Remarks on certain complemented subspaces on groups (Q762709)

A closed linear subspace E of a Banach space F is called complemented if there exists a bounded projection from F onto E. Consider first a locally compact Abelian group G and a weak *-closed translation invariant subspace E of \(L^{\infty}(G)\). It is due to Gilbert and Rosenthal, that E is complemented if and only if the spectrum of E, i.e. the intersection of E with the dual group \(\hat G\) of G, belongs to the coset-ring of \(\hat G.\) Using this, a criterion is proved for certain closed subspaces of \({\mathcal L}(L^ 1(G),L^{\infty}(G))\), the Banach space of all bounded linear operators from \(L^ 1(G)\) to \(L^{\infty}(G)\), to be complemented. No results seem to be known for non-Abelian roups G and arbitrary closed (but not weak *-closed) subspaces of \(L^{\infty}(G).\) Therefore, the author studies two specific ones, namely, the closed subalgebras AP(G) and WAP(G) of \(L^{\infty}(G)\) consisting of all continuous left almost periodic functions and left weakly almost periodic functions on G, respectively. he shows that for maximally almost periodic groups G, the following are equivalent: (i) G is finite, (ii) AP(G) is complemented in \(L^{\infty}(G)\), (iii) WAP(G) is complemented in \(L^{\infty}(G)\), and (iv) C(bG) is complemented in \(L^{\infty}(bG)\), where bG denotes the Bohr compactification of G. Note that, contrary to this, for a simple connected non-compact Lie group, AP(G) contains only the constant functions and hence is complemented.