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I am currently looking into the Segal-Machine for constructing spectra. I am working with his original article . The first thing that confuses me is the spectrum that arises from a $\Gamma$-space is not pointed. Should I just assume a disjoint basepoint?

The second question is regarding the $K$-Theory of a $C^*$-Algebra. I think if I have a $C^*$-Algebra $A$ and the category $\mathcal{C}$ of finitely generated projective modules over $A$ as a topological category, then the Segal Machine applied to $\mathcal{C}$ gives me the connective topological $K$-Theory-spectrum $K^{top}(A)$. I have been told that it's not too difficult to get the non-connective version from that with Bott periodicity.

Could somebody give me the rough outline of the argument? Thank you!

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  • $\begingroup$ Why do you say that the spectrum is not pointed? It seems pointed to me. What you get is a sequence of pointed spaces $\{X_n\}$ together with equivalences $X_n\xrightarrow{\sim} \Omega X_{n+1}$. $\endgroup$
    – Denis Nardin
    Commented Dec 8, 2016 at 16:11
  • $\begingroup$ I am just learning about spectra so I might totally be missing the point: The spectrum is given on page 295 as A(1), BA(1), B^2A(1)... .What are the basepoints? $\endgroup$
    – Max90
    Commented Dec 8, 2016 at 16:19
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    $\begingroup$ If A is a $\Gamma$-space, then $A(1)$ is a pointed space by definition: you have a map $*→A(0)→A(1)$ that gives the pointing (just pick any equivalence $*→A(0)$, it doesn't matter which since $A(0)$ is contractible). $\endgroup$
    – Denis Nardin
    Commented Dec 8, 2016 at 16:34

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