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).