Rapid decay and functional calculus in ${\mathrm{C}}^{\ast }$-probability spaces

Abstract

Let $(A,\rho )$ be a tracial ${\mathrm{C}}^{\ast }$-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.