A noncommutative generalization of the Amari-Chentsov \(\alpha\)-connection of P. Gibilisco and G. Pistone [Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1, 325–347 (1998; Zbl 0921.62004)] is formulated. For a semifinite von Neumann algebra \(M\) with a normal semifinite faithful trace \(\tau\), let \(\mathfrak M_\tau\) be the set of all density operators, namely positive elements \(\rho\) of the noncommutative \(L^ 1\) space \(L^ 1(M,\tau )\) with \(\tau (\rho )=1\) and with support \(1\) (in bijective correspondence with the set of all normal faithful states \(\tau_ \rho\) of \(M\)). Any subset \(N\) of \(\mathfrak M_ \tau\) which is a Banach manifold is called a statistical manifold.
For \(\alpha \in (-1,1)\) and \(p=2/(1-\alpha )\), a vector bundle \(\mathcal F^ {\alpha}\) on \(N\) is defined as the union of the set \(\mathcal F^ {\alpha}_\rho\) of all elements \(v\) of the noncommutative \(L^ p\) space \(L^ p(M,\tau_\rho)\) satisfying \(\text{ Re}\int vd\tau_\rho=0\), the union taken over all \(\rho\) in \(N\) (Definition 5.3).
The trivial connection on \(L^ p(M,\tau_\rho)\) induces the natural connection on its unit ball \(S^ p\) by the projection of tangent spaces. The \(\alpha\)-connection \(\nabla^\alpha\) on \(\mathcal F^\alpha\) is defined using the pullback of the natural connection on \(S^ p\) by the Amari embedding \[ A^\alpha\colon \rho\in N\rightarrow\rho^{1/p}\in S^ p. \] This construction of the \(\alpha\)-bundle-connection pair \((\mathcal F^\alpha,\nabla^\alpha)\) reduces to the Amari-Chentsov \(\alpha\)-bundle-connection pair for the commutative case (Theorem 5.1).