• On a Class of Smooth Frechet Subalgebras of $C^\ast$-Algebras

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/123/03/0393-0413

• # Keywords

Smooth subalgebra of a $C^∗$-algebra; spectral invariance; closure under functional calculus; Arens–Michael decomposition of a Frechet algebra; Banach $(D^∗_p)$-algebra; Frechet $(D^∗_\infty)$-algebra.

• # Abstract

The paper contributes to understanding the differential structure in a $C^∗$-algebra. Refining the Banach $(D^∗_p)$-algebras investigated by Kissin and Shulman as noncommutative analogues of the algebra $C^p[a,b]$ of 𝑝-times continuously differentiable functions, we investigate a Frechet $(D^∗\infty)$-subalgebra $\mathcal{B}$ of a $C^∗$-algebra as a noncommutative analogue of the algebra $C^\infty[a,b]$ of smooth functions. Regularity properties like spectral invariance, closure under functional calculi and domain invariance of homomorphisms are derived expressing $\mathcal{B}$ as an inverse limit over 𝑛 of Banach $(D^∗_n)$-algebras. Several examples of such smooth algebras are exhibited.

• # Author Affiliations

1. Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar, Gujarat 388 120, India

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019