Abstract. In this paper a certain kind of approximate identities in a Banach algebra is introduced. Many Banach algebras possess such approximate identities. In the case of the group algebras $L^1(G)$ the existence of such approximate identities gives a new characterization of [SIN]-groups. Examples and hereditary properties are also investigated.

AMS Subject Classification (1991): 46H05, 43A20

Received January 4, 1995, and in revised form April 27, 1995. (Registered under 5694/2009.)