On stability of restricted center properties and continuity of restricted center map under ${\mathrm{\ell }}_p$-direct sum

We study the stability of various restricted center properties and certain continuity properties of the restricted center map. We observe that restricted center property, property-$(P_1)$ and semi-continuity properties of the restricted center map are preserved under ${\mathrm{\ell }}_p$-direct sum $(1\le p<\mathrm{\infty })\mathrm{.}$ It is shown that property-$(P_2)$ is stable under finite ${\mathrm{\ell }}_p$-direct sum, but not under infinite ${\mathrm{\ell }}_p$-direct sum. Additionally, we introduce a notion called property-$(l{P}_2)$ as a sufficient condition for the continuity of the restricted center map. Further, the stability of property-$(l{P}_2)$ is established.