Curvature Formulas of Holomorphic Curves on \(C^*\)-Algebras and Cowen–Douglas Operators



For \(\Omega \subseteq \mathbb {C}\) a connected open set, and \({\mathcal {U}}\) a unital \(C^*\)-algebra, let \({\mathcal {I}} ({\mathcal {U}})\) and \({\mathcal {P}}({\mathcal {U}})\) denote the sets of all idempotents and projections in \({\mathcal {U}}\) respectively. A set \({\mathcal {P}}({\mathcal {U}})\) is called the Grassmann manifold of \(\mathcal {U}\) and \({\mathcal {I}} ({\mathcal {U}})\) is called the extended Grassmann manifold. If \(P:\Omega \rightarrow {\mathcal {P}}({\mathcal {U}})\) is a real-analytic \({\mathcal {U}}\)-valued map which satisfies \(\overline{\partial } PP=0\), then P is called a holomorphic curve on \({\mathcal {P}}({\mathcal {U}})\). In this note, we will define the formulas of curvature and it’s covariant derivatives for holomorphic curves on \(C^*\)-algebras. It can be regarded as a generalization of curvature and it’s covariant derivatives from complex geometry. By using the curvature formula, we give the unitarily and similarity classification theorems for the holomorphic curves and extended holomorphic curves on \(C^*\)-algebras respectively. Furthermore, we also give a description of the trace of the covariant derivatives of curvature for any Hermitian holomorphic vector bundles. Applications include the similarity of holomorphic Hermitian vector bundles and Cowen–Douglas operators.


\(C^*\)-algebras Holomorphic curves Curvatures Similarity classification Cowen–Douglas operators 

Mathematics Subject Classification

Primary 47C15 47B37 Secondary 47B48 47L40 



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Authors and Affiliations

  1. 1.Department of MathematicsHebei Normal UniversityShijiazhuangChina

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