Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality

On Pythagoras Theorem for Products of Spectral Triples

“…Theorem 5 generalizes to arbitrary spectral triples the results of [20], where the first triple was assumed to be unital, and the second one was the canonical spectral triple on C 2 . Let us recall that by unital spectral triple we mean that both the conditions ∃ e ∈ A and π(e) = 1 are satisfied.…”

“…These inequalities already appeared in [20,Prop. II.4], where one of the two spaces was assumed to be the two-point space C 2 and only pure states were considered.…”

“…Using (20) one checks that the pairing with K-theory is [F i ], [p j ] = δ ij , proving that we have a dual pair of generators of K-theory and K-homology.…”

“…Furthermore, in the last case Pythagoras theorem yields a metric interpretation of the Higgs field. Recently, it comes out in [20] that eq. (5) for is also true for the product of the Moyal plane by C 2 , but only between translated states, that is for δ x 1 , δ x 2 the two pure states of C 2 , δ x 1 = ϕ any state of the Moyal algebra and δ x 2 = ϕ • τ κ with τ κ , κ ∈ R 2 , the translation action of R 2 on the Moyal plane.…”

Metric Properties of the Fuzzy Sphere

“…Another noncommutative space where the distance between coherent states has already been studied is the Moyal plane [16,27,37]. In contrast with that example, whose distance is independent of the deformation parameter, here the distance depends on N .…”