12

W. G. BADE, H. G. DALES, Z. A. LYKOVA

splits strongly, and so there is a closed subalgebra D of 21/J with

21/j = £ e (i/J)

and X) = A. Define £ = q^CD)» where q : 21 -+ 21/J is the quotient map. Then £ is a

closed subalgebra of 21 containing J , and £/J = 1) = A.

Now consider the [commutative] extension of A:

By hypothesis, this extension splits strongly, and so there is a closed subalgebra 53 of £

with

£ = 05 0 J .

Clearly 03 is a closed subalgebra of 21 and 03 = A. We have

2l = £ + J = (03 + J) + I = 03 + J

and 03fl/ C £ f l J , so that 03 n I = 03f V = {0}. T h u s 21 = 03 0 / , and so £ splits

strongly.

(ii) This is the same as the proof of (i), save that the subalgebra 03 of £ is not now

known to be closed; we have £ = 03© J , and so 21 = 0 3 0 / . Thus £] splits algebraically. D

1.6. THEOREM. Let J2 ~ XX^5 -0 De a [commutative] extension of a Banach algebra 21,

and suppose that I/ rad / is finite-dimensional.

(i) Suppose that every [commutative] extension of A by rad I splits strongly Then

^2 splits strongly

(ii) Suppose that every [commutative] extension of A by rad / splits algebraically

Then ^T, splits algebraically

PROOF: We apply Proposition 1.5 with J = r a d / = / n rad2t, a closed ideal of 21.

Since I/J is a finite-dimensional, semisimple algebra, every extension of A by I/J splits

strongly by Proposition 1.4(h). The result now follows from Proposition 1.5. •

1.7. THEOREM. Let A be a [commutative] Banach algebra, and let m € N. Suppose

that every [commutative] nilpotent extension of dimension at most m splits strongly

(respectively, splits algebraically). Then every [commutative] extension of dimension at

most 77i splits strongly (respectively, splits algebraically).

PROOF: Let £](2l; /) be a [commutative] extension of A such that / is an ideal in 21

with dim I m. Then rad / is a nilpotent ideal in 21 with dim rad / m, and so every

[commutative] extension of A by rad / splits strongly (respectively, splits algebraically).

The result now follows from Theorem 1.6. •

It follows from Theorem 1.7 that, when considering whether finite-dimensional exten-

sions of a Banach algebra A split, either algebraically or strongly, it is sufficient to consider

nilpotent extensions.