On permanence of regularity properties II

In this article, we study the permanence of topological and algebraic dimension type properties of simple unital $C\text{sp}\ast$-algebras. When a pair of unital $C\text{sp}\ast$-algebras $(A,B)$ is associated by a $\ast$-homomorphism $\phi :A\to B$ which is tracially sequentially-split by order zero, we show that under suitable assumptions on either $A$ or $B$, or both $A$ and $B$ \begin{itemize} \item tracial $m$-comparison passes from $B$ to $A$; \item tracial $m$-almost divisibility passes from $B$ to $A$; \item tracial nuclear dimension less than $m$ passes from $B$ to $A$. \end{itemize}