On Lift Of Some Elements In Compact Quantum Metric Spaces

Let A and B be C*-algebras with corresponding *-seminorms L1 and L2 andφbe a *-homomorphism from A into B, this paper mainly deals with the following problem: For every element inφ(Af), under what kind of seminorm can we find a lift in Af which perserves the norms. And we get the following results:When L1 is Leibniz and continuous, for every element inφ(Af) which is self-adjoint, positive or positive and invertible, there exists the corresponding lift in Af which perserves the norms; When L1 and L2 are strongly Leibniz and lower semicontinuous, for every unitary element u inφ(Af) with sp(u)≠T, there exists a lift in Af. Besides, we list the sufficient and necessary conditions on which the lifts perserving the norms are all seminorm-finite.