Wavelet and Linear Algebra

Abstract     In this paper we introduce approximate $\phi$-biprojective Banach algebras, where $\phi$ is a non-zero character. We show that for SIN group $G$, the group algebra $L^{1}(G)$ is approximately $\phi$-biprojective if and only if $G$ is amenable, where $\phi$ is the augmentation character. Also we show that the Fourier algebra $A(G)$ over a locally compact $G$ is always approximately $\phi$-biprojective.

Abstract In this paper, we introduce new notions of $I$-biflatness and $I$-biprojectivity, for a Banach algebra $A$, where $I$  is a closed ideal of $A$. We show that $M(G)$ is $L^{1}(G)$-biprojective ($I$-biflat) if and only if $G$ is a compact group (an amenable group), respectively. Also, we show that, for a non-zero ideal $I$, if the Fourier algebra  $A(G)$ is $I$-biprojective, then $G$ is a discrete group. Some examples are given to show the differences between these new notions and the classical ones.