EXOTIC CYCLIC COHOMOLOGY CLASSES AND LIPSCHITZ ALGEBRAS arXiv:2204.00361v1 [math.KT] 1 Apr 2022 MAGNUS GOFFENG, RYSZARD NEST Abstract. We study the noncommutative geometry of algebras of Lipschitz continuous and Hölder continuous functions where non-classical and novel differential geometric invariants arise. Indeed, we introduce a new class of Hochschild and cyclic cohomology classes that pair non-trivially with higher algebraic K theory yet vanish when restricted to the algebra of smooth functions. Said cohomology classes provide additional methods to extract numerical invariants from Connes-Karoubi’s relative sequence in K -theory. 1. Introduction The algebras of Lipschitz continuous and Hölder continuous functions on a compact manifold are nonseparable Banach algebras, dense in the C ∗ -algebra of continuous functions. From a classical viewpoint, the low regularity often proves unwieldely. In the framework of noncommutative geometry and cyclic cohomology, certain geometric features resembling those arising from the dense subalgebra of differentiable functions persist. The algebra of differentiable functions forms a classical object, that is also well understood in the context of noncommutative differential geometry and cyclic cohomology [4, 6, 7, 16, 23] – indeed forming a key component in the archetypical noncommutative spectral geometry coming from a Dirac operator on the underlying manifold. In this paper, we study the noncommutative differential geometry arising from the algebras of Lipschitz- and Hölder continuous functions and novel features arising therefrom. The space of Lipschitz and Hölder functions on a compact metric space was carefully described in the book of Weaver [24]. We review some relevant parts of [24] and adopt a general view on the algebra of Hölder functions as having directional derivatives along a fibre bundle (with non-separable fibres) over the sphere bundle over the manifold. This forms a classical analogue of notions of non-measurability with respect to singular traces [17]. We provide results that describe the cyclic (co)homology groups of the algebra of Hölder continuous functions in relation to the de Rham homology of the manifold. The main results of this paper concern a new construction of exotic cyclic cohomology classes and Hochschild cohomology classes on the algebra of Hölder continuous functions. The cocycles are constructed from weakly summable Fredholm modules and relies on recent advances in the computation of singular traces [9, 17]. The construction is related to the Connes-Karoubi sequence [5] and Connes-Karoubi’s multiplicative characters, see also the work of Kaad [14, 15] and the work of Bunke [3] on Borel regulators. The cohomology classes pair non-trivially with algebraic K theory yet vanish on the algebra of smooth functions; indeed on any function of better than prescribed regularity. Date: April 4, 2022. 2020 Mathematics Subject Classification. 58J22, 26A16, 19D55, 18F25. MG was supported by the Swedish Research Council Grant VR 2018-0350. 1 2 MAGNUS GOFFENG, RYSZARD NEST Acknowledgements. We are grateful to Heiko Gimperlein, Jens Kaad, Adam Rennie, and Alexandr Usachev for creative discussions in which most of this work took shape. 2. Hölder and Lipschitz algebras Let us recall the space of Lipschitz and Hölder function of a compact metric space. At large, we follow [24] and summarize its salient points. We consider a compact metric space X , and let d denote the metric. We note that for α ∈ (0, 1), also dα is a metric. We let X̃ denote the space X ∪ {∞} topologized as a disjoint union. Since X is compact, X̃ is the one-point compactification of X . We metrize X̃ by declaring d(x, ∞) := 1 for x ∈ X . We identify C(X ) with an ideal in C(X̃ ) consisting of functions vanishing at the point ∞. For a compact metric space (X , d) we define the Lipschitz constant of a function f : X → to be | f (x) − f ( y)| | f |Lip(X ,d) := sup . d(x, y) x6= y C For α ∈ (0, 1), we can define the Hölder constant as | f |C α (X ,d) := | f |Lip(X ,dα ) . We note that for α > 1, dα need not be a metric but the Lipschitz constant nevertheless makes sense. The Hölder algebra is defined for α ∈ (0, 1) by C α (X ) := { f : X → 0 C : | f |C (X ,d) < ∞}. α We apply the convention that C (X , d) := C(X ). When the metric is clear from context, we simply write C α (X ). To have a notation similar for Hölder and Lipschitz algebras, we write C 0,α (X ) = Lip(X , dα ) for α ∈ (0, 1]. The reader should beware that for a Riemannian manifold (M , d g ) equipped with the geodesic distance associated with the Riemannian metric g , being Lipschitz continuous is a much weaker condition than being C 1 ; indeed C 0,1 (M ) = Lip(M , d g ) is non-separable while the space of C 1 -functions C 1 (M ) is separable. Proposition 2.1. The space C α (X , d) is a Banach ∗-algebra in the norm k f kC α (X ) := k f kC(X ) +| f |C α (X ) and if α ∈ (0, 1], C 0,α (X , d) ⊆ C(X ) is dense and closed under smooth functional calculus. ˙ {∞}, we define Definition 2.2. For a compact space X and X̃ = X ∪ (1) ∆X := {(x, x) ∈ X̃ × X̃ : x ∈ X̃ }; (2) ΩX := X̃ × X̃ \ ∆X and (3) X as the space of characters on C b (ΩX )/C0 (ΩX ). The projection onto the i :th coordinate defines a mapping πi : ΩX → X̃ , i = 1, 2. The ∗-monomorphism obtained as the composition π∗i X) C(X̃ ) −→ C b (ΩX ) → C(X is independent of i and the induced quotient mapping is denoted by qX : X → X̃ . The notation X is justified by it being a sort of “thickening” of X ∼ = ∆X inside X̃ × X̃ . This is made precise for a Riemannian manifold below in Proposition 2.10. Proposition 2.3. The space X is second countable if and only if X is finite. Proof. Since ΩX is second countable, the statement that X is second countable is equivalent to C b (ΩX ) being separable. The C ∗ -algebra C b (ΩX ) is separable if and only if ΩX is compact which holds if and only if ∆X is open, i.e. if X is finite.  Definition 2.4. For α > 0, and f ∈ C(X̃ ), we define δα ( f ) ∈ C(ΩX ) by f (x) − f ( y) . δα ( f )(x, y) := d(x, y)α EXOTIC CYCLIC COHOMOLOGY CLASSES 3 We introduce the notation π L := π∗1 : C(X̃ ) → C b (ΩX ) and πR := π∗2 : C(X̃ ) → C b (ΩX ). Proposition 2.5. It holds that C α (X ) = { f ∈ C(X ) : δα ( f ) ∈ C b (ΩX )}. The map δα : C α (X ) → C b (ΩX ) is a derivation for the bimodule structure on C b (ΩX ) defined from π L and πR , that is δα ( f1 f2 ) = δα ( f1 )πR ( f2 ) + π L ( f1 )δα ( f2 ). The map δα defines an equivalent norm on C α (X ), i.e. f , f1 , f2 ∈ C α (X ). kδα ( f )kC b (ΩX ) = max(| f |C α (X ) , k f kC(X ) ), R In particular, δα has closed range and µ 7→ ωµ (a) := Ω δα (a)dµ defines an isomorX phism ¨ « Z α ∗ ∼ C (X ) = M (ΩX )/Wα , where Wα := ∩a∈C α (X ) µ ∈ M (ΩX ) : δα (a)dµ = 0 . ΩX Proof. It follows from a formal algebraic manipulation that δα is a (π L , πR )-derivation if it is well defined. As such, it remains to prove that δα is an isometry for the norm f 7→ max(| f |C α (X ) , k f kC(X ) ). For f ∈ C α (X ), ‚ Œ kδα ( f )kC b (ΩX ) = max sup sup |δα ( f )(x, y)|, sup |δα ( f )(∞, y)| x∈X y∈X̃ \{x} = max  sup x, y∈X , x6= y y∈X  |δα ( f )(x, y)|, sup | f ( y)| = max(| f |C α (X ) , k f kC(X ) ). y∈X In the second step we used our conventions that f (∞) = 0 for f ∈ C(X ) and d( y, ∞) = 1 for all y ∈ X .  X ) as the composition of δα with the Definition 2.6. We define δ̃α : C α (X ) → C(X X ). quotient mapping C b (ΩX ) → C(X X ). Proposition 2.7. The mapping δ̃α is a derivation for the qX∗ -action of C α (X ) on C(X Proof. It is clear that δ̃α is a derivation for the qX∗ -action from the fact that δα is a (π L , πR )-derivation.  A metric space (X , d) is said to separate points uniformly [24, Definition 4.10] if there is a constant a > 1 such that for any x, y ∈ X there is an f ∈ ker(δ̃1 ) with kδ1 f kC b (ΩX ) ≤ a and | f (x) − f ( y)| = d(x, y). Proposition 2.8. The small Hölder algebra hα (X ) := ker(δ̃α ) satisfies • hα (X )∗∗ = C α (X ) as soon as (X , dα ) separates points uniformly. • If 0 < β < α < 1 then hα (X ) = C β (X ) C α (X ) . In particular, if X is a compact Riemannian manifold with its geodesic distance d and α ∈ (0, 1), then hα (X )∗∗ = C α (X ) and hα (X ) = C ∞ (X ) C α (X ) . Proof. The first statement can be found in [24, Theorem 4.38]. The second statement follows from [24, Theorem 2.52 and 8.27]. Finally, the last conclusion follows from the first two statements and a standard mollification argument.  4 MAGNUS GOFFENG, RYSZARD NEST Proposition 2.9. Consider the flip mapping σ : ΩX → ΩX , (x, y) 7→ ( y, x). The flip mapping induces a non-trivial involution σ̃ on X . Elements of δα (C α (X )) ⊆ C b (ΩX ) X ) are odd under σ∗ and σ̃∗ , respectively. and δ̃α (C α (X )) ⊆ C(X 2.1. The Hölder algebra on a manifold. Proposition 2.10. Let X be a Riemannian manifold equipped with its geodesic metric. Then there is a unique quotient mapping qS : X → SX factoring qX : X → X over the projection mapping πS : SX → X and making the following diagram commutative C 1 (X ) d (1) C(SX ) qS∗ Lip(X ) δ̃1 X ), C(X where the left vertical arrow is the inclusion and the top horizontal mapping is defined from the exterior derivative d : C 1 (X ) → C(X , T ∗ X ) when identifying sections in C(X , T ∗ X ) with functions on SX via the pairing of tangent vectors with cotangent vectors. Moreover, C 1 (X ) = { f ∈ C(X ) : δ̃1 ( f ) ∈ qS∗ C(SX )}. As a consequence of Proposition 2.10, we note that for v ∈ SX , and ω ∈ qS−1 (v) ⊆ X , the functional Lip(X ) ∋ f 7→ [δ̃1 ( f )](ω) extends the functional C 1 (X ) ∋ f 7→ v.d f (x), where x = πS (v), and we are using the inclusion SX → T X and the pairing of T X with T ∗X . Proof. Since dC 1 (X ) ⊆ C(SX ) generates C(SX ) as a C ∗ -algebra, the diagram (1) determines qS from dC 1 (X ) whenever qS factors over qX . Thus it suffices to construct qS . Pick a tubular neighborhood N ⊆ X̃ × X̃ of ∆X and a diffeomorphism ν : N ∼ = N ∗X ∗ ∗ with a normal bundle N X → X . Let R : N X \ X → SX denote the projection defined from the quotient of the + -action. After picking a χ ∈ Cc (N ∗ X , [0, 1]) with χ = 1 near the zero section X ⊆ N ∗ X , we can define a linear mapping τ : R R∗ C(SX ) → C b (ΩX ) by C 1 (SX ) − → C 1 (N ∗ X \ X ) ∩ C b (N ∗ X \ X )C0 (N ∗ X ) with the inclusion ν∗ C b (N ∗ X \X )C0 (N ∗ X ) − → C b (ΩX ). The mapping τ is readily verified to be multiplicative modulo C0 (ΩX ) and as such qS is well defined as the dual mapping to the composition τ X ). C(SX ) − → C b (ΩX ) → C(X The last identity in the proposition is easily proved from that whenever δ̃1 ( f ) ∈ qS∗ C(SX ) then all partial derivatives of f are continuous and f is by definition C 1 .  We consider the special case of the Riemannian manifold X = S 1 . We identify S with the unit circle in parametrized by a complex parameter z , |z| = 1, or sometimes denoted by w. We can refine the derivation δ1 by considering 1 C δS 1 ( f )(z, w) := f (z) − f (w) . z−w In a similar way, one defines δ̃S 1 : Lip(S 1 ) → C(SS 1 ). Then f ∈ C 1 (S 1 ) if and only if δ̃S 1 ( f ) ∈ qS∗ 1 C(S 1 ) and δS 1 ( f ) = qS∗ 1 ( f ′ ). In this particular instance, the non-separable algebra L ∞ (S 1 ) provides a more natural object to us when studying Lip(S 1 ) rather than C(SS 1 ). Proposition 2.11. Let X be a compact Riemannian manifold and x 0 ∈ X a point. Define C0α (X \ {x 0 }) := C α (X ) ∩ C0 (X \ {x 0 }) and define dαx 0 ∈ C0α (X \ {x 0 }) by dαx 0 (x) := d(x 0 , x)α . Then both of the spaces EXOTIC CYCLIC COHOMOLOGY CLASSES 5 • C α (X )/dαx C0α (X \ {x 0 }) 0 • C0α (X \ {x 0 })/(C0α (X \ {x 0 }))2 are infinite-dimensional. This proposition shows that any infinitesimal approach to Hölder geometries require infinite dimensional model spaces. Proof. Pick a small enough ǫ > 0 so that B2ǫ (x 0 ) is diffeomorphic to a euclidean ball B2 (0) ⊆ n . We pick a diffeomorphism B2 (0) \ B1 (0) ∼ = B2 (0) \ {0} that extends to a uniformly Lipschitz mapping on B2 (0). We extend this diffeomorphism to a diffeomorphism X \ Bǫ (x 0 ) ∼ = X \ {x 0 } that extends by continuity to a Lipschitz mapping on X . We denote the image of C ∞ (X \ Bǫ (x 0 )) under pullback along this diffeomorphism by V0 ⊆ C b (X \{x 0 }). Define Vx 0 := dαx 0 V0 ⊆ C0α (X \{x 0 }). The proposi- R tion follows if we can show that Vx 0 /(Vx 0 ∩ (C0α (X \ {x 0 }))2 ) is infinite-dimensional. We restrict our attention to the case n > 1 where the existence of an injection C ∞ (S n−1 )/ 1 → Vx 0 /(Vx 0 ∩ (C0α (X \ {x 0 }))2 ) proves the result. The case n = 1 goes similarly.  C Remark 2.12. Proposition 2.11 provides a reason for why one can expect the cyclic cohomology of Hölder spaces to be complicated: the ordinary methods of localization applying in presence of higher regularity (cf. [23, Chapter 3]) are not applicable. 2.2. Operator algebra structure. The Hölder algebra admits additional structure. We equip it with the structure of an operator ∗-algebra. Recall that an operator algebra is a Banach algebra which is matrix normed and a completely bounded multiplication. Equivalently, an operator algebra is isomorphic to a closed subalgebra of a C ∗ -algebra. For more details on operator algebras, see [1]. The notion of an operator ∗-algebra was developed in [2]. The following two propositions are clear from construction. Proposition 2.13. Let X be a compact metric space. The mapping  ‹ π L (a) 0 , iα : C α (X ) → C b (ΩX , M2 ( )), a 7→ δα (a) πR (a) C realizes C α (X ) as a closed subalgebra of a C ∗ -algebra and induces an operator ∗-algebra structure on C α (X ). Proposition 2.14. Let M be a compact Riemannian manifold and fix a first order differential operator D acting between two hermitean vector bundles E, F → M such that σ2 (D∗ D)(x, ξ) = |ξ|2 id E coincides with the metric on M . The mapping  ‹ a 0 i D : Lip(M ) → L ∞ (M , End(E ⊕ F )), a 7→ . [D, a] a realizes Lip(M ) as a weak *-closed subalgebra of L ∞ (M , End(E ⊕ F )). 3. Cyclic (co)homology We shall be interested in the cyclic cohomology of Hölder algebras. In this section we recall the construction of cyclic (co)homology of a unital Banach algebra. We write ˜ for the projective tensor product. The ⊗alg for the algebraic tensor product and ⊗ constructions’ dependence on the Banach structure is through the tensor product, and applies to any choice of tensor product making products and flip maps continuous. For a Banach algebra A we write ˜ Ck (A) := A⊗k+1 and C k (A) := Ck (A)∗ . 6 MAGNUS GOFFENG, RYSZARD NEST In particular, C k (A) consists of k+1-linear functionals c on A such that |c(a0 , . . . , ak )k ® ka0 k · · · kak k. We consider the cyclicity operator Λ : Ck (A) → Ck (A), Λ(a0 ⊗ a1 ⊗ · · · ⊗ ak ) := (−1)k a1 ⊗ · · · ⊗ ak ⊗ a0 , and the Hochschild boundary operator b : Ck (A) → Ck−1 (A), b(a0 ⊗ a1 ⊗ · · · ⊗ ak ) := k−1 X j=0 (−1) j a0 ⊗ · · · ⊗ a j a j+1 ⊗ · · · ⊗ · · · ⊗ ak + + (−1)k ak a0 ⊗ a1 ⊗ · · · ⊗ ak−1 . We write Ckλ (A) := Ck (A)/(1 − Λ)Ck (A) and define Cλk (A) ⊆ C k (A) as the sub-complex of Λ-invariant elements. Definition 3.1. Let A be a unital Banach algebra. The Hochschild homology of A is defined as the homology of (C∗ (A), b) and is denoted by H H∗ (A). The Hochschild cohomology of A is defined as the cohomology of (C ∗ (A), b) and is denoted by H H ∗ (A). The cyclic homology of A is defined as the homology of (C∗λ (A), b) and is denoted by H C∗λ (A). The cyclic cohomology of A is defined as the cohomology of (Cλ∗ (A), b) and is denoted by H Cλ∗ (A). We use the notation H C∗λ (A) to distinguish the model we use from that of the bicomplex normally used to define H C∗ (A), even if the two produce isomorphic results. We sometimes consider reduced (co)-homology groups, i.e. ker(b)/im (b) , which we indicate by an additional (super)subscript red. For instance, H C∗λ,red (A) := ker(b : λ λ Ckλ (A) → Ck−1 (A))/bCk+1 (A). 3.1. Connes’ SBI-sequence, Chern character and character formula. Cyclic (co)homology comes with a rich structural theory, see for instance [4, 7, 16]. The following result of Connes relates cyclic (co)homology with Hochschild (co)homology, a result proven using the cyclic bicomplex. Theorem 3.2 (Connes [7], Chapter III.1.γ). Let A be a Banach algebra. Then there is a long exact sequence in homology B S I B S I λ λ λ (A) − → ··· , (A) − → H C∗−1 → H H∗+1 (A) − → H C∗+1 (A) − → H C∗λ (A) − ··· − → H C∗+2 and a long exact sequence in cohomology. B S I B S I ··· − → H Cλ∗−2 (A) − → H Cλ∗ (A) − → H H ∗ (A) − → H Cλ∗−1 (A) − → H Cλ∗+1 (A) − → ··· , The sequence from Theorem 3.2 is called the SBI-sequence. Definition 3.3. A bounded Fredholm module on A is a triple (π, H , F ) where π is a bounded representation of A on the Hilbert space H and F is a bounded operator such that (2) [F, b], b(F ∗ − F ), b(F 2 − 1) ∈ K(H ), ∀b ∈ π(A) ∪ π(A)∗ . If b(F ∗ − F ) = b(F 2 − 1) = 0, we say that (π, H , F ) is a strict Fredholm module. When H is equipped with a Z/2-grading, in which F is odd and A acts as even operators, we say that (π, H , F ) is an even Fredholm module, or of parity 0. Otherwise we say that (π, H , F ) is an odd Fredholm module, or of parity 1. EXOTIC CYCLIC COHOMOLOGY CLASSES 7 We recall the space of Schatten class operators L p (H ) = {T ∈ K(H ) : Tr(|T | p ) < ∞}, and the space of weak Schatten class operators L (p,∞) (H ) = {T ∈ K(H ) : µk (T ) = O(k−1/p )}. Here µk denotes the k:th singular value. See more in [22] or [7, Chapter IV, Appendix C]. A Fredholm module (π, H , F ) of A is said to be p-summable if [F, b] ∈ L p (H ), b(F ∗ − F ), b(F 2 − 1) ∈ L p/2 (H ), ∀b ∈ π(A) ∪ π(A)∗ . Similarly, one defines the notion of a (p, ∞)-summable Fredholm module. Examples of (p, ∞)-summable Fredholm modules can be found below in Section 6. We write STr : L 1 (H ) → for the supertrace when H is /2-graded. For notational convenience, we use the same notation also for the trace when H is ungraded. For an extended limit ω, i.e. a state ω ∈ ℓ∞ ( )∗ annihilating c0 ( ), we let STrω : L 1 (H ) → denote the associated super-Dixmier trace with the same convention as above if H is ungraded. Write ¨ 0, n even, #n := 1, n odd. C Z N C N Theorem 3.4. Let (π, H , F ) be a strict n + 1-summable bounded Fredholm module of parity #n. Then the cochain chnC C (F ).(a0 , . . . , an ) := cn STr (F [F, a0 ] · · · [F, an ]) , is a cyclic n-cocycle. Here ¨ (−1)n(n−1)/2 Γ (n/2 + 1), cn = p 2i(−1)n(n−1)/2 Γ (n/2 + 1), for n even, for n odd, is choosen so that the class [chnC C (F )] ∈ H Cλn (A) satisfies that S[chnC C (F )] = [chn+2 (F )]. CC The reader can find a proof of Theorem 3.4 in [7, Chapter IV.1.β ]. Let K∗top (A) denote the topological K -theory of A. The class [chnC C (F )] ∈ H Cλn (A) is called the Connes-Chern character of (π, H , F ) and is of interest as it in the pairing K∗top (A) × H Cλ∗ (A) → C, computes the pairing with the K -homology class [F ] ∈ K ∗ (A) under the index pairing Z 〈·, ·〉 : K∗top (A) × K ∗ (A) → . Definition 3.5. An unbounded Fredholm module on A is a triple (π, H , D) where π is a bounded representation of A on the Hilbert space H and D is a densely defined self-adjoint operator on H such that (1) For any b ∈ π(A) ∪ π(A)∗ , we have that bDom (D) ⊆ Dom (D) and [D, b] is bounded in the norm of H . (2) For any b ∈ π(A) ∪ π(A)∗ , the operator b(i ± D)−1 is compact on H . When H is equipped with a /2-grading, in which D is odd and A acts as even operators, we say that (π, H , D) is an even Fredholm module, or of parity 0. Otherwise we say that (π, H , D) is an odd Fredholm module, or of parity 1. If for any b, we have b(i ± D)−1 ∈ L p (H ) or b(i ± D)−1 ∈ L (p,∞) (H ) we say that (π, H , D) is p-summable or (p, ∞)-summable, respectively. Z 8 MAGNUS GOFFENG, RYSZARD NEST It follows from the results of [20] that if (π, H , D) is an unbounded Fredholm module, then (π, H , D|D|−1 ) is a bounded Fredholm module. In fact, the operator estimate of [20] implies that if (π, H , D) is p-summable or (p, ∞)-summable, then so is (π, H , D|D|−1 ). Theorem 3.6. Let (π, H , D) be an (n, ∞)-summable unbounded Fredholm module of parity #n. Fix an extended limit ω. Then the cochain
n Hochω (D).(a0 , . . . , an ) := STrω a0 [D, a1 ] · · · [D, an ](1 + D2 )−n/2 n is a Hochschild n-cocycle. The class [Hochω (D)] ∈ H H n (A) is independent of ω and satisfies that n [Hochω (D)] = I [chnC C (D|D|−1 )]. Theorem 3.6 is called Connes’ character formula and its proof can be found in [7, Chapter IV.2.γ]. Corollary 3.7. Let (π, H , D) be an (n, ∞)-summable unbounded Fredholm module of parity #n. Assume that [Hochn (D)] 6= 0 ∈ H H n (A). Then F := D|D|−1 does not make (π, H , F ) into an n − 1-summable bounded Fredholm module of parity #n. Proof. If (π, H , F ) is an n − 1-summable bounded Fredholm module of parity #n, then by Theorem 3.2 and 3.6, (D|D|−1 )] = 0. [Hochn (D)] = I [chnC C (D|D|−1 )] = I S[chn−2 CC  3.2. The Connes-Karoubi sequence and pairings with cohomology. For Banach algebras, the pairing of Connes’ SBI sequence with algebraic K -theory, and a relative group for the map into topological K -theory, was studied by Connes-Karoubi [5]. We recall it here and provide an application to obstructions against finite summability of Fredholm modules therenext. Theorem 3.8 (Connes-Karoubi [5]). Let A be a Banach algebra. Then there is a commuting diagram with exact rows: top . . . −−−−→ Kn+1 (A) −−−−→   chn+1 y S Knrel (A)   chrel n y −−−−→ Knalg (A) −−−−→ Kntop (A) −−−−→ . . .     1 chn y n! Dn y . 1 nB I λ λ . . . −−−−→ H Cn+1 (A) −−−−→ H Cn−1 (A) −−−−→ H H n (A) −−−−→ H Cnλ (A) −−−−→ . . . Here K∗alg (A) denotes the algebraic K -theory of A and K∗rel (A) is the relative group for the forgetful map K∗alg (A) → K∗top (A) considered in [5]. In the same spirit as in Corollary 3.7, Theorem 3.8 provides an obstruction in algebraic K -theory to finite summability. N Corollary 3.9. Let n ∈ , and (π, H , F ) be a bounded Fredholm module of parity #n for a Banach algebra A. Assume that there is an element x ∈ im(Knalg (A) → Kntop (A)), such that the index pairing is nontrivial: 〈x, [(π, H , F ))〉 6= 0. Then F does not define an n − 1-summable bounded Fredholm module of parity #n. EXOTIC CYCLIC COHOMOLOGY CLASSES 9 Proof. We can assume that (π, H , F ) is n + 1-summable because if not the statement of the corollary holds trivially. Let y ∈ Knalg (A). By Connes-Karoubi’s theorem we have that  1  n 1 I chC C (F ) .Dn ( y). 〈 j∗ y, [(π, H , F ))〉 = chnC C (F ).ch( j∗ y) = chnC C (F ). [I Dn ( y)] = n! n! If F is n − 1-summable bounded Fredholm module of parity #n, then I chnC C (F ) = I Schn−2 (F ) = 0. Therefore the assumption of the corollary contradicts n−1-summability. CC  4. Properties of maps induced from C α ,→ C β We are interested in the cyclic (co)homology groups of the algebra of Hölder continuous functions. In large degrees, much can be said using the inclusion C ∞ ,→ C α and the Connes-Hochschild-Kostant-Rosenberg theorem. We study large degrees in Subsection 4.1. In low degrees, less is known and we give results showing that C ∞ ,→ C α provides little information. We study low degrees in Subsection 4.2. 4.1. The map C ∞ (X ) → C α (X ) in large degrees. The cyclic (co)homology groups of C ∞ (X ) were computed by Connes [7, Chapter III.2.α] using the Hochschild-KostantRosenberg theorem and the SBI-sequence. Theorem 4.1 (Connes-Hochschild-Kostant-Rosenberg theorem). Let X be a compact manifold. Let Ωk (X ) denote the space of smooth k-forms on X with the exterior differential d. Let Ωk (X ) denote the space of k-currents and Zk (X ) ⊆ Ωk (X ) the space of closed k-currents. Then there are natural isomorphisms H Ckλ (C ∞ (X )) ∼ = Ωk (X )/dΩk−1 (X ) ⊕ k−2 j ⊕∞ H j=1 dR (X ) and H Cλk (C ∞ (X )) ∼ = Zk (X ) ⊕ dR ⊕∞ H (X ) j=1 k+2 j The SBI-sequence in the case of C ∞ (X ) can be described further; in terms of negative cyclic homology this was done in [16, Chapter 5.1.12]. Using the ConnesHochschild-Kostant-Rosenberg theorem we can deduce a relation between classical de Rham (co)homology groups and large degree cyclic (co)homology groups of the algebra of Hölder continuous functions. Theorem 4.2. Let X be a compact manifold, α ∈ (0, 1], and k > n/α. • The induced map H Ckλ (C ∞ (X )) → H Ckλ (C α (X )), is injective. • The induced map H Cλk (C α (X )) → H Cλk (C ∞ (X )), is surjective. Proof. By the Connes-Hochschild-Kostant-Rosenberg theorem, the pairing H Ckλ (C α (X ))× H Cλk (C α (X )) → is perfect for k > n so the injectivity of H Ckλ (C ∞ (X )) → H Ckλ (C α (X )) follows from the surjectivity of H Cλk (C α (X )) → H Cλk (C ∞ (X )). Since the Chern character induces an isomorphism ch∗ : K∗ (X )⊗ → H∗dR(X ) and Poincaré duality K ∗ (T ∗ X ) → K∗ (X ) is an isomorphism whose range at the level of cycles consists of bounded Fredholm modules defined from zeroth order pseudo-differential operators, the theorem now follows from [11, Theorem 3.5], see also [10].  C R 10 MAGNUS GOFFENG, RYSZARD NEST 4.2. Approximating the diagonal and the inclusion C β ,→ C α if β > 2α. Theorem 4.3. Let X be a compact metric space. Then the space of Hochschild cocycles satisfies Z1 (C β (X )) ⊆ B1 (C α (X )) C α (X 2 ) , for β > α. If X is a compact Riemannian manifold and β > 2α, the same is true with C α (X )⊗min 2 replacing C α (X 2 ). To prove this theorem, we need a lemma. We note that for the minimal tensor product, the condition β > 2α can at best be relaxed to β ≥ 2α in general as per the remark after Proposition 4.6. Lemma 4.4. Let X be a compact metric space. Then there is a sequence (∆ j ) j∈N ⊆ Lip(X ) approximating the diagonal in the sense that i) ∆ j (x, x) = 1 for x ∈ X ; ii) k∆ j kLip(X ×X ) = O( j); iii) supp(∆ j ) ⊆ {(x, y) ∈ X × X : d(x, y) < 1j }. If X is a Riemannian manifold, we can take (∆ j ) j∈N ⊆ Lip(X ) such that additionally iv) k(δ1 ⊗ δ1 )(∆ j )kC b (ΩX ×ΩX ) = O( j 2 ). Proof. Take a positive, decreasing function χ ∈ Cc∞ [0, 1) with χ(t) = 1 in a neighbourhood of t = 0. It is a short computation to verify that ∆ j (x, y) := χ( jd(x, y)), satisfies the stated properties.  Proof of Theorem 4.3. Let s denote the map Ck (A) → Ck+1 (A), s(x) := 1 ⊗ x . Define the linear maps T j : C1 (C β (X )) → C α (X 3 ), For x = P l f l,0 T j (x) := (∆ j ⊗ 1) · s(x), j∈ N. ⊗ f l,1 , we compute that X bT j (x) = x + ∆ j · ( f l,0 f l,1 ⊗ 1 − 1 ⊗ f l,0 f l,1 ). l As such, the first part of the theorem is proved upon showing that ∆ j (1⊗ f − f ⊗1) → 0 in C α (X × X ) uniformly in f ∈ C β (X ) if β > α. The second statement in the minimal tensor product is proven upon showing that ∆ j (1 ⊗ f − f ⊗ 1) → 0 in C α (X ) ⊗min C α (X ) uniformly in f ∈ C β (X ) if β > 2α. The second statement is proven along similar lines as the first, using the injectivity of the minimal tensor product and the embedding iα from Proposition 2.13, and is therefore omitted. Write U j = {(x, y) ∈ X × X : d(x, y) < 1j }. We have that k · kC α = k · kC 0 + | · |C α . We estimate that k∆ j (1 ⊗ f − f ⊗ 1)kC 0 (X ×X ) = k∆ j (1 ⊗ f − f ⊗ 1)kC 0 (U j ) ≤ k1 ⊗ f − f ⊗ 1kC 0 (U j ) ® | f |C β j −β . Note that 1− α α β |1 ⊗ f − f ⊗ 1|C β (U j ) ≤ j β −1 k f kC β (X ) , |1 ⊗ f − f ⊗ 1|C α (U j ) ≤|1 ⊗ f − f ⊗ 1|C 0 (U ) j and by a similar trick |∆ j |C α (U j ) ® j α . EXOTIC CYCLIC COHOMOLOGY CLASSES 11 We can therefore estimate that |∆ j (1 ⊗ f − f ⊗ 1)|C α (X ×X ) = |∆ j (1 ⊗ f − f ⊗ 1)|C α (U j ) ≤ ≤|∆ j |C α (U j ) k1 ⊗ f − f ⊗ 1kC 0 (U j ) + k∆ j kC 0 (U j ) |1 ⊗ f − f ⊗ 1)|C α (U j ) ® ®k f kC β (X ) j −γ , for ‹  α γ = min 1 − , β − α . β We have that γ > 0 if β > α. We conclude that k∆ j (1 ⊗ f − f ⊗ 1)kC α (X ×X ) ® k f kC β (X ) j −γ , and the theorem follows.  Remark 4.5. If the linear map T j from the proof of Theorem 4.3 would be i) continuous C1 (C β (X )) → C2 (C α (X )) in the projective norm and ii) ∆ j (1 ⊗ f − f ⊗ 1) → 0 in ˜ α (X ) uniformly in f ∈ C β (X ) for β > 2α, then the map H H1red (C β (X )) → C α (X )⊗C red H H1 (C α (X )) is the zero map as soon as β > 2α. The argument dualizes to show the 1 1 vanishing also of H Hred (C α (X )) → H Hred (C β (X )). In particular, we could conclude that H C1red (C β (X )) → H C1red (C α (X )) is the zero map as soon as β > 2α. The next result provides an example when the map H C1red (C β (X )) → H C1red (C α (X )) is non-zero in the range 2α > β > α. Proposition 4.6. Let α ∈ (1/2, 1]. Then it holds that [z ⊗ z −1 ] 6= 0 in H C1red (C α (S 1 )). In fact, there is a cyclic 1-cocycle c1 continuous on C α (S 1 ) for α > 1/2 with [c1 ].[z ⊗ z −1 ] 6= 0. Proof. Let F denote the phase of the Dirac operator on S 1 . The cyclic 1-cocycle Z 1 c(a, b) := adb = Tr(F [F, a][F, b]), 2π S 1 extends by continuity to C α (S 1 ) for α > 1/2. For details, see [7, Chapter III.2.α] or [10]. A direct computation shows that [c1 ].[z ⊗ z −1 ] = −1.  The formula c(a, b) = Tr(F [F, a][F, b]) shows that c1 is continuous in the minimal tensor product for C α (S 1 ) for α > 1/2. In particular, the inclusion Z1 (C β (S 1 )) ⊆ C α (S 1 )⊗min C α (S 1 ) B1 (C α (S 1 )) fails for α < β < 2α when α > 1/2. In light of Remark 4.5 we make the following conjecture. We find the problem interesting seeing that Proposition 4.6 would then demonstrate a cutoff behaviour. Conjecture 4.7. Let α ∈ (0, 1/2). Then it holds that [z ⊗ z −1 ] = 0 in H C1red (C α (S 1 )). 5. Singular Chern characters of weakly summable Fredholm modules We now introduce a novel class of cocycles in cyclic and Hochschild cohomology using singular traces. They are associated with weakly summable Fredholm modules. Examples thereof will be computed below in Section 6. 12 MAGNUS GOFFENG, RYSZARD NEST Definition 5.1. Let (π, H , F ) be a strict (p + 1, ∞)-summable Fredholm module of parity #p, for a p ∈ , on a Banach algebra A. Fix an extended limit ω and let Trω denote the associated Dixmier trace on L 1,∞ . Define the p + 1-cochain
hω (a0 , a1 , . . . , a p+1 ) = pSTrω Fa0 [F, a1 ] · · · [F, a p+1 ] , N and the p-cochain
cω (a0 , a1 , . . . , a p ) = STrω F [F, a0 ] · · · [F, a p ] . Proposition 5.2. The p + 1-cochain hω is a Hochschild p + 1-cocycle and the p-cochain cω is a cyclic p-cocycle. Proof. One computes that (3)
bhω (a0 , a1 , . . . , a p+1 , a p+2 ) = pSTrω [F, a p+2 ]a0 [F, a1 ] · · · [F, a p+1 ] = 0. The last equality follows from the fact that [F, A] ⊆ L (p+1,∞) (H ) and L (p+1,∞) (H ) p+1 ⊆ L 1 (H ) on which any singular trace vanishes. That cω is a cyclic p-cocycle follows from standard considerations as in [7, Chapter III].  The fact that we are using singular traces manifests itself through the identity (3) ensuring that hω is a Hochschild p + 1-cocycle. This fact leads us to non-classical features such as the following theorem. Theorem 5.3. Let (π, H , F ) be a (p + 1, ∞)-summable Fredholm module of parity #p, p for a p ∈ , on a Banach algebra A. The classes [hω ] ∈ H H p+1 (A) and [cω ] ∈ H Cλ (A) satisfy N p B[hω ] = p[cω ] ∈ H Cλ (A) p+2 S[cω ] = 0 ∈ H Cλ (A). Proof. By definition,
B hω (a0 , a1 , . . . , a p ) = hω (1, a0 , a1 , . . . , a p ) = pSTrω F [F, a0 ] · · · [F, a p ] = pcω (a0 , a1 , . . . , a p ). Since [cω ] ∈ im (B), S[cω ] = 0 follows.  q,∞ Write L0 (H ) for the closure of the finite rank operators in L q,∞ (H ). By definition, continuous singular traces vanish on L01,∞ (H ). Proposition 5.4. Let (π, H , F ) be a strict (p + 1, ∞)-summable Fredholm module of parity #p, for a p ∈ , on a Banach algebra A. Assume that A0 ⊆ A is a Banach subalgebra such that N p+1,∞ [F, a] ∈ L0 (H ), ∀a ∈ A0 . Let j : A0 ,→ A denote the inclusion. Then the classes [hω ] ∈ H H p+1 (A) and [cω ] ∈ p H Cλ (A) satisfy j ∗ [hω ] = 0 ∈ H H p+1 (A0 ) p j ∗ [cω ] = 0 ∈ H Cλ (A0 ). Proof. The proposition follows from the definition of hω and the fact that Hölder es′ q,∞ timates give us the inclusion L0 (H )L q ,∞ (H ) ⊆ L01,∞ (H ) for any two conjugate ′ exponents q and q .  From Theorem 3.8 and 5.3 we can deduce the following. EXOTIC CYCLIC COHOMOLOGY CLASSES 13 Theorem 5.5. Let (π, H , F ) be an (n, ∞)-summable Fredholm module of parity #n−1, for an n ∈ , on a Banach algebra A. Then we have a commuting diagram N Knrel (A) Knalg (A) chrel n 1 n! 1 nB λ H Cn−1 (A) Dn H H n (A) cω C. hω The reader should compare Theorem 5.5 to the pairing of Connes-Karoubi’s multiplicative character with the Connes-Karoubi sequence first studied in [5], and further developed by Kaad [14, 15]. Connes-Karoubi’s multiplicative character only maps H H n (A) → / . The quotient by is needed to make the map well defined since the Connes-Chern character of a Fredholm module does not lift to a Hochschild cocycle, a situation quite different from that of Theorem 5.3. CZ Z 6. Examples of nontrivial singular Chern characters on Hölder algebras In this section we will compute the singular Chern characters in examples and show that it produces non-trivial classes in cyclic cohomology. A result from [9] allows us to produce an abundance of examples of weakly summable Fredholm modules on Hölder algebras. We write Ψ 0 (X ; E) ⊆ B (L 2 (X ; E)) for the ∗-algebra of order zero classical pseudodifferential operators on a vector bundle E → X , for more details on pseudodifferential operators see [13, 21]. Theorem 6.1. Let X be a compact n-dimensional Riemannian manifold, P ∈ Ψ 0 (X ; E) a classical order 0 pseudodifferential operator with P = P 2 = P ∗ and take α ∈ (0, 1]. Let π denote the pointwise action of C 0,α (X ) on L 2 (X ; E). Then (π, L 2 (X ; E), 2P −1) is a strict (n/α, ∞)-summable Fredholm module for C 0,α (X ). The proof of this result follows immediately from [9, Theorem 2.1], but as discussed in [9] this result was likely known before to experts. To compute with the associated singular cocycles, we will make use of the following result that follows from [9, Section 2.2] (see especially Theorem 2.15, Lemma 2.21 and Proposition 2.26 of [9]). Theorem 6.2. Let X be a compact n-dimensional Riemannian manifold, P ∈ Ψ 0 (X ; E) a classical order 0 pseudodifferential operator with P = P 2 = P ∗ and p ≥ n − 1 an odd integer. Let (ek )∞ ⊆ L 2 (M ; E) be an ON-basis consisting of eigenfunctions of some k=1 positive order elliptic pseudodifferential operator. n n p The associated classes [hω ] ∈ H H p+1 (C 0, p+1 (X )) and [cω ] ∈ H Cλ (C 0, p+1 (X )) can be computed from the identity PN 〈(2P − 1)ek , a0 [P, a1 ] · · · [P, a p+1 ]ek 〉 L2 (M ;E) k=1 p+1 hω (a0 , . . . a p+2 ) = p2 lim . N →ω log(2 + N ) Moreover, if p > n − 1, the two cocycles hω and cω vanish when restricted to the small n Hölder algebra h p+1 (X ). In the case of Lipschitz functions on manifolds, p = n − 1, the Hochschild n-cocycle hω is computed by rather classical means. For p > n − 1, the case of Hölder algebras, we arrive at some non-classical expressions below. 14 MAGNUS GOFFENG, RYSZARD NEST Theorem 6.3. Let X be a compact n-dimensional Riemannian manifold and P ∈ Ψ 0 (X ; E) a classical order 0 pseudodifferential operator with P = P 2 = P ∗ . Let π denote the pointwise action of Lip(X ) on L 2 (X ; E). Then (π, L 2 (X ; E), 2P − 1) is a strict (n, ∞)-summable Fredholm module. Moreover, the Hochschild n-cocycle hω is given by Z 2n+1 hω (a0 , a1 . . . , an ) = Tr E ((2p − 1)a0 {p, a1 } · · · {p, an }) dxdξ, (2π)n S ∗ X where p is the principal symbol of P and {·, ·} denotes the Poisson bracket. For a detailed proof of the theorem, see [9, Theorem 2.29]. The idea in the proof is to use the work of Rochberg-Semmes [19] to estimate k[P, a]kL n,∞ ® kakW 1,n and to use that C ∞ (X ) ⊆ Lip(X ) is dense in the W 1,n -topology. Therefore, hω is determined by its restriction to C ∞ (X ). In this case, the computation reduces to the Connes’ trace formula [6]. 6.1. Non-triviality in H Cλ1 (C 1/2 (S 1 )). Let us consider the Fredholm module defined from the Dirac operator on the circle. In this case, F denotes the phase of the Dirac operator. We have that F = 2P−1, where P is the Szegö projector. In the Fourier basis en (z) = (2π)−1/2 z n we have Fen = sign(n)en . We use the convention that sign(0) = 1. We can also describe P = (F + 1)/2 from the Cauchy integral operator Z f (w)dw 1 . P f (z) = lim r→1− 2πi w − rz 1 S The following construction will be used heavily to prove non-triviality of cyclic cohomology classes. Proposition 6.4. Let α ∈ (0, 1). The map N Wα : ℓ∞ ( ) → C α (S 1 ), ∞ X k [Wα (c)](z) := ck 2−αk z 2 . k=0 is continuous and an isomorphism onto its range. The proposition follows from Littlewood-Paley theory, see [18] for a discrete version thereof adapted to the circle. A direct proof of Proposition 6.4 can be found in [25, Theorem 4.9, Chapter II]. Recall also from [9, Theorem 3.5], that (4) Trω (P W1/2 (c1 )(1 − P)[W1/2 (c2 )]∗ P) = ω ◦ log2 (c1 c2∗ ). This computation can be derived from Theorem 6.2 using the Fourier basis. Theorem 6.5. Let P denote the Szegö projector on S 1 , π denote the pointwise action of C 1/2 (S 1 ) on L 2 (S 1 ) and consider the (2, ∞)-summable bounded Fredholm module (π, L 2 (S 1 ), 2P − 1) for C 1/2 (S 1 ). (1) The associated cyclic cohomology class [cω ] ∈ H Cλ1 (C 1/2 (S 1 )) is non-trivial and vanishes on h1/2 (S 1 ). (2) The associated Hochschild cohomology class [hω ] ∈ H H 2 (C 1/2 (S 1 )) pairs nonalg trivially with K2 (C 1/2 (S 1 )). Proof. By Theorem 5.5, the theorem follows if we can prove that hω ◦ D2 (x) 6= 0 for alg alg an x ∈ im(K2rel (C 1/2 (S 1 )) → K2 (C 1/2 (S 1 ))). Following [14], we take x ∈ K2 (C 1/2 (S 1 )) W1/2 (c1 ) W1/2 (c2 )∗ to be the Steinberg symbol of e and e and compute from Equation (4) alg top ∗ that hω ◦ D2 (x) = ω ◦ log2 (c1 c2 ). Clearly, x ∈ ker(K2 (C 1/2 (S 1 )) → K2 (C 1/2 (S 1 ))) = EXOTIC CYCLIC COHOMOLOGY CLASSES 15 N alg im(K2rel (C 1/2 (S 1 )) → K2 (C 1/2 (S 1 ))) and we can take c1 , c2 ∈ ℓ∞ ( ) such that ω ◦ log2 (c1 c2∗ ) 6= 0.  From the proof of Theorem 6.5 and [12], we can even conclude the following generalization of [9, Proposition 3.15]. Corollary 6.6. The convex set of cyclic cohomology classes {[cω ] : where ω is an extended limit} ⊆ H Cλ1 (C 1/2 (S 1 )), associated with the Szegö projector on S 1 , spans a subspace containing a copy of (ℓ∞ ( )/c0 ( ))∗ . The copy of (ℓ∞ ( )/c0 ( ))∗ is separated by its pairing with K2rel (C 1/2 (S 1 )). N N N N 6.2. Non-triviality in H Cλ3 (C 1/4 (S 1 )). Let us now consider the case of even lower regularity, that by Theorem 6.1 results in a higher summability degree. We have the following theorem. Theorem 6.7. Let P denote the Szegö projector on S 1 , π denote the pointwise action of C 1/2 (S 1 ) on L 2 (S 1 ) and consider the (4, ∞)-summable bounded Fredholm module (π, L 2 (S 1 ), 2P − 1) for C 1/4 (S 1 ). The associated cyclic cohomology class [cω ] ∈ H Cλ3 (C 1/4 (S 1 )) and the associated Hochschild cohomology class [hω ] ∈ H H 4 (C 1/2 (S 1 )) are non-trivial and vanish on alg h1/4 (S 1 ). They pair non-trivially with K4rel (C 1/4 (S 1 )) and K4 (C 1/4 (S 1 )), respectively, as in Theorem 5.3. Proof. Let us start with some preliminary computations. We note that by Theorem 5.3, it suffices to prove that cω (x) 6= 0 for some cyclic 3-cycle x in the range of chrel . 4 1/4 1 For a0 , a1 , a2 , a3 ∈ C (S ), we introduce the notation a0 ∧ a1 ∧ a2 ∧ a3 := X σ∈S3 (−1)σ a0 ⊗ aσ(1) ⊗ aσ(2) ⊗ aσ(3) , being the relative Chern which is a cyclic 3-cycle for C 1/4 (S 1 ) in the range of chrel 4 character of a higher Loday symbol, see [15, Section 5.3]. We compute using Theorem 6.2 that for a0 , a1 , a2 , a3 ∈ C 1/4 (S 1 ), such that a0 , a1∗ , a2 , a3∗ admit holomorphic extensions to the unit disc, we have 1 cω (a0 ∧ a1 ∧ a2 ∧ a3 ) =Trω (Pa0 (1 − P)a1 Pa2 (1 − P)a3 P)− 2 − Trω (Pa0 (1 − P)a3 Pa2 (1 − P)a1 P) = N ∞ XX
1 ka0,k a2,m a1,−m a3,−k − a1,−k a3,−m , N →ω log(N + 2) k=0 m=k = lim where a j,k denotes the k:th Fourier coefficient of a j . Consider c0 , c1 , c2 , c3 ∈ ℓ∞ ( ) defined from c2 ( j) = c3 ( j) = 1 and c0 ( j) = c1 ( j) = (−1) j . The functions a0 = W1/4 (c0 ), a1 = W1/4 (c1 )∗ , a2 = W1/4 (c2 ), and a3 = W1/4 (c3 )∗ satisfy that a0 , a1∗ , a2 , a3∗ admit holomorphic extensions to the unit disc, N 16 MAGNUS GOFFENG, RYSZARD NEST and the discussion above applies to give 1 cω (W1/4 (c0 )∧W1/4 (c1 )∗ ∧ W1/4 (c2 ) ∧ W1/4 (c3 )∗ ) = 2 N X ∞ X
1 = lim 2l/2−m/2 (−1)m+l − 1 = N →ω◦log2 log(2)N l=0 m=l  N  X 2−1/2 2−1/2 1 = + = − lim N →ω◦log2 log(2)N 1 + 2−1/2 1 − 2−1/2 l=0   p 1 2 2 2−1/2 2−1/2 =− =− 6= 0. + log(2) 1 + 2−1/2 1 − 2−1/2 log(2)  6.3. Non-triviality in H Cλ2 (C 2/3 (S 1 × S 1 )). Let us end with a higher-dimensional computation, on the two-torus S 1 × S 1 . In even dimensions, we need an even analogue of Theorem 6.2 which is proven ad verbatim to the odd case. We introduce the notation EN ⊆ 2 for the set of multiindices ∈ EN such that | |2 ≤ λN – the N :th eigenvalue of the Laplacian. Z k k Theorem 6.8. Let F denote the phase of the flat Dirac operator on S 1 ×S 1 , let π denote the pointwise action of C 2/3 (S 1 × S 1 ) on L 2 (S 1 × S 1 , 2 ) and consider the (3, ∞)summable even bounded Fredholm module (π, L 2 (S 1 × S 1 , 2 ), F ) for C 2/3 (S 1 × S 1 ). The associated cyclic 2-cocycle cω ∈ Zλ2 (C 2/3 (S 1 × S 1 )) is given by C C cω (a0 , a1 , a2 ) =Trω (U[U ∗ , a0 ][U, a1 ][U ∗ , a2 ]) − Trω (U ∗ [U, a0 ][U ∗ , a1 ][U, a2 ]) = X X 4i = lim ρ( , , )a0,−n a1,m a2,n−m, N →ω log(2 + N ) k∈E m,n∈Z2 kmn N where (a j,m)m∈Z2 denotes the Fourier coefficients of a j ∈ C 2/3 (S 1 × S 1 ), and for , , ∈ 2 kmn Z k m, n) = m ×|nn++k(||mm−+nk)|× k + |kn||k×+kn| + |kk||k×+mm| . ρ( , The reader can note that ρ defines a smooth function on degree 0. Proof. We use the Fourier basis ek (z) = (2π)−1 z k , operator F decomposes as  ‹ 0 U F= , U∗ 0 R6 \ {0} homogeneous of k ∈ Z2 , for L 2 (S 1 × S 1 ). The where U is the partial isometry on L 2 (S 1 × S 1 ) that in the Fourier basis acts as k +ik U e(k1 ,k2 ) = |k1 +ik2 | e(k1 ,k2 ) . We can identify the multiindex = (k1 , k2 ) ∈ 2 with the 1 2 complex number k = k1 + ik2 , so U ek = |kk| ek . k Z Using an even analogue of Theorem 6.2, and a degree argument we have that cω (a0 , a1 , a2 ) =Trω (U[U ∗ , a0 ][U, a1 ][U ∗ , a2 ]) − Trω (U ∗ [U, a0 ][U ∗ , a1 ][U, a2 ]) = X X 4i = lim ρ( , , )a0,−n a1,m a2,n−m, N →ω log(2 + N ) k∈E m,n kmn N EXOTIC CYCLIC COHOMOLOGY CLASSES 17 where 1 k ρ(k, m, n) = 2i |k|  k k+n k+n k+m k+m k − − − |k| |k + n| |k + n| |k + m| |k + m| |k|  k k+n k+m k k+n k+m =Im = + + |k| |k + n| |k + m| |k| |k + n| |k + m| m × n + (m − n) × k n×k k×m = + + . |n + k||m + k| |k||k + n| |k||k + m|   ‹  = Here we use the notation z × w = Im(z̄w) = −w × z and that for complex numbers z, w, u of modulus 1, w(w̄ − z̄)(z − u)(ū − w̄) = 2iIm(wz̄ + uw̄ + zū) = 2i(z × w + w × u + u × z).  Conjecture 6.9. 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Vol. I, Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (2002). Magnus Goffeng, Centre for Mathematical Sciences, University of Lund, Box 118, 221 00 LUND, Sweden Ryszard Nest, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København, Denmark Email address: magnus.goffeng@math.lth.se, rnest@math.ku.dk