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So here is a construction outlined in Higson's note on index theory, pg 46


Let $\mathcal{K}$ denote a $C^*$ algebra of graded compact operators on a graded hilbert space $H=H_0\oplus H_1$. Let $\mathcal{S}=C_0(\Bbb R)$. Let $A$ and $B$ be (not necessarily unital) $C^*$-algebras.

Let $\phi:\mathcal{S} \otimes A \rightarrow B \otimes \mathcal{K}$ be a $*$-algebra homomoprhism.

We want to construction a $K$-theory map $\phi_*:K(A) \rightarrow K(B)$.


He argues the existence as follow, beginning with:

Let $C$ be image of $\mathcal{S} \otimes A$ under $\phi$. Then we obtain homomoprhisms $\phi_{\mathcal{S}}, \phi_A$ of $\mathcal{S}$ and $A$ into the multiplier algebra of $C$.

How does this work? I do not see how this even follows from universal property of multiplier algebras.

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Since $\phi\colon \mathcal{S}\otimes A\to C$ is non-degenerate, there exists a unique strictly continuous extension $\tilde \phi\colon M(\mathcal{S}\otimes A)\to M(C)$. Moreover, the universal property of the tensor product gives canonical maps $\iota_{\mathcal{S}}\colon\mathcal{S}\to M(\mathcal{S}\otimes A)$, $\iota_A\colon A\to M(\mathcal{S}\otimes A)$. The $\ast$-homomorphisms $\phi_{\mathcal{S}}$ and $\phi_A$ are given by the obvious compositions of the previous maps.

On a more technical level, the maps $\phi_{\mathcal{S}}$ and $\phi_A$ are given by $$ \phi_{\mathcal{S}}(f)=\lim_\lambda \phi(f\otimes e_\lambda)\\ \phi_A(a)=\lim_j \phi(g_j\otimes a), $$ where $(g_j)$ and $(e_\lambda)$ are approximate units for $\mathcal{S}$ and $A$ respectively and the limits are taken in the strict topology (of course one has to justify that they exist).

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  • $\begingroup$ Thank you so much. Currently I only know the definition of multiplier algebra as given in nlab: ncatlab.org/nlab/show/multiplier+algebra , assuming its existence to. It seems to me you are using a lot more results then I know. Is there a reference for the results you wrote ? i.e. 1. How is $\tilde{\phi}$ constructed? 2. What is the universal property of tensor product here? 3. How does one obtian the formula for $\phi_S, \phi_A$. $\endgroup$ – CL. Mar 5 at 13:58
  • $\begingroup$ Then you may want to take a look at Section II.7.3. in Blackadar's book on operator algebras (not the one about K-theory). I don't see an easy way to prove this statement using only the universal property of the multiplier algebra. $\endgroup$ – MaoWao Mar 5 at 14:06
  • $\begingroup$ Thanks a lot. Sorry, but I still don't see where the maps $i_S, i_A$ come from. $\endgroup$ – CL. Mar 5 at 14:54
  • $\begingroup$ This is essentially contained in Theorem II.9.2.1. (remember that the multiplier algebra can be characterized as double centralizer of a non-degenerate faithful representation). $\endgroup$ – MaoWao Mar 5 at 15:00
  • $\begingroup$ I am still working through certain definitions, hope you don't mind if i come back in future. $\endgroup$ – CL. Mar 5 at 18:40

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