Appendix: Twisted Rapid Decay
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Appendix: Twisted Rapid Decay by Indira Chatterji, Department of Mathematics, Cornell University, Ithaca NY 14853, USA. email: Throughout this appendix, ? is a finitely generated group, endowed with a length function , and ? is a multiplier on ?. We adopt the notations used in the first paragraph of the paper. Definition 0.1. We will say that the group ? has ?-twisted Rapid Decay property (with respect to the length ) if H∞ (?, ?) ? C ? r (?, ?). We just say that the group ? has the Rapid Decay property (with respect to the length ), if it has the ?-twisted Rapid Decay property (with respect to the length ) for the constant multiplier 1. For short, we shall say that a group ? has property ?-RD if there esists a length function with respect to which ? has the ?-twisted Rapid Decay property. Remark 0.2. In the context of noncommutative geometry, the reduced C?-algebra C?r (?, ?) represents the space of continuous functions on a noncommutative mani- fold, and H∞ (?, ?) the space of of smooth functions on the same noncommutative manifold. This comes from the abelian case, where using Fourier transforms, one easily sees that C?r (Z n) ?= C(Tn) and that H∞ (Z n) ?= C∞(Tn) (for the word length associated to the generating set S
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Appendix: TwistedRapid Decay by Indira Chatterji, Department of Mathematics, Cornell University, Ithaca NY 14853, USA. email: indira@math.cornell.edu Throughout this appendix, Γ is a finitely generated group, endowed with a length function`, andσWe adopt the notations used in the firstis a multiplier on Γ. paragraph of the paper. Definition 0.1.We will say that the group Γ hasσ-twisted Rapid Decay property (with respect to the length`)if ∞ ∗ H(Γ, σ)⊆C(Γ, σ). ` r We just say that the group Γ has theRapid Decay property (with respect to the length`), if it has theσ-twisted Rapid Decay property (with respect to the length `For short, we shall say that a group Γ has) for the constant multiplier 1.property σ-RDif there esists a length function`with respect to which Γ has theσ-twisted Rapid Decay property. ∗ Remark0.2.In the context of noncommutative geometry, the reducedC-algebra ∗ C(Γ, σ) represents the space ofcontinuousfunctions on a noncommutative mani-r ∞ fold, andH(Γ, σ) the space of ofsmoothfunctions on the same noncommutative ` manifold. Thiscomes from the abelian case, where using Fourier transforms, one ∗n n∞n∞n ∼ ∼ tC easily sees thar(Z) =C(T) and thatH(Z) =C(T) (for the word ` n length associated to the generating setS={(±1,0, . . .), . . . ,(0, . . . ,±1)}ofZ). The (σ-twisted) Rapid Decay property can be rephrased as the desirable property that every smooth function on the noncommutative manifold is also a continuous function. Proposition 0.3.Letσbe a multiplier onΓand`be a length function onΓ. The following are equivalent: (1) Γhasσ-twisted Rapid Decay (with respect to the length`). (2)There existsC, s>0such that for anyf∈C(Γ, σ) kfkop≤Ckfks. (3)There exists a polynomialPsuch that for anyf∈C(Γ, σ)andfsupported in a ball of radiusr kfkop≤P(r)kfk`Γ. 2 (4)There exists a polynomialPsuch that for anyf, g∈C(Γ, σ)andfsupported in a ball of radiusr kf∗σgk`Γ≤P(r)kfk`Γkgk`Γ. 2 22 ∞ Proof.(1)⇔(2) As in the case of untwisted Rapid Decay, the inclusionH(Γ, σ)⊆ ` ∗ ∞2∗2 C(Γ, σ) is continuous since both inclusionsH(Γ, σ)⊆`Γ andC(Γ, σ)⊆`Γ r `r ∞ are continuous.SinceH(Γ, σ)aFisecr´tshecepanoisulcniehtoftyuiinntcohe,t ` ∞ ∗ H(Γ, σ)⊆C(Γ, σThe converse is obvious) rephrases as the statement of (2). ` r s+1s sinceH(Γ)⊆H(Γ). ` ` (2)⇒(3)⇒(4) Takef∈C(Γ, σ) supported in a ball of radiusr, then s X s 2 2s kfkop≤Ckfks=C|f(γ)|(1 +`(γ))≤C(1 +r)kfk`Γ. 2 γ∈Γ 1
