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I am currently reading up on K-theory and I am a bit confused by the definition of the topological K-Theory-spectrum for C*-algebras. It is mentioned in many articles as basic, but I can only find one definition which I am confused by. It goes like this: Let A be a $C^*$-algebra. The topoligcal K-spectrum $K^{top}(A)$ is the spectrum where every second space is $$GL(A)=\text{colim}GL_n(A).$$ The spectrum is then given by homotopy-equivalences $$K_0(A) \times BGL(A)\rightarrow \Omega GL(A)$$ and $$GL(A) \rightarrow \Omega (BGL(A))=\Omega (K_0(A) \times BGL(A))$$ that arise from the Bott periodicity theorem.

I want to understand how this agrees with the definition of a spectrum as can be found on Wikipedia. So if every second space is GL(A), what are the other spaces. What are the base points and the structure maps? Thank you

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The odd-level spaces are given by $\Omega GL(A)$. The structure map $\Sigma K(A)_{2n-1} \to K(A)_{2n}$ is adjoint to the identity map $K(A)_{2n-1} = \Omega GL(A) \xrightarrow{=} \Omega GL(A) = \Omega K(A)_{2n}$. The structure map $\Sigma K(A)_{2n} \to K(A)_{2n+1}$ is adjoint to the map $$K(A)_{2n} = GL(A) \to \Omega(K_0(A) \times BGL(A)) \to \Omega(\Omega GL(A)) = \Omega K(A)_{2n+1}$$ where the first map is the second homotopy equivalence you listed and the second is the loop of the first homotopy equivalence.

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