algebra $e^{C^{\ast }(\mathcal{A},p)}$.