Hi,
Below is link to the paper where is that statement. Remark after proof of second Theorem.
I didn't see that mistake befor, of course it should be
.
And You are right it is algebra of all continuous functions on the unit interval.
I'm sorry for that mistake.
PS. Here is link to the original paper http://matwbn.icm.edu.pl/ksiazki/sm/sm29/sm29126.pdf
Here the Banach algebra of all real valued bounded continuous functions on . For let . Then one easily proves that . Though the infimum and the supremum may not be attained, one can choose in the range of such that . Now applying the Intermediate Value Theorem, one concludes that there is some such that is in the range of . Consequently, is not a continuous function, in fact it is not defined at . Thus . Since is chosen arbitrary, we see that the result is true for all