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 * and a submultiplicative seminorm p with p(a*) = p(a) for ${a\in \mathcal{A}}$. Then ${\mathtt{ball}(\mathcal{A}, p) := \{ a \in \mathcal{A} \: p(a) < 1\}}$ is an involutive subsemigroup. For the case where ${\mathcal{A}}$ is unital, our main result asserts that a function ${\varphi \: \mathtt{ball}(\mathcal{A},p) \to B(V)}$, 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 ${\mathtt{ball}(\mathcal{A},p)}$ is open. If ${\eta_\mathcal{A} \: \mathcal{A} \to C^{*}(\mathcal{A},p)}$ 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 ${\varphi = \Phi \circ \Gamma}$, where ${\Phi \: e^{C^{*}(\mathcal{A}, p)} \to B(V)}$ 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 the unitary group ${{\rm U}(\mathcal{A})}$ with bounded analytic extensions to ${\mathtt{ball}(\mathcal{A},p)}$ in terms of representations of the C*-algebra ${e^{C^{*}(\mathcal{A}, p)}}$.