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



k_0 group for $M(J\otimes K)=0$
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 $[p] + [q] = [q] \Rightarrow [p] = [0]$.


