We investigate generalized amenability and biflatness properties of various (operator) Segal algebras in both the group algebra, $L^1(G)$, and the Fourier algebra, $A(G)$, of a locally compact group~$G$.

Let $\mathcal H$ be a complex separable Hilbert space and ${\mathcal L}({\mathcal H})$ denote the collection of bounded linear operators on ${\mathcal H}$. In this paper, we show that for any operator $A\in{\mathcal L}({\mathcal H})$, there exists a stably finitely (SI) decomposable operator $A_\epsilon$, such that $\|A-A_{\epsilon}\|<\epsilon$ and ${\mathcal{\mathcal A}'(A_{\epsilon})}/\operatorname{rad} {{\mathcal A}'(A_{\epsilon})}$ is commutative, where $\operatorname{rad}{{\mathcal A}'(A_{\epsilon})}$ is the Jacobson radical of ${{\mathcal A}'(A_{\epsilon})}$. Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen-Douglas operators given by C. L. Jiang.