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In the paper Cohomology of Maximal Ideal Space, there is a corollary about if $M$ is a compact orientable n-dimensional manifold, then $C(M,\mathbb{C})$ cannot be generated by fewer than n+1 elements.

I am confused by that, consider $S^{1}$, it is only generated by 1 elements only,namely $e^{i\theta}$. Is the definition of "generator" different in this context?

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    $\begingroup$ Don't you also need $e^{-i\theta}$? $\endgroup$
    – Olivier Bégassat
    Commented Oct 20, 2021 at 14:03
  • $\begingroup$ Usually(at least for me), I said the *-algebra generated by n elements means the elements are polynomials of n generators and their adjoint. But I guess I figure out what is the confusion. $\endgroup$
    – Ken.Wong
    Commented Oct 20, 2021 at 14:10
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    $\begingroup$ Their notion of "generated" is explained in the first paragraph of the paper: they only look at polynomials in the generators, without the adjoints. $\endgroup$
    – user126256
    Commented Oct 20, 2021 at 14:15

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