Nonlinear completely positive maps and dilation theory for real involutive algebras

Abstract

A real seminormed involutive algebra is a real associative algebra $\mathcal{A}$ endowed with an involutive antiautomorphism $\ast$ and a submultiplicative seminorm $p$ with $p(a^{\ast })=p(a)$ for $a\in \mathcal{A}$. Then $\mathrm{ball}(\mathcal{A},p):=\{a\in \mathcal{A}:p(a)<1\}$ is an involutive subsemigroup. For the case where $\mathcal{A}$ is unital, our main result asserts that a function $\varphi :\mathrm{ball}(\mathcal{A},p)\to B(V)$, $V$ 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 $\mathrm{ball}(\mathcal{A},p)$ is open. If ${\eta }_{\mathcal{A}}:\mathcal{A}\to C^{\ast }(\mathcal{A},p)$ is the enveloping $C^{\ast }$-algebra of $(\mathcal{A},p)$ and $e^{C^{\ast }(\mathcal{A},p)}$ is the $c_0$-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 $\varphi =\Phi \circ \Gamma$, where $\Phi :e^{C^{\ast }(\mathcal{A},p)}\to B(V)$ 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 $\mathrm{ball}(\mathcal{A},p)$ in terms of representations of the $C^{\ast }$-algebra $e^{C^{\ast }(\mathcal{A},p)}$.

arXiv comment44 pages