# homotopy

• Jul 19th 2010, 03:10 AM
Mauritzvdworm
homotopy
$\varphi:A\rightarrow K(H)\otimes_{*}A,a\mapsto p\otimes a$
show that the above map is a *-homomorphism with A a C*-algebra, H a Hilbert space and p a rank-one projection in K(H).

If we let $\psi_t=\text{Ad}u_t\otimes_{*}\text{Id}_{A}:K(H)\o times_{*}A\rightarrow K(H)\otimes_{*}A$
where $(\text{Ad}u)(x)=uxu^*$ and $u_t=e^{itv}$ with v a self-adjoint operator in B(H)

Show that $\varphi_t=\psi_t\varphi:A\rightarrow K(H)\otimes_{*}A$ is a homotopy
• Jul 23rd 2010, 06:09 AM
Mauritzvdworm
linearity of $\varphi$ is easy
$\varphi(a+b)=p\otimes (a+b)=p\otimes a+p\otimes b$

so is multiplicativity
$\varphi(ab)=p\otimes ab=pp\otimes ab=(p\otimes a)(p\otimes b)$

so is preservation of the adjoint
$\varphi(a^*)=p\otimes a^*=p^*\otimes a^*=(p\otimes a)^*$
This shows that $\varphi$ is indeed a *-homomorphism

To show that $\varphi_t$ is a homotopy we need to be a little more creative

suppose $q\in K(B)$ in another rank-one projection then there exits unitary element $u\in B(H)$ such that $q=upu*$

now let $a\in A$ and let us consider what is happening at t=0
$\varphi_0(a)=\psi_0\varphi(a)=(\text{Id}_{K(H)}\ot imes_{*}\text{Id}_A)(p\otimes a)=p\otimes a=\varphi(a)$

now consider what is happening at t=1
$\varphi_1(a)=\psi_1\varphi(a)=(\text{Ad}u\otimes_* \text{Id}_A)(p\otimes a)=upu^*\otimes_* a=q\otimes a$