On orientation and dynamics in operator algebras. I (Q1270332)

Let \(A\) be a unital JB-algebra. A \(C^*\)-product (\(W^*\)-product) compatible with \(A\) is an associative product \(xy\) on the complex linear space \(A+iA\) which induces the given Jordan product on \(A\) and organizes \(A+iA\) to a \(C^*\)-algebra (von Neumann algebra) with the involution \((a+ ib)^*= a- ib\) and the norm \(\| x\|=\| x^* x\|^{1/2}\). It is clear that a JB-algebra \(A\) is the selfadjoint part of a \(C^*\)-algebra if and only if there exists a \(C^*\)-product compatible with \(A\) on \(A+ iA\), similarly in the JBW-context. The main theorem of the paper proves: A unital JB-algebra \(A\) is isomorphic to the selfadjoint part of a \(C^*\)-algebra if and only if there exists a dynamical correspondence on \(A\). Moreover, each dynamical correspondence gives the ``Lie part'' of a unique \(C^*\)-product compatible with \(A\), and each \(C^*\)-product compatible with \(A\) appears in this way. The same conclusions hold with ``JBW'' in place of ``JB'' and ``\(W^*\)'' or ``von Neumann'' in place of ``\(C^*\)''. The paper also explains how the associative product is determined by a general notion of orientation which is related to dynamics. This concept of orientation bridges an approach in Connes' characterization of the natural cone of a von Neumann algebra and an authors' characterization of the state space of a \(C^*\)-algebra.