Journal of Operator Theory

Volume 66, Issue 2, Fall 2011 pp. 301-334.

Differential structures in $C^*$-algebras

Authors: Subhash J. Bhatt (1), Atsushi Inoue (2), and Hidekazu Ogi (3)
Author institution: (1) Sardar Patel University, Vallabh Vidyanagar 388120, India
Fukuoka University, Fukuoka, 814-0180, Japan(2)
(3) Fukuoka Institute of Technology, Fukuoka, 811-0295, Japan

Summary: The paper elaborates the general approach to the differential structure of $C^*$-algebras proposed by Blackadar and Cuntz. The smoothness properties of differential Fr\'{e}chet algebras defined by (not necessarily $\ell^1$-sum\-mable) differential norms are investigated. They are used, by taking appropriate projective limits and inductive limits, to construct and investigate classes of non-commutative smooth algebras describing differential structures defined by differential norms. A large number of examples are discussed exhibiting unified nature of general theory.

Keywords: differential seminorms, derived norms, (strongly) smooth algebras, $C^k$-completion, $C^\infty$-algebras

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