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Differential Banach algebra norms and smooth subalgebras of $$C^*$$- algebras. (English) Zbl 0813.46036
The classical Gel’fand-Neumark theory establishes a complete duality between the category of commutative $$C^*$$-algebras and that of the locally compact spaces. In this paper the authors consider the general problem of refining this duality to treat subcategories of the latter with additional structure, the most transparent example being that of compact manifolds. There the differentiable structure is stored in the knowledge of the space of smooth functions as a subalgebra of the algebra of continuous ones (in both cases we consider those functions which vanish at infinity in the non-compact case). Motivated by this and other situations they introduce the concept of a differential seminorm which (in spite of its name) is a suitable mapping into the positive cone of the $$\ell^ 1$$-algebra. Such seminorms arise in various situations e.g. from derivations and from homomorphisms from an algebra into a suitable normed graded algebra. After studying such objects from an abstract point of view, they introduce smooth subalgebras of $$C^*$$-algebras those which are complete under a suitable family of such seminorms. The basic properties of such subalgebras are studied in the final section.
The authors conclude their introduction with the following programmatical statement: “Even though some of our definitions are still somewhat experimental, we believe that we essentially have developed the right framework for smooth algebra in the non-commutative setting. The groundwork laid in this paper is only a beginning of a reasonable theory of smooth structures on non-commutative algebras and leads to many intriguing and difficult questions which we have to leave open”.
Reviewer: J.B.Cooper (Linz)

##### MSC:
 46H05 General theory of topological algebras 46L05 General theory of $$C^*$$-algebras 46L87 Noncommutative differential geometry