SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS

7

1.1. DEFINITION. Let A be an algebra. Then an algebraic extension of A is a short

exact sequence Y2 = X X ^ -0 °f algebras and homomorphisms

J2 • 0—/-^2l -^A—^Q.

The algebraic extension J ] is:

(i) radical if / C rad 21;

(ii) finite-dimensional if / is a finite-dimensional ideal;

(iii) nilpotent if / is a nilpotent ideal;

(iv) singular if J2 = 0;

(v) commutative if 21 is a commutative algebra;

(vi) annihilator if 21/ = 721 = 0.

The extension splits algebraically if there is a homomorphism 9 : A —• 21 such that

7r o 0 = iA. Two extensions 5Z(^i5-0 an(^ XX^2 5^) are equivalent if there is an

isomorphism t/ : 2lx — • 2l2 such that \j)(x) = x (x € I) and 7T2 O ^ = TT\ .

Of course, by the definition of a short exact sequence, TT is an epimorphism, I is a

non-zero ideal in 21, i is the natural embedding, and L(I) = ker TT.

Certainly a finite-dimensional, radical extension is nilpotent, a singular extension is

nilpotent, and a nilpotent extension is radical. Let XX21;I) be a radical extension of A.

For each (p G $A , we have p | I = 0, and so there is a natural identification of $% with

JA ; we shall make this identification.

A homomorphism 0 : A —» 21 such that TT o 0 = iA is often called a splitting homo-

morphism.

The following definition is essentially that given by Palmer in [Pa2, Definition 1.2.9].

1.2. DEFINITION. Let yl be a Banach algebra. Then an extension of ^4 is a short exact

sequence Yl = XX^5^) which is an algebraic extension of A such that 21 is a Banach

algebra and TT is a continuous epimorphism. The extension ^ is admissible if there is a

continuous linear map Q : A —» 21 such that TT o Q — iA. The extension sp^s strongly

if there is a continuous homomorphism 0 : A —* 21 such that IT o Q = iA.

For such an extension, I is a closed ideal in 21. We sometimes say that $^(21; J)

is an extension of A by the Banach algebra I. Certainly a necessary condition for an

extension ^ to split strongly is that it be admissible, but, in general, this is not a sufficient

condition. In another terminology, an extension of A is admissible if and only if it splits

in the category of Banach spaces.

There are many extensions of Banach algebras which are not admissible. The easiest

example is

0—c

0

—+ e°° —*£oc/co—0;