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
where and with v a self-adjoint operator in B(H)
Show that is a homotopy
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
where and with v a self-adjoint operator in B(H)
Show that is a homotopy
linearity of is easy
so is multiplicativity
so is preservation of the adjoint
This shows that is indeed a *-homomorphism
To show that is a homotopy we need to be a little more creative
suppose in another rank-one projection then there exits unitary element such that
now let and let us consider what is happening at t=0
now consider what is happening at t=1