Premium
Continuity of Derivations, Intertwining Maps, and Cocycles from Banach Algebras
Author(s) -
Dales H. G.,
Villena A. R.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610700001770
Subject(s) - bimodule , mathematics , banach algebra , pure mathematics , bilinear interpolation , extension (predicate logic) , algebra over a field , banach space , discrete mathematics , computer science , programming language , statistics
Let A be a Banach algebra, and let E be a Banach A ‐bimodule. A linear map S : A → E is intertwining if the bilinear map ( a , b ) ↦ ( δ 1 S ) ( a , b ) : = a ċ S b ‐ S ( a b ) + S a ċ b , A × A → E ,is continuous, and a linear map D : A → E is a derivation if δ 1 D =0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous. The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A ‐bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [ 1 , p. 36]. Indeed, we prove a somewhat stronger result involving left‐ (or right‐) intertwining maps.