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

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- July 19th 2010, 03:10 AMMauritzvdwormhomotopy

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 - July 23rd 2010, 06:09 AMMauritzvdworm
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