## Extensions and isomorphisms for the generalized Fourier algebras of a locally compact group.(English)Zbl 1019.43001

Let $$\Gamma$$ be a locally compact abelian group and denote by $$\widehat{\Gamma}$$ the dual group which is also locally compact and abelian. The Fourier transform maps the Banach algebra $$L^1(\Gamma)$$ onto the dense subalgebra $$A_2(\widehat{\Gamma})$$ of $$C_0(\widehat{\Gamma})$$ (the continuous functions on $$\widehat{\Gamma}$$, vanishing at infinity). It follows by Plancherel’s theorem that $$A_2(\widehat{\Gamma})$$ consists precisely of those elements of $$C_0(\widehat{\Gamma})$$ that are convolutions of two $$L^2$$ functions. This observation motivates the definition of the following Banach algebra for an arbitrary, not necessarily abelian, locally compact group $$G$$: let $$p\in (1,\infty)$$ and put $$\check{f_k}(g):=f_k(g^{-1})$$, $$g\in G$$; then $$A_p(G):=\{u\in C_0(G)\mid u=\sum_{k=1}^\infty h_k*\check{f_k} , f_k\in L^p(G) , h_k\in L^q(G) , 1/p+1/q=1 , \sum_{k=1}^\infty \|f_k\|_p \|h_k\|_q<\infty\}$$.
In this rather technical paper the author studies mainly finite-dimensional extensions and order isomorphisms of the Banach algebra $$A_p(G)$$. Also order isomorphisms for the multiplier algebra $$B_p(G)$$ are analysed. In Section 2 the author introduces very briefly the notation and terminology. Further results and motivation may be found, for example, in P. Eymard [Sem. Bourbaki 1969/70, No. 367, 55-72 (1971; Zbl 0264.43006)], W. G. Bade, H. G. Dales and Z. A. Lykova [Mem. Am. Math. Soc. 656 (1999; Zbl 0931.46034)] or V. Runde [Lectures on amenability. Springer Verlag (2002; Zbl 0999.46022)]. One of the results in Section 3 states that if $$G$$ is amenable, then all finite-dimensional extensions of $$A_p(G)$$ split strongly. A partial converse statement, implying the amenability of $$G$$, is stated in Theorem 3.21. In Section 4 different results concerning order isomorphisms are proved. For example, a characterization of order isomorphisms for the pointwise order of the $$B_p(G)$$ algebras is given (cf. Theorem 4.18).

### MSC:

 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. 43A20 $$L^1$$-algebras on groups, semigroups, etc. 43A07 Means on groups, semigroups, etc.; amenable groups

### Citations:

Zbl 0264.43006; Zbl 0931.46034; Zbl 0999.46022
Full Text:

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