Johnson pseudo-contractibility and pseudo-amenability of θ -Lau product of Banach algebras

Given Banach algebras $ A $ and $ B $ with $ \theta\in\Delta(B) $. We shall study the Johnson pseudo-contractibility and pseudo-amenability of $ \theta $-Lau product $ A\times_{\theta} B $. We show that if $ A\times_{\theta} B $ is Johnson pseudo-contractible, then $ A $ is Johnson pseudo-contractible and has a bounded approximate identity and $ B $ is Johnson pseudo-contractible. In some particular cases complete characterization of Johnson pseudo-contractibility of $ A\times_{\theta} B $ are given. Also, we show that pseudo-amenability of $ A\times_{\theta} B $ implies approximate amenability of $ A $ and pseudo-amenability of $ B $.