Noncommutative Kahler structure on C*-dynamical systems

Notions of noncommutative complex and Kahler structure have been introduced by Frohlich et al. (1999), in the context of supersymmetric quantum theory. Here we show that whenever a C*-dynamical system (A, G, alpha, tau) equipped with a faithful G-invariant trace tau, where G is an even dimensional abelian Lie group, determines a spectral triple, the smooth dense subalgebra A(infinity) inherits a noncommutative Kahler structure. In particular, whenever T-2n acts ergodically on the algebra, it inherits a noncommutative Kahler structure. This produces a class of examples of noncommutative Kahler manifolds. As a corollary, we obtain that all the noncommutative even dimensional tori are noncommutative Kahler manifolds. We explicitly compute the space of complex differential forms and study holomorphic vector bundles on all noncommutative even dimensional tori. (C) 2019 Elsevier B.V. All rights reserved.