# Induced $K$-theory maps between $C^*$ algebras.

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.

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).
• 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 $\phi_S, \phi_A$. – CL. Mar 5 at 13:58
• Thanks a lot. Sorry, but I still don't see where the maps $i_S, i_A$ come from. – CL. Mar 5 at 14:54