Enveloping Dual Banach Algebras and Approximate Properties

Suppose that <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>A</a:mi></a:math> is a Banach algebra and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>F</c:mi><c:mfenced open="(" close=")" separators="|"><c:mrow><c:mi>A</c:mi></c:mrow></c:mfenced></c:math> is its enveloping dual Banach algebra, we show that <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"><h:mi>F</h:mi><h:mfenced open="(" close=")" separators="|"><h:mrow><h:mi>A</h:mi></h:mrow></h:mfenced></h:math> is approximately contractible (approximately amenable) if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M4"><m:mi>A</m:mi></m:math> has the same property. Also, we study the relation between the pseudoamenability of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M5"><o:mi>F</o:mi><o:mfenced open="(" close=")" separators="|"><o:mrow><o:mi>A</o:mi></o:mrow></o:mfenced></o:math> and the pseudoamenability of the second dual <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M6"><t:msup><t:mrow><t:mi>A</t:mi></t:mrow><t:mrow><t:mi>∗</t:mi><t:mi>∗</t:mi></t:mrow></t:msup></t:math> and we also characterize approximate biflatness and approximate biprojectivity of <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M7"><v:mi>F</v:mi><v:mfenced open="(" close=")" separators="|"><v:mrow><v:mi>A</v:mi></v:mrow></v:mfenced></v:math> associated with approximate biflatness and approximate biprojectivity of the second dual <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" id="M8"><ab:msup><ab:mrow><ab:mi>A</ab:mi></ab:mrow><ab:mrow><ab:mi>∗</ab:mi><ab:mi>∗</ab:mi></ab:mrow></ab:msup></ab:math> .