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This is 1.5.8 on page 15 of Dixmier's _C* Algebra_:

Let $A$ be a C*-algebra. For each positive integer $n$, we have $A = A^n$. It is enough to show that each hermitian element $x$ of $A$ is a product of $n$ elements of $A$. Now, if $f_1, \cdots, f_n$ are continuous real-valued functions of a real variable such that $f_1(t)f_2(t) \cdots f_n(t) =t$, and $f_1(0) = \cdots = f_n(0)=0$ then $x=f_1(x) \cdots f_n(x)$.

Here, an element $x$ in a C*-algebra is hermitian if $x^* =x$.

I don't why can the author assume the existence of these functions $f_1, \cdots, f_n$. Would anyone please give me a reason or a method to construct them? Thanks a a lot.

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It seems to me there are infinitely many choices. – Siminore Sep 24 '12 at 8:49
Actually, he doesn't necessarily assume the existence fo such $f_k$ at all: "if $f_1, \ldots, f_n$ are ..." – Hagen von Eitzen Sep 24 '12 at 11:27
up vote 3 down vote accepted

We can take $f_j(t)=|t|^{1/n}$ for $1\leq j\leq n-1$ and $f_n(t)=|t|^{1/n}\operatorname{sgn}(t)$, where $$\operatorname{sgn}(t)=\begin{cases}1&\mbox{if }t>0;\\ -1&\mbox{if }t<0;\\ 0&\mbox{if }t=0. \end{cases}$$

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Thank you very much. – ShinyaSakai Oct 7 '12 at 11:51

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