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.
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