homotopy

• Jul 19th 2010, 02:10 AM
Mauritzvdworm
homotopy
$\displaystyle \varphi:A\rightarrow K(H)\otimes_{*}A,a\mapsto p\otimes a$
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 $\displaystyle \psi_t=\text{Ad}u_t\otimes_{*}\text{Id}_{A}:K(H)\o times_{*}A\rightarrow K(H)\otimes_{*}A$
where $\displaystyle (\text{Ad}u)(x)=uxu^*$ and $\displaystyle u_t=e^{itv}$ with v a self-adjoint operator in B(H)

Show that $\displaystyle \varphi_t=\psi_t\varphi:A\rightarrow K(H)\otimes_{*}A$ is a homotopy
• Jul 23rd 2010, 05:09 AM
Mauritzvdworm
linearity of $\displaystyle \varphi$ is easy
$\displaystyle \varphi(a+b)=p\otimes (a+b)=p\otimes a+p\otimes b$

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

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

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

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

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

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