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\(B^*\)-algebra representations in a quaternionic Hilbert module. (English) Zbl 0546.46063

A Hilbert Q-module (closed vector space over quaternion multipliers Q with quaternion-valued scalar product and positive definite norm) is a quantum theory [C. Piron, ”Foundations of Quantum Physics,” New York (1976; Zbl 0333.46050)] given a somewhat ad hoc definition of the quantum state which is justified through the Gleason theorem. In this paper, the authors start with a \(B^*\) algebra A over the reals, which contains a subalgebra \(A_ 0\). It is shown that this isomorphism is an isometry. A two sided quaternion linear mapping \(\rho\) is then defined, for which \(\rho (A_ qaA_{q'})=q\rho (a)q',\) when \(A_ q\) belongs to \(A_ 0\), q to Q, and a to A. In case \(\rho\) is positive and \(\rho (I)=1\), it is called a state. It is shown that linearity of \(\rho\) implies that \(\rho (a^*)=\rho (a)^*,\) and in case it is positive, the Schwarz inequality \(|\rho (a^*b)|^ 2=\rho (a^*a)\rho (b^*b)\) is valid. The following theorems are proven:
1. Let A be a \(B^*\) algebra over the reals which contains a subalgebra \(A_ 0 {}^*\)-isomorphic to the real quaternions Q, and \(\rho\) a two sided quaternion linear state on A. Then, there exists a representation \(\Pi_{\rho}\to {\mathcal B}({\mathcal H}_{\rho})\), where \({\mathcal H}_{\rho}\) is a Hilbert Q-module, and \({\mathcal B}({\mathcal H}_{\rho})\) is the set of bounded Q-linear operators on \({\mathcal H}_{\rho}\). 2. Under the above conditions, let \(\lambda\) be a two sided quaternion linear functional defined on a subspace \(Y\subset A\) which is an \(A_ Q\)-module, and which is bounded by the real function \(p(x)=\| x\|\). Then, \(\lambda\) has a two-sided quaternion linear extension \(\Lambda\) which is also bounded by p(x), and coincides with \(\lambda\) on Y (Hahn-Banach).

MSC:

46N99 Miscellaneous applications of functional analysis
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
81T08 Constructive quantum field theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)

Citations:

Zbl 0333.46050
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References:

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