ω ℒ {\omega }_{{\mathcal{ {\mathcal L} }}} -biprojective and ω ¯ \overline{\omega } -contractible Banach algebras

Abstract For a given Banach algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> </m:math> {\mathcal{ {\mathcal M} }} and a continuous endomorphism <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> \omega on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> </m:math> {\mathcal{ {\mathcal M} }} , we define <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> </m:math> {\mathcal{ {\mathcal M} }} to be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℒ</m:mi> </m:mrow> </m:msub> </m:math> {\omega }_{{\mathcal{ {\mathcal L} }}} -biprojective and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mo stretchy="true">¯</m:mo> </m:mrow> </m:mover> </m:math> \overline{\omega } -contractible. We then explore the relationship between them. Additionally, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>l</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> <m:mrow> <m:mo>∨</m:mo> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {l}^{1}\left({{\mathbb{N}}}_{\vee }) is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℒ</m:mi> </m:mrow> </m:msub> </m:math> {\omega }_{{\mathcal{ {\mathcal L} }}} -biprojective Banach algebra. Finally, we examine the concept of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> \omega -pseudo amenability and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> \omega -approximate biprojectivity in Banach algebras. We demonstrate that for every unital Banach algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℳ</m:mi> </m:math> {\mathcal{ {\mathcal M} }} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> \omega -approximate biprojectivity and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> \omega -pseudo contractibility coincide.