Among several versions of the concept of cohomology of BANACH algebras, two are the most popular. The first and oldest is the so called continuous (or "ordinary") cohomology introduced by KAMOWITZ in 1962. The second is the so called normal cohomology introduced by KADISON and RINGROSE in the context of operator algebras and reflecting essentially the operator framework. Normal cohomology defined nine years later than ordinary one was initially used a! an effective tool to its computing. Afterwards in the paper of CONNES [l] about the characterization of hyperfinite von NEUMANN algebras in terms of vanishing of their normal cohomology and in several other papers, normal cohomology was demonstrated t o have considerable interest in itself.
Various definitions of oohomology of BANACH algebras are based on the classic purely algebraic concept of HOCHSCHTLD cohomology of associative algebras and they differ from one another by different topological demands to cochains and coefficients. In fifties the founding fathers of homological algebra -CARTAN, EILENBERG, MACLANE -have shown that cohomologies of various algebraic systems, and HOCHSCHTLD cohomology as an important example, are special cases of the unified notion of functor Ext. Later it was proved that an analogical result is valid for the continuous cohomology of BANAUH algebras: t,he latter is a special case of the so called "BANACH Ext" (a similar concept, adapted to BANACH struc-