k_0 group for $M(J\otimes K)=0$

Question

Let J be any C*-algebra and K be the C*-algbra of compact operators on a separable, infinite dimensional Hibert space.

How to show $K_0(M(J\otimes K))=0$, where M denotes Multiplier algebra

Answer 1

This is a variation of an argument that is called the Eilenberg swindle. It can be found for example in Blackadars book "K-theory for operator algebras" (see Proposition 12.2.1) and works as follows:

Note that, since $J \otimes K$ is an essential ideal in $M(J) \otimes_{\alpha} M(K) = M(J) \otimes_{\alpha} B(H)$, the latter algebra embeds into $M(J \otimes K)$ (this map is not an isomorphism in most cases -- see for example Proposition 3.8 in the paper "Multipliers of C* algebras" by Akemann, Pedersen and Tomiyama). Nevertheless, this enables us to fiddle around in $B(H)$ to get elements in the stable multiplier algebra.

Now construct a sequence of partial isometries $v_k$ such that $v_k^*v_i = \delta_{k,i}$, where $\delta_{k,i}$ is $1$ for $i = k$ and $0$ else. Moreover $v_k$ should satisfy $\sum_k v_k v_k^* = 1$. If $e_0, e_1, \dots, e_i, \dots$ denotes an orthonormal basis of $H$, we could take for example

$v_1(e_i) = e_{2i}$

$v_2(e_i) = e_{4i+1}$

$v_3(e_i) = e_{8i+3}$

$v_4(e_i) = e_{16i + 7}$

and so on. Now let $p \in M(J \otimes K)$ be any projection and set $q = \sum_i v_i p v_i^*$ and

$w = \begin{pmatrix} 0 & 0 \\\\ v_1 & \sum_{i=1}^{\infty} v_{i+1}v_i^* \end{pmatrix} \cdot \begin{pmatrix} p & 0 \\\\ 0 & q \end{pmatrix}$

and convince yourself that the sum in the lower right corner of the first factor converges with respect to the strict topology.

Now we have

$ww^* = \begin{pmatrix} 0 & 0 \\\\ 0 & q \end{pmatrix}$

and

$w^*w = \begin{pmatrix} p & 0 \\\\ 0 & q \end{pmatrix}$.

Therefore $p \oplus q$ is Murray-von Neumann equivalent to $q$ and in $K_0(M(J \otimes K))$ we have $[p] + [q] = [q] \Rightarrow [p] = [0]$.