### Abstract

For 1 < p < 2 and n > 1, let A_{p}(R^{n}) denote the Figa- Talamanca-Herz algebra, consisting of functions of the form (FORMULA PRESENTED) with (FORMULA PRESENTED). We show that if 2n/(n + 1) < p < 2, then the subalgebra of radial functions in A_{p}(R^{n}) is strictly larger than the subspace of functions with expansions subject to the additional condition that /& and gk are radial for all k. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix G Î [FIA]B and consider^{B}A_{P}(G) the subalgebra of B-invariant elements of A_{P}(G). In particular, we show that the dual of B A_{P}(G) is equal to the space of bounded, right-translation invariant operators on L_{P}(G) which commute with the action of B.

Original language | English |
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Pages (from-to) | 665-674 |

Number of pages | 10 |

Journal | Transactions of the American Mathematical Society |

Volume | 286 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1984 |

Externally published | Yes |

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### Keywords

- B-characters
- Central function
- Compact semisimple Lie group
- Convolution operator
- Figa-Talamanca-Herz algebra
- Fourier transform
- Multiplier
- Radial function
- [FIA]g

### Cite this

*Transactions of the American Mathematical Society*,

*286*(2), 665-674. https://doi.org/10.1090/S0002-9947-1984-0760979-0