C^∞ functions on the Cantor set, and a smooth m-convex Fréchet subalgebra of O₂

Larry B. Schweitzer

Vol. 184 (1998), No. 2, 349–365

DOI: 10.2140/pjm.1998.184.349

Abstract

We construct a nuclear, spectral invariant, dense Fréchet subalgebra C^∞(K) of the commutative algebra C(K) of continuous complex valued functions on the Cantor set K. The construction uses the group structure of the 2-adic integers on K.

We then use a smooth crossed product construction to get a dense, nuclear Fréchet subalgebra 𝒪₂ of the Cuntz algebra O₂. We prove the general result that a tempered action of a locally compact group on a strongly spectral invariant dense Fréchet subalgebra of a Banach algebra is automatically m-tempered, and obtain the m-convexity of 𝒪₂ as a special case.

Milestones

Received: 10 October 1994

Revised: 18 October 1996

Published: 1 June 1998