Beltita D, Neeb KH (2015)
Publication Type: Journal article, Original article
Publication year: 2015
Publisher: Springer Verlag (Germany)
Book Volume: 83
Pages Range: 517 - 562
Journal Issue: 4
DOI: 10.1007/s00020-015-2244-3
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 c0-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 \({e^{C^{*}(\mathcal{A}, p)}}\).
APA:
Beltita, D., & Neeb, K.H. (2015). Nonlinear completely positive maps and dilation theory for real involutive algebras. Integral Equations and Operator Theory, 83(4), 517 - 562. https://dx.doi.org/10.1007/s00020-015-2244-3
MLA:
Beltita, Daniel, and Karl Hermann Neeb. "Nonlinear completely positive maps and dilation theory for real involutive algebras." Integral Equations and Operator Theory 83.4 (2015): 517 - 562.
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