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On Bures distance and \(*\)-algebraic transition probability between inner derived positive linear forms over \(W^*\)-algebras. (English) Zbl 0961.46043
Let \(\nu_1,\nu_2\in M_+^*\) be two positive linear forms on a \(W^*\)-algebra \(M\) and let \(\nu_i^{a_i}\), \(i=1,2\), denote the inner derived positive linear forms, \(\nu_i^{a_i}(\cdot)=\nu_i(a_i^*\cdot a_i)\), where \(a_1,a_2\in M\). The authors obtain variational expressions for the Bures distance between \(\nu_1^{a_1}\) and \(\nu_2^{a_2}\) and for the transition probability \(P_M(\nu_1^{a_1},\nu_2^{a_2})\).

MSC:
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
46L10 General theory of von Neumann algebras
58B10 Differentiability questions for infinite-dimensional manifolds
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
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