For a Banach algebra B let L(B), R(B), M(B) denote the Banach algebras of left, right, double multipliers (actually the discussion takes place with only \(\| ab\| \leq K\| a\| \| b\|)\). Assume B is faithful \((acb=0\) for all a, b implies \(c=0)\) and that there exists a continuous injection B’\(\to M(B)\), where B’ is the dual space of B. The authors show that these assumptions have far-reaching consequences, e.g., B’ is an ideal in M(B), M(B’)\(\cong M(B)\cong L(A)\cong R(A)\) for a (concretely identified) Banach algebra A with A’ a Banach space isomorphic to M(B). Further developments concern weakly almost periodic functionals, consequences of the presence of bounded approximate identities and the identification of some Arens second duals. The final section analyzes the examples \(B=\ell^ p\), \(C_ p\), \(L^ p(G)\) (with G compact), and \(B_ A(X)\) (almost-finite-rank operators on reflexive X); this section is an excellent source reference for these standard examples in Banach algebra theory.