Topological K-Theory-Spectrum for $C^*$-Algebras

Question

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

Answer 1

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.