Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am trying to show that the $K$-theory groups of the following $C^*$-algebra $A$ vanish:

Let $\mathcal{H}$ be a separable Hilbert space. Now consider the subalgebra of $\mathcal{B}(\mathcal{H}\oplus\mathcal{H})$ given by those matrices

\begin{pmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{pmatrix}

where $T_{11}, T_{12}$ and $T_{21}$ are compact and $T_{22}$ is arbitrary but bounded. Schematically this is the algebra

$$A = \begin{pmatrix} \mathcal{K} & \mathcal{K} \\ \mathcal{K} & \mathcal{B} \end{pmatrix} $$

where $\mathcal{K}$ denotes the compact operators of $\mathcal{H}$.

As is said I want to show $K_0(A) = 0 = K_1(A)$.

Unfortunately I do not really know what tools to use here. Any suggestions are appreciated.

share|cite|improve this question
+1 for a question on k-theory. There are not that many. – Raskolnikov Apr 25 '12 at 17:20
to bad there aren't. but thanks anyway :) – mland Apr 25 '12 at 18:24
up vote 6 down vote accepted

Hints: Consider the ideal $$I = \begin{pmatrix} \mathcal{K} & \mathcal{K} \\ \mathcal{K} & \mathcal{K} \end{pmatrix}=\mathcal{K}\otimes\mathbb M_2\mathbb C\cong\mathcal{K} $$ in $A$. What is $A/I$? Write down the six-term exact sequence in K-theory for the extension $$ 0\to I\to A\to A/I\to 0. $$ Plug in what you know about the K-theory of $I$ and $A/I$. The desired result follows.

share|cite|improve this answer
of course. thanks. – mland Apr 25 '12 at 18:23

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.