Rapid decay and functional calculus in ${\rm C}^*$-probability spaces

Abstract

Let $(A,\rho)$ be a tracial ${\rm C}^*$-probability space with the rapid decay property relative to a filtration $L$. We show that the associated Sobolev algebra $H_L^{\infty}(A,\rho)$ is closed under the smooth functional calculus of $A$. Some consequences include norm estimates, the ideal separation property, and the fact that the inclusion $H_L^{\infty}(A,\rho)\subset A$ induces an isomorphism in $K$-theory.