Twisted Lie Group C*-Algebras as Strict Quantization

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Abstract

A nonzero 2-cocycle Γ∈ Z2(ℊ, ℝ) on the Lie algebra ℊ of a compact Lie group G defines a twisted version of the Lie–Poisson structure on the dual Lie algebra ℊ*, leading to a Poisson algebra C∞ (ℊ*(Γ). Similarly, a multiplier c∈ Z2(G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C*-algebra C*(G,c)

Further to some superficial yet enlightening analogies between C∞ (ℊ*(Γ)) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Planck's constant ħ assumes values in (ℤ\{0})−1. This means that there exists a continuous field of C*-algebras, indexed by ħ ∈ 0 ∪ (ℤ\{0})−1, for which 𝔄0= C0(ℊ*) and 𝔄ħ=C*(G,c) for ħ ≠ 0, along with a cross-section of the field satisfying Dirac's condition asymptotically relating the commutator in 𝔄ħ to the Poisson bracket on C∞(ℊ*(Γ)). Note that the ‘quantization’ of ħ does not occur for Γ=0.

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