Approximate biprojectivity of certain semigroup algebras

In this paper, we investigate the notion of approximate biprojectivity for semigroup algebras and for some Banach algebras related to semigroup algebras. We show that $\ell^{1}(S)$ is approximately biprojective if and only if $\ell^{1}(S)$ is biprojective, provided that $S$ is a uniformly locally finite inverse semigroup. Also for a Clifford semigroup $S$, we show that approximate biprojectivity $\ell^{1}(S)^{**}$ gives pseudo amenability of $\ell^{1}(S)$. We give a class of Banach algebras related to semigroup algebras which is not approximately biprojective

A Conditional Expectation on the Tensor Product of Exel-Laca algebras

We show that the ultragraph $C^*$-algebra $C^*(\mathcal{G}_1\times\mathcal{G}_2)$ can be embedded in $C^*(\mathcal{G}_1)\otimes C^*(\mathcal{G}_2)$ as a $*$-subalgebra. We then use this fact to investigate the existence of a conditional expectation on the tensor product of Exel-Laca algebras onto a certain subalgebra