In this paper, let A be an infinite-dimensional stably finite unital simple separable $\mathrm {C^*}$ -algebra. Let $B\subset A$ be a centrally large subalgebra in A such that B has uniform property $\Gamma $ . Then we prove that A has uniform property $\Gamma $ . Let $\Omega $ be a class of stably finite unital $\mathrm {C^*}$ -algebras such that for any $B\in \Omega $ , B has uniform property $\Gamma $ . Then we show that A has uniform property $\Gamma $ for any simple unital $\mathrm {C^*}$ -algebra $A\in \rm {TA}\Omega $ .

We show that the following ${{K}_{0}}$-monoid properties of ${{C}^{*}}$-algebras in the class $\Omega $ are inherited by simple unital ${{C}^{*}}$-algebras in the class $TA\Omega :$ (1) weak comparability, (2) strictly unperforated, (3) strictly cancellative.