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B)a) = R , (b )L (a ) = b L 'L (a ) L L '(a )b = L '(a )L (b ) = L '(a L (b )) = L '(L (a )b ) = L 'L {a )b . A similar calculation for R >R shows that (c3) implies ( c i ) when A satisfies A a = {0 }. I f A a = { 0 } and (C3 ) are both true, then the identities L (a )b = L(ab) = aL (b) = R (a )b and bL(a) — R (b )a = R(ba) = b R (a) for all o, b G A imply (C2 ). I f A 2 = A and (C3 ) both hold, then the identity L (ab) = aL (b ) = i2(d)6 = R (ab) for all a, b G A implies (c2). O A linear operator T = L = R G C (A 1) satisfying condition (c) in the above theorem is an example o f a multiplier.

This proposition explains why we call B f ( X ) the ideal o f finite-rank operators. Note that the equations in (21) show that 8 f ( X , y ) is a type o f generalized ideal in the complex of spaces B ( X , y ) as X and y vary. 7. 2 D o u ble Centralizers and Extensions A double centralizer (sometimes called a double multiplier) of an algebra A is a pair of maps of A into itself. The set of double centralizers of A forms a unital algebra T>(A) under natural operations. We will begin by introducing the regular representation which defines a homomorphism of the original algebra onto an ideal in T>(A).

Hence the map is surjective. Finally we prove the results about semidirect and direct products. First, suppose t can be written a s r = i r o w f o r u 6 CHom(C, V ( A ) ) . , the extension is semidirect). Conversely, if the splitting map x: C —» B is given, we may define ui to be 6 a The statement about when the extension is a direct product is even easier to check. □ We have discussed extensions o f Banach algebras because this is the case in which we are most interested. However the reader can easily strip away all the norm and continuity assumptions from the above discussion and obtain an improvement of Hochschild’s original theory [1947] for extensions of algebras.

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