# Approximate biprojectivity and biflatness of some algebras over certain semigroups

Document Type: Original Article

Authors

1 Department of Mathematics‎, University of ‎Tabriz‎, ‎Tabriz‎, ‎Iran.

2 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.

Abstract

‎We investigate (bounded) approximate biprojectivity of $l^1(S)$ for uniformly locally finite inverse semigroups‎. ‎As a consequence‎, ‎we show that when $S=\mathcal{M}(G‎, ‎I)$ is the Brandt inverse semigroup‎, ‎then $l^1(S)$ is (boundedly) approximately biprojective if and only if $G$ is amenable‎. ‎Moreover‎, ‎we study biflatness and (bounded) approximate biprojectivity of the measure algebra $M(S)$ of a topological Brandt semigroup‎.

Keywords

### References

O. Yu. Aristov, On approximation of at Banach modules by free modules, Sb. Math. 196 (2005), no. 11-12, 1553--1583

Y. Choi, Biatness of 1-semilattice algebras, Semigroup Forum 75 (2007), no. 2, 253--271.

H. G. Dales, F. Ghahramani and A. Yu. Helemskii, The amenability of measure algebras, J. London Math. Soc. (2) 66 (2002), no. 1, 213--226.

F. Ghahramani and R. J. Loy, Generalized notions of amenability, J. Funct. Anal. 208 (2004), no. 1, 229--260.

F. Ghahramani and Y.Zhang, Pseudo-amenable and pseudo-contractible Banach algebras, Math. Proc. Camb. Phil. Soc. 142 (2007), no. 1, 111--123.

A. R. Medghalchi and H. Pourmahmood-Aghababa, Figa-Talamanca-Herz algebras for restricted inverse semigroups and Clifford semigroups, J. Math. Anal. Appl. 395 (2012), no. 2, 473--485.

H. Pourmahmood-Aghababa, Approximately biprojective Banach algebras and nilpotent ideals, Bull. Aust. Math. Soc. 87 (2013), no. 1, 158--173.

P. Ramsden, Biatness of semigroup algebras, Semigroup Forum 79 (2009), no. 3, 515--530.

V. Runde, Biatness and biprojectivity of the Fourier algebra, Arch. Math. (Basel) 92 (2009), no. 5, 525--530.

H. Samea, Topological Brandt semigroups, Semigroup Forum 86 (2013), no. 2, 404--412.

G. L. Sleijpen, Convolution measure algebras on semigroups, PhD thesis, Katholieke Universiteit, The Netherlands 1976.

Y. Zhang, Nilpotent ideals in class of Banach algebras, Proc. Amer. Math. Soc. 127 (1999), no. 11, 3237--3242.