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Given a cycle of the form $(\pi,H,0)$ in $KK^G(A,A)$, when is it equivalent to the identity cycle $1_A=(i_A,A,0)$?

The operator $T=0$, and $\pi:A \rightarrow L(H)$ may be injective.

Any criterions how to decide this?

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  • $\begingroup$ I know $l^2(G)$ is a $C_0(G)$-$C_0(G)$ bi-module but how to make it a Hilbert $C_0(G)$-module? What is the inner product? $\endgroup$
    – Zhaoting Wei
    Commented Apr 28, 2015 at 16:56
  • $\begingroup$ I have edited it. $\endgroup$
    – hans
    Commented Apr 28, 2015 at 17:03

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