On approximate left φ-biprojective Banach algebras

Let A be a Banach algebra. We introduce the notions of approximate left φ-biprojective and approximate left character biprojective Banach algebras, where φ is a non-zero multiplicative linear functional on A. We show that for a SIN group G, the Segal algebra S(G) is approximate left φ1-biprojective if and only if G is amenable, where φ1 is the augmentation character on S(G). Also we show that the measure algebra M(G) is approximate left character biprojective if and only if G is discrete and amenable. For a Clifford semigroup S, we show that (l^1(S)) is approximate left character biprojective if and only if (l^1(S)) is pseudo-amenable. We study the hereditary property of these notions. Finally we give some examples to show the differences of these notions and the classical ones

A note on the simplicity of C*-algebras of edge-colored graphs

We modify the definitions of hereditary and saturated subsets and Condition (L) for edge-colored graphs. We then give three necessary conditions for the simplicity of C*-algebras of edge-colored graph

On left φ\varphi-biflat Banach algebras

summary:We study the notion of left $\varphi$-biflatness for Segal algebras and semigroup algebras. We show that the Segal algebra $S(G)$ is left $\varphi$-biflat if and only if $G$ is amenable. Also we characterize left $\varphi$-biflatness of semigroup algebra $l^{1}(S)$ in terms of biflatness, when $S$ is a Clifford semigroup