## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 21, Number 3 (2014), 523-534.

### Zero products preserving maps from the Fourier algebra of amenable groups

#### Abstract

Let $G$ be a locally compact amenable group. The goal of this paper is to investigate the problem of surjective zero products preserving maps from the Fourier algebra of $G$ into a completely contractive Banach algebra. We show that if $B$ is a completely contractive Banach algebra which is faithful and factors weakly, then every surjective completely bounded linear map from $A(G)$ into $B$ which preserves zero products is a weighted homomorphism. Moreover an equivalent condition is given for such a map to be a homomorphism. In particular, this result implies that if $B$ is a commutative C$^*$-algebra or a matrix space and $T:A(G)\rightarrow B$ is a continuous surjective linear map which preserves zero products, then $T$ is a weighted homomorphism and there is an equivalent condition for $T$ to be a homomorphism.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 3 (2014), 523-534.

**Dates**

First available in Project Euclid: 11 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1407765887

**Digital Object Identifier**

doi:10.36045/bbms/1407765887

**Mathematical Reviews number (MathSciNet)**

MR3250776

**Zentralblatt MATH identifier**

1314.47055

**Subjects**

Primary: 47B48: Operators on Banach algebras

Secondary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

**Keywords**

locally compact group amenable group Fourier algebra operator spaces set of spectral synthesis Weighted homomorphism

#### Citation

Farsani, Jafar Soltani. Zero products preserving maps from the Fourier algebra of amenable groups. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 3, 523--534. doi:10.36045/bbms/1407765887. https://projecteuclid.org/euclid.bbms/1407765887