Abstract
For a compact metric space (M, d), \({\mathrm {Lip}}M\) denotes the Banach algebra of all complex-valued Lipschitz functions on (M, d). Motivated by a classical result of de Leeuw, we give a canonical construction of a compact Hausdorff space \({\hat{M}}\) and a continuous surjection \(\pi :{\hat{M}} \rightarrow M\) which may viewed as a metric analogue of the unit sphere bundle over a Riemannian manifold. It is shown that, for each \(n \ge 1\) the continuous Hochschild cohomology \({\mathrm {H}}^{n}({\mathrm {Lip}}M, C({\hat{M}}))\) has the infinite rank as a \({\mathrm {Lip}}M\)-module, if the metric space (M, d) admits a local geodesic structure, for example, if M is a compact Riemannian manifold or a non-positively curved metric space. Here \(C({\hat{M}})\) denotes the algebra of all complex-valued continuous functions on \({\hat{M}}\). On the other hand, if the coefficient \(C({\hat{M}})\) is replaced with C(M), then it is shown that \({\mathrm {H}}^{1}({\mathrm {Lip}}M,C(M)) = 0\) for each compact Lipschitz manifold M.
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1 Introduction, main result and preliminaries
For a Banach algebra A and a Banach A-bimodule X, let \(C^{n}(A,X)\) be the continuous n-cochains of A to X
with \(C^{0}(A,X) = X\). The coboundary operator \(\delta ^{n}:C^{n}(A,X) \rightarrow C^{n+1}(A,X)\) is defined by
for \(f \in C^{n}(A,X)\) and \(a_{1}, \ldots , a_{n+1} \in A\). Then \(\delta ^{n+1}\circ \delta ^{n} = 0\) and \(Z^{n}(A,X) = {\mathrm {Ker}}~\delta ^{n} \supset B^{n}(A,X) = {\mathrm {Im}}~\delta ^{n-1}\). The continuous Hochschild cohomology of A with coefficient X is defined by \({\mathrm {H}}^{n}(A,X) = Z^{n}(A,X)/B^{n}(A,X)\) (see [1, 5, 6]). When A is a commutative Banach algebra, \(C^{n}(A,X)\) is a left A-module by the action
and the coboundary operator \(\delta ^{n}:C^{n}(A,X) \rightarrow C^{n+1}(A,X)\) is an A-module homomorphism, which induces a left A-module structure on \({\mathrm {H}}^{n}(A,X)\).
For a Banach algebra A and a Banach A-bimodule X, a bounded linear operator \(D:A \rightarrow X\) is called a derivation if it follows the Leibniz rule:
The space of all continuous derivations \(A \rightarrow X\) is denoted by \({\mathfrak {D}}(A,X)\). An inner derivation is a derivation \(D:A \rightarrow X\) defined by \(Da = a \cdot x - x \cdot a ~(a \in A)\) for some \(x \in X\). The first cohomology \({\mathrm {H}}^{1}(A,X)\) is isomorphic to the space of derivations modulo the inner derivations.
The present paper studies continuous Hochschild cohomologies of Lipschitz algebras over compact metric spaces. For a compact metric space (M, d), let \({\mathrm {Lip}}M\) be the Banach algebra of all complex-valued Lipschitz functions \(f:M \rightarrow {\mathbb {C}}\) with the norm
where \(\Vert f\Vert _{\infty } = \sup _{p \in M}|f(p)|\), the sup norm, and
the Lipschitz constant of f. In a previous paper [8] the author proved that, for each \(n\ge 1\), \({\mathrm {H}}^{n}({\mathrm {Lip}}M, {\mathbb {C}})\) is an infinite dimensional \({\mathbb {C}}\)-linear space when M contains a certain point-sequence which converges to a point \(p \in M\). Here \({\mathbb {C}}\) is endowed with a \({\mathrm {Lip}}M\)-bimodule structure given by:
The above result relies only on the local geometry of M at p and a question arises whether the same holds if the coefficient \({\mathbb {C}}\) is replaced with an appropriate continuous function algebra over M with a \({\mathrm {Lip}}M\)-module structure. The present paper gives an answer to the question.
For a compact metric space (M, d), let \({\tilde{M}} = M \times M {\setminus } \Delta M\), where \(\Delta M = \{ (x,x)~|~ x \in M\} \subset M \times M\). Let \(\beta {\tilde{M}}\) be the Stone–Čech compactification of \({\tilde{M}}\) (see [20]). Since \(M\times M\) is another compactification of \({{\tilde{M}}}\), there exists a continuous surjection \(\pi :\beta {\tilde{M}} \rightarrow M \times M\) such that \(\pi | \pi ^{-1}({\tilde{M}}): \pi ^{-1}({\tilde{M}}) \rightarrow {\tilde{M}}\) is a homeomorphism. Let
The restriction \(\pi |{\hat{M}}\) is also denoted by \(\pi :{\hat{M}} \rightarrow \Delta M\). In what follows we identify the space \(\Delta M\) with M via the diagonal map \(\Delta _{M}:M \rightarrow \Delta M\) and the map \((\Delta _{M})^{-1}\circ \pi \) is also denoted by \(\pi :{\hat{M}} \rightarrow M\). As will be explained in Sect. 3, the space \({\hat{M}}\) may be regarded as an analogue of the unit sphere bundle of the tangent bundle over a Riemannian manifold. For a point \(\omega \in {\hat{M}}\), a point derivation \(D_{\omega }:{\mathrm {Lip}}M \rightarrow {\mathbb {C}}\) is defined as an analogue of the directional derivative of smooth functions.
The Banach space \(C({\hat{M}})\) of all complex-valued continuous functions on \({\hat{M}}\) with the sup norm admits a Banach \({\mathrm {Lip}}M\)-bimodule structure given by
Our first result is on the continuous Hochschild cohomology \({\mathrm {H}}^{*}({\mathrm {Lip}}M,C({\hat{M}}))\). A map \(\gamma :[a,b] \rightarrow M\) of the interval [a, b] to a metric space (M, d) is called a geodesic if \(d(\gamma (s),\gamma (t)) = |s-t|\) for each \(s,t\in [a,b]\). By abuse of terminology the image of \(\gamma \), denoted by \({\mathrm {Im}}\gamma \), is also called a geodesic.
Definition 1.1
A metric space (M, d) is said to satisfy the condition (G) if there exists a positive number \(\delta >0\) such that
\((*)\) for each \(x, y \in M\) with \(d(x,y) \le \delta \), there exists a unique geodesic \(\gamma _{xy}:[0,d(x,y)] \rightarrow M\) such that \(\gamma _{xy}(0) = x, \gamma _{xy}(d(x,y)) = y.\)
Besides Riemannian manifolds, all CAT(\(\kappa \)) metric spaces (see [3]) are examples of spaces satisfying the condition (G).
Theorem 1.2
Let (M, d) be a compact metric space satisfying the condition (G). Then for each \(n \ge 1\), the cohomology \({\mathrm {H}}^{n}({\mathrm {Lip}}M,C({\hat{M}}))\) has the infinite \({\mathrm {Lip}}M\)-rank in the sense that, for each \(N \ge 1\), there exist \({\mathrm {Lip}}M\)-linearly independent N elements in \({\mathrm {H}}^{n}({\mathrm {Lip}}M,C({\hat{M}}))\).
The main result of [8] may be viewed as a local version of the above theorem. The above theorem should also be compared with the homological dimension theorems of Ogneva [14, 15], Kleshchev [10] and Pugach [18]; the global homological dimension of the Frechét algebra \(C^{\infty }(M)\) of the smooth functions on a smooth manifold M is equal to \(\dim M\) [14, 15], while the global homological dimension of \(C^{n}(M)\) of the Banach algebra of the \(C^n\)-functions on M is infinity for each \(n,~1 \le n < \infty \). A long standing open problem is to decide the global homological dimension of \(C([0,1]) = C^{0}([0,1])\) [5, Chap.V, section 2.5].
Our proof is conceptually motivated by the classical Hochschild–Kostant–Rosenberg theorem [13, 16, 17]. The space \({\mathfrak {D}}({\mathrm {Lip}}M,C({\hat{M}}))\) of all derivations \({\mathrm {Lip}}M \rightarrow C({\hat{M}})\) is a \({\mathrm {Lip}}M\)-module under the action
We take the n-fold exterior product \(\wedge _{{\mathrm {Lip}}M}^{n}{\mathfrak {D}}({\mathrm {Lip}}M, C({\hat{M}}))\) of the \({\mathrm {Lip}}M\)-module \({\mathfrak {D}}({\mathrm {Lip}}M, C({\hat{M}}))\), define a homomorphism \( \Omega _{n}{:}\wedge _{{\mathrm {Lip}}M}^{n}{\mathfrak {D}}({\mathrm {Lip}}M, C({\hat{M}})) \rightarrow {\mathrm {H}}^{n}({\mathrm {Lip}}M, C({\hat{M}})) \) by
and prove that the image \({\text {Im}}\Omega _{n}\) contains arbitrarily large number of \({\mathrm {Lip}}M\)-linearly independent elements of \({\mathrm {H}}^{n}({\mathrm {Lip}}M, C({\hat{M}}))\) when the space M satisfies the condition (G). The notion of alternating n-cocycle due to Johnson [7] plays the crucial role in the proof.
The above idea naturally leads to the study of the cohomology with C(M)-coefficient \({\mathrm {H}}^{n}({\mathrm {Lip}}M,C(M))\). The situation is rather different than that of the smooth-function setup and we prove the following theorem. A homeomorphism \(h:S_{1}\rightarrow S_{2}\) between metric spaces \((S_{1},d_{1})\) and \((S_{2},d_{2})\) is called a bi-Lipschitz homeomorphism (a lipeomorphism in [11]) if h and \(h^{-1}\) are both Lipschitz maps. A topological embedding \(\alpha :D \rightarrow M\) of a metric space D into a metric space M is called a bi-Lipschitz embedding if \(\alpha :D \rightarrow {\mathrm {Im}}\alpha \) is a bi-Lipschitz homeomorphism. Throughout \({\mathbb {R}}^{m}\) is assumed to be endowed with the standard Euclidean metric. Let \(D^{m} = \{ x \in {\mathbb {R}}^{m}~|~\Vert x \Vert \le 1\}\) and \({\mathrm {int}}D^{m} = \{x\in D^{m}~|~\Vert x \Vert <1\}\).
Theorem 1.3
Let (M, d) be a compact metric space such that, for each point \(p \in M\), there exists a bi-Lipschitz embedding \(\alpha : D^{m(p)} \rightarrow M\) of \(D^{m(p)}\) into M (m(p) may depend on p) such that \(p \in \alpha (D^{m})\) and \(\alpha ({\mathrm {int}}D^{m(p)})\) is open in M. Then we have
In particular the conclusion holds for each compact Lipschitz manifold M.
Theorem 1.2 is proved in Sect. 2 and Theorem 1.3 is proved in Sect. 3 after developing the sphere-bundle-analogue mentioned above.
The rest of this section fixes notation and recalls some basic results. For a compact metric space (M, d), let \(\pi :{\hat{M}}\rightarrow \Delta M\) be the map defined in (1.4). For a Lipschitz function \(f:M \rightarrow {\mathbb {C}}\), let \(\Phi _{f}:{\tilde{M}} \rightarrow {\mathbb {C}}\) be the function defined by
By the Lipschitz condition, \(\Phi _{f}\) is a bounded continuous function on \({\tilde{M}}\) and hence admits the unique extension, called the de Leeuw map [2, 4, 19, 22]
to the Stone-Čech compactification of \({\tilde{M}}\) which restricts to the map
on the space \({\hat{M}}\). This defines a pairing \({\hat{\Phi }}:{\hat{M}}\times {\mathrm {Lip}}M \rightarrow {\mathbb {C}}\) by
such that
It is convenient to introduce the notation
The map \({\hat{\Phi }}\) (or \(D_{\omega }\) in the above notation) induces two maps
defined by
Observe that (1.8) guarantees that \(T(\omega ) \in ({\mathrm {Lip}}M)^{*}\) for each \(\omega \in {\hat{M}}\). The map D is a \(\Vert \cdot \Vert _{L}-\Vert \cdot \Vert _{\infty }\)-bounded linear operator and T is continuous if \(({\mathrm {Lip}}M)^{*}\) is endowed with the weak*-topology. We use the map D in the proof of Theorem 1.2 and T will be used in the discussion on the space \({{\hat{M}}}\) in Sect. 3. It follows from the proof of [19, Theorem 9.8] that \(D:{\mathrm {Lip}}M \rightarrow C({\hat{M}})\) satisfies
that is, D is a derivation of \({\mathrm {Lip}}M\) to the \({\mathrm {Lip}}M\)-module \(C({\hat{M}})\) (cf. 1.5). A point derivation\(D:{\mathrm {Lip}}M \rightarrow {\mathbb {C}}\) at a point \(p\in M\) is a bounded linear functional on \({\mathrm {Lip}}M\) such that
The space of all point derivations at p is denoted by \({\mathfrak {D}}_{p}({\mathrm {Lip}}M)\). The next result, which also follows from of [19, Theorem 9.8], explains the role of the operator defined by (1.9).
Theorem 1.4
(cf. [19, Theorem 9.8] ) Let (M, d) be a compact metric space and let \(\pi :{\hat{M}} \rightarrow M\) be the map defined in (1.4).
- 1.
For each \(p \in M\) and for each \(\omega \in \pi ^{-1}(p) \subset {\hat{M}}\), \(D_{\omega }:{\mathrm {Lip}}M \rightarrow {\mathbb {C}}\) is a continuous point derivation at p.
- 2.
The weak \(*\)-closure of the linear span of \(\{ D_{\omega }~|~\omega \in \pi ^{-1}(p) \}\) is equal to the space \({\mathfrak {D}}_{p}({\mathrm {Lip}}M)\).
We use the classical extension theorem of McShane [12].
Theorem 1.5
[12] Let (K, d) be a metric space and let E be a subset of K. For each bounded real-valued Lipschitz function \(f:E \rightarrow {\mathbb {R}}\), there exists a Lipschitz function \(F:K \rightarrow {\mathbb {R}}\) such that
- 1.
\(F|E = f\),
- 2.
\(\Vert F \Vert _{\infty } = \Vert f \Vert _{\infty }\) and \(L(F) = L(f)\).
Next we recall the notion of alternating cocycles due to Johnson. Let \({\mathfrak {S}}_{n}\) be the nth symmetric group. For a Banach algebra A and a Banach A-bimodule X, the continuous n-cochains \(C^{n}(A,X)\) is an \({\mathfrak {S}}_n\)-module by the action
An n-chain F is said to be alternating if \(\sigma F = ({\mathrm {sgn}}\sigma )F\), where \({\mathrm {sgn}}~\sigma \) denotes the signature of \(\sigma \in {\mathfrak {S}}_{n}\). The subspace of all continuous alternating n-cocycles is denoted by \(Z^{n}_{\mathrm {alt}}(A,X)\). An n-chain \(F \in C^{n}(A,X)\) is called an n-derivation if
for each \(i=1, \ldots , n\) and for each \(a_{1}, \ldots , a_{i-1}, a_{i+1}, \ldots , a_{n}, b_{i}, c_{i} \in A\).
Theorem 1.6
[7, Theorem 2.3, Propostion 2.9, Corollary 2.10] Let A be a commutative Banach algebra and let X be a symmetric Banach A-bimodule.
- 1.
An n-cochain \(F \in C^{n}(A,X)\) is an alternating n-cocycle if and only if it is an alternating n-derivation.
- 2.
The restriction \(q_{n}| Z^{n}_{\mathrm {alt}}(A,X): Z^{n}_{\mathrm {alt}}(A,X) \rightarrow H^{n}(A,X)\) of the natural quotient map \(q_{n}:Z^{n}(A,X) \rightarrow H^{n}(A,X)\) to \(Z^{n}_{\mathrm {alt}}(A,X)\) is injective.
2 Proof of Theorem 1.2
This section is devoted to prove Theorem 1.2. The proof is divided into several steps. In Step 1, we give a construction of derivations \({\mathrm {Lip}}M \rightarrow C({\hat{M}})\). Step 2 supplies a construction of Lipschitz functions associated with a convergent point-sequence of M. Step 3 proves the theorem for \(n=1\) and the proof for \(n >1\) will be given in Step 4.
We start with a general discussion on maps induced on the Stone–Čech compactification of a space. Let M be a compact metric space and let \(\pi :\beta {\tilde{M}} \rightarrow M\times M\) be the continuous surjection defined in (1.4) with the restriction \(\pi :{\hat{M}} \rightarrow M\) (recall the identification \(M \approx \Delta M\)). Let N be a closed, hence compact, neighborhood of the diagonal set \(\Delta M\) and let \(F:N \rightarrow N\) be a continuous map such that \(F(\Delta M) = F^{-1}(\Delta M) = \Delta M\). Let \({\tilde{N}} = N {\setminus } \Delta M\) and let \({\tilde{F}}= F|{\tilde{N}}:{\tilde{N}} \rightarrow {\tilde{N}}\) be the restriction of F. The map \({{\tilde{F}}}\) admits a unique extension \(\beta {{{\tilde{F}}}}:\beta {\tilde{N}} \rightarrow \beta {\tilde{N}}\). Since N is another compactification of \({{\tilde{N}}}\), there exists the canonical continuous surjection \(\pi _{N}:\beta {{{\tilde{N}}}} \rightarrow N\) such that \(\pi _{N}|\pi _{N}^{-1}({\tilde{N}}): \pi _{N}^{-1}({\tilde{N}}) \rightarrow {\tilde{N}}\) is a homeomorphism. Notice that \(\beta {{\tilde{F}}}\) is the unique map such that
Lemma 2.1
-
1.
We have the inclusion
$$\begin{aligned} {\hat{M}} = \pi ^{-1}(\Delta M) \subset \beta {\tilde{N}} \subset \beta {\tilde{M}} \end{aligned}$$and \(\pi _{N} = \pi |\beta {\tilde{N}}\).
-
2.
\(\pi _{N} \circ \beta {\tilde{F}} = F \circ \pi _{N}.\)
-
3.
The restriction \(\beta {\tilde{F}}|{\hat{M}}\) of \(\beta {{\tilde{F}}}\) to \({{\hat{M}}}\) induces a map \({\hat{F}}:{\hat{M}} \rightarrow {\hat{M}}\) such that \(\pi \circ {\hat{F}} = (F|\Delta M) \circ \pi \).
Proof
-
1.
Since N is closed in M, \({\tilde{N}}\) is closed in \({{\tilde{M}}}\) and by [20, Proposition 1.48], the Stone-Čech compactification \(\beta {\tilde{N}}\) is the closure of \({{\tilde{N}}}\) in \(\beta {\tilde{M}}\): \(\beta {\tilde{N}} = {\mathrm {cl}}_{\beta {\tilde{M}}}{\tilde{N}}\). In particular \(\beta {\tilde{N}} \subset \beta {\tilde{M}}\) and we have \(\pi _{N} = \pi |\beta {\tilde{N}}\). It follows from this that \(\pi ^{-1}(\Delta M) \subset \beta {\tilde{N}}\).
-
2.
We have from (2.1) that \(\pi _{N} \circ \beta {\tilde{F}}|\pi _{N}^{-1}({\tilde{N}}) = {\tilde{F}} \circ \pi _{N}|\pi _{N}^{-1}({\tilde{N}})\) and the desired equality follows from the denseness of \(\pi _{N}^{-1}({\tilde{N}})\) in \(\beta {\tilde{M}}\).
-
3.
is a direct consequence of (1) and (2).
\(\square \)
For a map \(F:N \rightarrow N\) as above, we define a bounded linear map \(F^{*}D:{\mathrm {Lip}}M \rightarrow C({\hat{M}})\) by
Lemma 2.2
If \(F|\Delta M = {\mathrm {id}}_{\Delta M}\), then the operator \(F^{*}D:{\mathrm {Lip}}M \rightarrow C({\hat{M}})\) is a derivation.
Proof
It suffices to verify the Leibniz rule. Fix Lipschitz functions \(f,g \in {\mathrm {Lip}}M\) and a point \(\omega \in {\hat{M}}\). We have, by (1.11), the assumption \(F|\Delta M = {\mathrm {id}}_{\Delta M}\) and (3) of Lemma 2.1, the following equalities:
Recalling the \({\mathrm {Lip}}M\)-module structure of \(C({\hat{M}})\) ((1.5)) we obtain the conclusion. \(\square \)
Proof of Theorem 1.2
Step 1. Let (M, d) be a compact metric space satisfying the condition (G) with a positive number \(\delta >0\) that meets the condition \((*)\) of Definition 1.1. We may and will assume that \(\delta < 1\). Let
and for each \((x,y) \in W\), let \(\gamma _{xy}\) be the unique geodesic joining x with y. In what follows it is convenient to take the parametrization of \(\gamma _{xy}\) as
Also let \(m_{xy} = \gamma _{xy}(0)\), the midpoint of x and y. For \(w(x,y) = d(x,y)/2\), the above parametrization of \(\gamma _{xy}\) is given by
We make a convention that \(\gamma _{xx} = m_{xx} = \{x\}\) and \(w(x,x) = 0\). Let \(\kappa :[0,\delta ] \rightarrow [0,1]\) be the function defined by
It satisfies
The argument in Step 1 depends only on (2.4) and the explicit form (2.3) will be used in later steps. Let \(H:W \rightarrow W\) be the map defined by
Let \(\xi (x,y) = \gamma _{xy}(-w(x,y)\kappa (w(x,y)))\) and \(\eta (x,y) = \gamma _{xy}(w(x,y)\kappa (w(x,y)))\) so that \(H(x,y) = (\xi (x,y),\eta (x,y)).\) The map H has the following properties.
- (a)
For each \((x,y) \in W\), we have
- (a.1)
the points \(\xi (x,y), \eta (x,y)\) are on the geodesic \(\gamma _{xy}\),
- (a.2)
\(m_{\xi (x,y)\eta (x,y)} = m_{xy}\),
- (a.3)
\(w(\xi (x,y),\eta (x,y)) = d(\xi (x,y), m_{xy}) = d(\eta (x,y),m_{xy}) = \kappa (w(x,y))w(x,y)\),
- (a.4)
\(\gamma _{\xi (x,y)\eta (x,y)} = \gamma _{xy}|[-\kappa (w(x,y))w(x,y),\kappa (w(x,y))w(x,y)]\).
- (a.1)
- (b)
\(H|\Delta M = {\mathrm {id}}_{\Delta M},~H|\partial W = {\mathrm {id}}_{\partial W}\) and \(H^{-1}(\Delta M) = H(\Delta M) = \Delta M\),
- (c)
If \(d(x,y) < \delta \), then \(\lim _{n\rightarrow \infty }H^{n}(x,y) = (m_{xy},m_{xy})\), where \(H^{n}\) denotes the n-fold iteration of H.
\(\square \)
Proof
(a.1)–(a.3) are direct consequences of the definition. (a.4) follows from the uniqueness of the geodesic joining \(\xi (x,y)\) and \(\eta (x,y)\). (b) follows from the definition (2.5) and (2.4). Note that \(d(x,y) = \delta \) if \((x,y) \in \partial W\). To verify (c) let \(w^{i} = w(H^{i}(x,y))\). By induction we can see directly that \(w^{i+1} < w^{i}\) and \(\kappa (w^{i+1}) < \kappa (w^{i})\) due to (2.4). Then we see from (a.3) that
Since \(w(x,y) = d(x,y)/2 \le \delta /2 < 1\), we have \(\kappa (w^{1}) = \kappa (w(x,y)) <1\) and \(\lim _{i}w^{i} = 0\). This and (a.2) imply the condition (c).
We apply Lemma 2.1 to the map \(H:W \rightarrow W\) defined on the closed neighbourhood W of \(\Delta M\) and obtain a sequence of linear operators
We see from Lemma 2.2 and the condition (b) that \((H^{n})^{*}D\) is a derivation. Our goal is to prove that the above forms a \({\mathrm {Lip}}M\)-linearly independent sequence of derivations.
Step 2. Fix a point p of M and take a geodesic \(\gamma :[0,\delta ] \rightarrow M\) such that \(\gamma (0) = p\). Take a sequence \(S_{0} = \{x_{k}, y_{k}~|~k \ge 1\} \) of points on the geodesic \({\mathrm {Im}}\gamma \) which satisfies the following conditions:
- (d.1)
\(\lim _{k}x_{k} = \lim _{k}y_{k} =p\), \(x_{k}\ne y_{k}\) for each k,
- (d.2)
\(d(x_{1},p) < \delta \) and, for each \(k\ge 1\), \(d(x_{k+1},p)< d(y_{k},p) < d(x_{k},p)\),
- (d.3)
for each \(k\ge 1\), \(d(x_{k+1},y_{k+1}) < d(x_{k},y_{k}).\)
For a fixed integer \(N \ge 1\), we examine the sequence \(\{H^{\nu }(x_{k},y_{k})~|~k \ge 1,~1 \le \nu \le N\}\) of points of W. The following statements are consequences of (a)–(c) above and will be used later.
- (e)
For each k, the geodesic \(\gamma _{x_{k}y_{k}}\) is the geodesic segment in \(\gamma \) joining \(x_{k}\) and \(y_{k}\), denoted by \(\overline{x_{k}y_{k}}\) for simplicity.
- (f)
For \(i \ge 0\), let \((x_{k}^{i},y_{k}^{i}) = H^{i}(x_{k},y_{k})\) with \((x_{k}^{0},y_{k}^{0}) = (x_{k},y_{k})\). Then the points \(x_{k}^{i+1}\) and \(y_{k}^{i+1}\) are on the geodesic \(\overline{x_{k}^{i}y_{k}^{i}}\) so that \(d(x_{k}^{i},m_{k}) \downarrow 0\) and \(d(y_{k}^{i},m_{k}) \downarrow 0\) as \(i \rightarrow \infty \).
The next lemma describes a general procedure to find a Lipschitz function that detects the derivation \((H^{i})^{*}D\).
Lemma 2.3
(cf. [8, Lemma 2.2]) Under the above notation, for each \(N \ge 1\) and for each \(i \in \{1,\ldots , N\}\), there exist an integer \(k_{0} \ge 1\) and a real-valued Lipschitz function \(f \in {\mathrm {Lip}}M\) such that
- 1.
\(L(f) = 1\),
- 2.
for each \(k \ge k_{0}\) we have \(|\Phi _{f}(x_{k}^{i},y_{k}^{i})| \ge 1/4\) for each \(i=1,\ldots , N\),
- 3.
for each \(k \ge k_{0}\) and for each \(j \in \{1,\ldots ,N\}\) with \(j \ne i\), we have \(\Phi _{f}(x_{k}^{j},y_{k}^{j}) = 0\).
Proof
First we make some preliminary estimates on the distance \(d(x_{k}^{i},y_{\ell }^{j})\). Let \(d_{k} = d(x_{k},y_{k}),\)\(w_{k} = w(x_{k},y_{k}) = d_{k}/2\) and \(m_{k} = m_{x_{k}y_{k}}\). Also for \(j \ge 1\), let \(w_{k}^{j} = d(x_{k}^{j},y_{k}^{j})/2 = d(x_{k}^{j},m_{k}) = d(y_{k}^{j},m_{k})\). Under this notation we have
In fact, \(w_{k}^{1} = \kappa (w(x_{k},y_{k}))w(x_{k},y_{y}) = w_{k}^{2}/\delta \), and \(w_{k}^{j+1}= \kappa (w_{k}^{j})w_{k}^{j} = \delta ^{-1}(w_{k}^{j})^{2}\), from which (2.6) follows by an induction. Let
We have by (2.6)
Let \(r_{k} = w_{k}/\delta \). We use (2.8) to see
for each \(j \ge 1\). Since \(w_{k} = d(x_{k},y_{k})/2 < d(x_{k},p)/2 \le \delta /2\), we see \(0< r_{k} < 1\) and thus, for each \(j\ge 1\), we obtain
Also by (d.1) we see \(\lim _{k}w_{k} = 0\). Take a large \(k_{0} \ge 1\) such that
Fix an integer \(N \ge 1\), let \(S_{k}^{N} = \{x_{k}^{j},y_{k}^{j}~|~0 \le j \le N\}\) and \(S^{N} = \cup _{k\ge k_{0}}S_{k}^{N} \cup \{p\}\). We fix \(i \in \{1, \ldots , N\}\) and define a function \(f:S^{N} \rightarrow [0,\infty )\) by:
We first verify that the function f is a Lipschitz function on \(S^N\) with the Lipschitz constant 1 which satisfies the condition (2) and (3).
In order to estimate
we may assume that \(k \le \ell \). First we observe
and by (2.9)
For j with \(0 \le j \le i-2\), we see \(d(x_{k}^{i},x_{k}^{j}) = d(x_{k}^{i},x_{k}^{i-1})+d(x_{k}^{i-1}, x_{k}^{j}) \ge d(x_{k}^{i},x_{k}^{i-1})\) by (f). Hence we have by (2.12),
Similarly by using (2.11) we have for j with \(i+2 \le j \le N\),
Next we estimate \(\Phi _{f}(x_{k}^{i},x_{\ell }^{j})\) for \(\ell > k\). By definition \(|\Phi _{f}(x_{k}^{i},x_{\ell }^{i})| = \frac{|\varepsilon _{k}^{i}d_{k} - \varepsilon _{\ell }^{i}d_{\ell }|}{d(x_{k}^{i},x_{\ell }^{i})},\) and we see
In fact, we have, by (2.8), \(\varepsilon _{k}^{i} d_{k} = \delta r_{k}^{2^i}(1-r_{k}^{2^i})\) and \(\varepsilon _{\ell }^{i}d_{\ell } = \delta r_{\ell }^{2^i}(1-r_{\ell }^{2^i})\). Also by (2.9) we have \(r_{\ell } = w_{\ell }/\delta \le w_{k}/\delta = r_{k} \le 1/2\) and hence \(r_{\ell }^{2^i} \le r_{k}^{2^i} \le 1/2\), from which we obtain the desired inequality.
Also by (d.2) we have \(d(x_{k}^{i},x_{\ell }^{i}) = d(x_{k}^{i},x_{k}^{i+1}) + d(x_{k}^{i+1},x_{\ell }^{i}) \ge d(x_{k}^{i},x_{k}^{i+1})\). Hence we obtain, by (2.11),
Similarly we have for \(\ell > k\),
Combining (2.11)–(2.16), we obtain \(L(f) = 1\) on \(S^N\).
In order to prove (2), we estimate \(\Phi _{f}(x_{k}^{i},y_{k}^{i}) = \frac{\varepsilon _{k}^{i}d_{k}}{d(x_{k}^{i},y_{k}^{i})}\). First we see
Using (2.8) with \(d_{k} = 2w_{k}\), we compute
Hence we obtain, by \(w_{k} = \delta r_{k}\) (see 2.9),
and by (2.17), we have
Thus we obtain
Using \(r_{k}^{2^i} =(\frac{w_{k}}{\delta })^{2^{i}} \le \frac{w_{k}}{\delta } \le 1/2\) we see that the last term of the above is at least 1 / 4. Hence we obtain
which proves (2). (3) directly follows from the definition (2.10). Finally we apply Theorem 1.5 to the above f to obtain a Lipschitz extension \({\bar{f}}:M \rightarrow {\mathbb {R}}\) such that \(L({\bar{f}}) = L(f) = 1\), the desired condition (1). The function \({\bar{f}}\) satisfies (2) and (3) as well. This completes the proof of lemma. \(\square \)
Step 3. We prove the theorem for \(n=1\). Since \(C({\hat{M}})\) is a symmetric \({\mathrm {Lip}}M\)-module, we have \({\mathrm {H}}^{1}({\mathrm {Lip}}M,C({\hat{M}})) = {\mathfrak {D}}({\mathrm {Lip}}M,C({\hat{M}}))\). In order to prove that \({\mathfrak {D}}({\mathrm {Lip}}M,C({\hat{M}}))\) has the infinite \({\mathrm {Lip}}M\)-rank, we take the map \(H:M \times M \rightarrow M \times M\) in Step 1, fix an integer \(N\ge 1\) and consider the N derivations
and assume that, for \(\varphi _{1},\ldots , \varphi _{N} \in {\mathrm {Lip}}M\), the equality
holds for each \(\omega \in {\hat{M}}\) and for each \(f \in {\mathrm {Lip}}M\). We fix \(i \in \{1,\ldots , N\}\) and show that \(\varphi _{i} \equiv 0\). Pick an arbitrary point \(p \in M\), take a geodesic \(\gamma \), choose a sequence \(\{x_{k},y_{k}~|~k \ge 1\}\) of points on \(\gamma \) such that
and apply Lemma 2.3 to find an integer \(k_{0} \ge 1\) and a Lipschitz function f satisfying the conditions of the lemma.
Let \(\omega \) be an accumulation point of the set \(\{ (x_{k},y_{k})~|~k \ge k_{0}\} \subset \beta {\tilde{W}}\). Then \(\pi (\omega )\), as a point of \(M \times M\), is an accumulation point of the set \(\{(x_{k},y_{k})~|~k\ge k_{0}\} \subset M\times M\), that is, the singleton (p, p). Recalling the identification \(M \approx \Delta M\) via the diagonal map, we have \(\pi (\omega ) = p\). Also \({\hat{H}}^{j}(\omega ) = (\beta {\tilde{H}}^{j})(\omega ) = (\beta {\tilde{H}})^{j}(\omega )\) is an accumulation point of \(\{H^{j}(x_{k},y_{k}) = (x_{k}^{j},y_{k}^{j})~|~k \ge k_{0}\}\). This and the conditions 2 and 3 of Lemma 2.3 imply
Therefore from (2.19) we have
which shows \(\varphi _{i}(p) = 0\) as required.
This completes the proof of the theorem for \(n=1\).
Step 4. This step finishes the proof of theorem, proving the case \(n > 1\), by carrying out the idea stated in Sect. 1. Rather than considering the homomorphism \(\Omega _n\) in (1.6), we proceed directly as follows. Let \(Z_{\mathrm {alt}}^{n}({\mathrm {Lip}}M,C({\hat{M}}))\) be the space of the alternating n-cocycles on \({\mathrm {Lip}}M\) with coefficient \(C({\hat{M}})\). By Theorem 1.3 we have an injection \(Z_{\mathrm {alt}}^{n}({\mathrm {Lip}}M,C({\hat{M}})) \rightarrow {\mathrm {H}}^{n}({\mathrm {Lip}}M,C({\hat{M}}))\) and thus it suffices to prove that \(Z_{\mathrm {alt}}^{n}({\mathrm {Lip}}M,C({\hat{M}}))\) has the infinite \({\mathrm {Lip}}M\)-rank.
Fix an arbitrary integer \(N\ge 1\). For \(\nu =1,\ldots , N\) and \(i=1,\ldots , n\), let
and define the n-cochain \(d_{\nu } \in C^{n}({\mathrm {Lip}}M,C({\hat{M}}))\) by
It follows from the definition that \(d_{\nu }\) is an alternating cochain. By Lemma 2.2, \(D_{{\hat{H}}_{\nu ,i}}(\omega )\) is a derivation, from which it follows that \(d_{\nu }\) is an n-derivation. Thus by (1) of Theorem 1.6 we see that \(d_{\nu }\) is an alternating cocycle: \(d_{\nu } \in Z_{\mathrm {alt}}^{n}({\mathrm {Lip}}M,C({\hat{M}}))\).
Assume that, for \(\varphi _{1},\ldots , \varphi _{N} \in {\mathrm {Lip}}M\), the equality
holds for each \(\omega \in {\hat{M}}\) and for each \(a_{1},\ldots ,a_{n} \in {\mathrm {Lip}}M\). We fix \(\mu \in \{1,\ldots , N\}\) and show \(\varphi _{\mu }\equiv 0\). Take an arbitrary point p of M and choose a geodesic \(\gamma \) and a sequence \(\{x_{k},y_{k}~|~k \ge 1\}\) as in (2.20). Applying Lemma 2.3 we obtain an integer \(k_{0} \ge 1\) and a sequence \(\{f_{j}~|~1\le j \le n\}\) of Lipschitz functions such that
Let \(\omega \) be an accumulation point of \(\{(x_{k},y_{k})~|~k \ge k_{0}\} \subset {\tilde{W}}\). As in Step 3, we see \(\pi (\omega ) = p\) and \({\hat{H}}_{\nu ,i}(\omega )\) is an accumulation point of \(\{H_{\nu ,i}(x_{k},y_{k})~|~k \ge k_{0}\}\) for each \(\nu \) and i with \(1\le \nu \le N,~1\le i \le n\). Thus by (2.23) and (2.24) we find a nonzero \(c_{i}\) such that
Also by (2.25) \(D_{{\hat{H}}_{\nu ,i}(\omega )}f_{j} = 0\) for each \(\nu \ne \mu \). Hence by (2.22) we have
which implies \(\varphi _{\mu }(p) = 0\) as desired.
This completes Step 4 and hence completes the proof of the theorem. \(\square \)
3 The space \({\hat{M}}\) and Proof of Theorem 1.3
Here we compare the point derivation \(D_{\omega }\) for a point \(\omega \in {\hat{M}}\) [see (1.9) and Theorem 1.4] with the derivation by tangent vectors of compact smooth manifolds. The comparison indicates that the space \({\hat{M}}\) may be regarded, to certain extent, as a Lipschitz analogue of the unit sphere bundle of a Riemannian manifold.
Let (M, g) be a compact Riemannian manifold with the metric d induced by g. By the compactness of M, there exists a \(\delta >0\) such that, for each pair p, q of points of M with \(d(p,q) \le \delta \), there exists a unique geodesic \(\gamma _{pq}:[0,d(p,q)] \rightarrow M\) such that
As in (2.2), let \(W =\{(p,q) \in M\times M~|~d(p,q) \le \delta \}\) and let \({\tilde{W}} = W {\setminus } \Delta M\). By Lemma 2.1, we have the inclusion \({\hat{M}} \subset \beta {\tilde{W}} \subset \beta {\tilde{M}}\) and the canonical surjection \(\pi _{W}:\beta {\tilde{W}} \rightarrow W\) is the restriction of \(\pi :\beta {\tilde{M}} \rightarrow M\times M\). In what follows \(\pi _{W}\) is simply denoted by \(\pi :\beta {\tilde{W}} \rightarrow W\). Let \(\tau :TM \rightarrow M\) be the tangent bundle of M and let \(SM =\{ v \in TM~|~\Vert v \Vert = 1\}\), the unit sphere bundle. We define a map \(V:{\tilde{W}} \rightarrow SM\) by
By the uniqueness of the geodesic \(\gamma _{qp}\) (3.1), the map V is a well-defined continuous map to the compact space SM and hence extends uniquely to the Stone-Čech compactification: \(\beta V: \beta {\tilde{W}} \rightarrow SM\) which restricts to:
As in Sect. 1, let \(\Delta _{M}:M \rightarrow \Delta M \subset M\times M\) be the diagonal map. We have
Lemma 3.1
We have the equality
Proof
For a point \(\omega \in {\hat{M}} \subset \beta {\tilde{W}}\) there exists a net \((p_{\alpha },q_{\alpha })_{\alpha }\) of points of \({\tilde{W}}\) such that \(\lim _{\alpha }(p_{\alpha },q_{\alpha }) = \omega \) in \(\beta {\tilde{W}}\). By the continuity of \(\beta V\) we have
Noticing \(\tau ({\dot{\gamma }}_{q_{\alpha }p_{\alpha }}(0)) = q_{\alpha }\), we have
On the other hand \(\pi (\omega ) = \lim _{\alpha }(p_{\alpha },q_{\alpha }) = (\lim _{\alpha } p_{\alpha }, \lim _{\alpha }q_{\alpha })\). Since \(\omega \in \pi ^{-1}(\Delta M)\) we have by [19, Lemma 9.6] that \(\lim _{\alpha }p_{\alpha } = \lim _{\alpha }q_{\alpha }\). Hence we have \(\Delta _{M}(\tau ({\hat{V}}(\omega ))) = \pi (\omega )\), as desired. \(\square \)
In Sect. 1 the map \(T:{\hat{M}} \rightarrow ({\mathrm {Lip}}M)^{*}\) was defined by \((T(\omega ))(f) = D_{\omega }f\) for \(\omega \in {\hat{M}},~f \in {\mathrm {Lip}}M\). The map is continuous when \(({\mathrm {Lip}}M)^{*}\) is endowed with the weak*-topology. Restricting \(T(\omega )\) to the subspace \(C^{1}(M)\) of \({\mathrm {Lip}}M\) consisting of the \(C^1\)-functions on M we obtain a composition
which is continuous when \((C^{1}(M))^{*}\) is endowed with the weak*-topology. On the other hand we have a map \(\theta :SM \rightarrow (C^{1}(M))^{*}\) given by
See [21, 1.21] for the action of tangent vectors on \(C^{1}\)-functions. The map \(\theta \) is related to the map T by the next lemma. For \(\xi \in (C^{1}(M))^{*}\) and \(f \in C^{1}(M)\), \(\xi (f)\) is also denoted by \(\langle \xi ,f \rangle \).
Lemma 3.2
-
1.
\(\theta \circ {\hat{V}} = T\), that is, for each \(\omega \in {\hat{M}}\) and for each \(f \in C^{1}(M)\), we have
$$\begin{aligned} D_{\omega }f = {\hat{V}}(\omega )f. \end{aligned}$$ -
2.
\({\text {Im}}\theta = {\text {Im}}T\).
Proof
-
1.
For a point \(\omega \in {\hat{M}}\) take a net \(((p_{\alpha },q_{\alpha }))_{\alpha }\) of points of \({{\tilde{W}}}\) such that \(\omega = \lim _{\alpha }((p_{\alpha },q_{\alpha }))\). By the continuity of \(\theta \), we have, for each \(f \in C^{1}(M)\),
$$\begin{aligned} \langle (\theta \circ {\hat{V}}) (\omega ),f \rangle= & {} \langle \theta (\lim _{\alpha }V(p_{\alpha },q_{\alpha })),f \rangle \\= & {} \lim _{\alpha } \langle \theta ({\dot{\gamma }}_{q_{\alpha }p_{\alpha }}(0)),f\rangle = \lim _{\alpha } {\dot{\gamma }}_{q_{\alpha }p_{\alpha }}(0)f \\= & {} \lim _{\alpha } \frac{d}{dt}\big |_{t=0}f(\gamma _{q_{\alpha }p_{\alpha }}(t)). \end{aligned}$$The complex-valued function f is written as \(f = u+iv\) for real-valued \(C^1\)-functions u and v. Applying the mean value theorem to u and v, we have
$$\begin{aligned} f(p_{\alpha }) - f(q_{\alpha })= & {} f(\gamma _{q_{\alpha }p_{\alpha }}(d(q_{\alpha },p_{\alpha })))-f(\gamma _{q_{\alpha }p_{\alpha }}(0)) \nonumber \\= & {} \left( \frac{d(u \circ \gamma _{q_{\alpha }p_{\alpha }})}{dt}(\rho _{\alpha }) + i\frac{d(v \circ \gamma _{q_{\alpha }p_{\alpha }})}{dt}(\sigma _{\alpha })\right) d(p_{\alpha },q_{\alpha })\nonumber \\ \end{aligned}$$(3.4)for some \(\rho _{\alpha }, \sigma _{\alpha } \in (0,d(p_{\alpha },q_{\alpha }))\). Since \(\omega \in {\hat{M}}\) we have again by [19, Lemma 9.6] that \(\lim _{\alpha }d(p_{\alpha },q_{\alpha }) = 0\). By (3.4) we have
$$\begin{aligned} \begin{aligned} \Phi _{f}(p_{\alpha },q_{\alpha }) = \frac{d(u \circ \gamma _{q_{\alpha }p_{\alpha }})}{dt}(\rho _{\alpha }) + i\frac{d(v \circ \gamma _{q_{\alpha }p_{\alpha }})}{dt}(\sigma _{\alpha }) \end{aligned} \end{aligned}$$(3.5)Taking the limit in (3.5) and using \(\lim _{\alpha }(\frac{d}{dt}u \circ \gamma _{q_{\alpha }p_{\alpha }})(\rho _{\alpha }) = (\frac{d}{dt}u \circ \gamma _{q_{\alpha }p_{\alpha }})(0)\), \(\lim _{\alpha }(\frac{d}{dt}v \circ \gamma _{q_{\alpha }p_{\alpha }})(\sigma _{\alpha }) = (\frac{d}{dt}v \circ \gamma _{q_{\alpha }p_{\alpha }})(0)\), we have
$$\begin{aligned} D_{\omega }f= & {} {\hat{\Phi }}_{f}(\omega ) = \lim _{\alpha }\Phi _{f}(p_{\alpha },q_{\alpha }) \\= & {} \lim _{\alpha } \frac{d(f \circ \gamma _{q_{\alpha }p_{\alpha }})}{dt}(0) = {\hat{V}}(\omega )f. \end{aligned}$$This proves (1).
-
2.
From (1) we see \({\mathrm {Im}}T \subset {\mathrm {Im}}\theta \). In order to prove the reverse inclusion, let \(v \in S_{p}M\) with \(\Vert v \Vert = 1\) and take the geodesic \(\gamma _{v}:[0,\delta ] \rightarrow M\) such that
$$\begin{aligned} \gamma _{v}(0) = p,\quad {\dot{\gamma }}_{v}(0) = v. \end{aligned}$$Note that the point \((\gamma _{v}(t),p)\) is in \({\tilde{W}}\) for each \(t \in (0,\delta ]\). Using \(\Vert {\dot{\gamma }}_{v}\Vert \equiv 1\), we see \(d(\gamma _{v}(t),p) = t\). Thus for each \(f\in C^{1}(M)\) and for each \(t \in (0,\delta ]\),
$$\begin{aligned} \Phi _{f}((\gamma _{v}(t),p))= & {} \frac{f(\gamma _{v}(t)) - f(\gamma _{v}(0))}{d(\gamma _{v}(t),p)} \\= & {} \frac{1}{d(\gamma _{v}(t),p)} \frac{d(f\circ \gamma _{v})}{dt}(\rho _{t}) \cdot d(\gamma _{v}(t),p) \\= & {} \frac{d(f\circ \gamma _{v})}{dt}(\rho _{t}) \end{aligned}$$for some \(\rho _{t} \in (0,t)\). Let \(\omega \in {\hat{M}}\) be an accumulation point of \(\{(\gamma _{v}(t),p)~|~t \in (0,\delta ]\}\). Since the net \((\frac{d(f\circ \gamma _{v})}{dt}(\rho _{t}))_{t\in (0,\delta ]}\) converges to \(\frac{d(f\circ \gamma _{v})}{dt}(0) = {\dot{\gamma }}(0)f = vf\), we have
$$\begin{aligned} D_{\omega }f = {\hat{\Phi }}_{f}(\omega ) = vf, \end{aligned}$$as desired.
\(\square \)
In view of these lemmas we may regard \({{\hat{M}}}\) as a Lipschitz-analogue of the unit sphere bundle SM of a Riemannian manifold M. Let \(\Gamma (M,SM)\) be the space of smooth sections of the bundle \(\tau :SM\rightarrow M\)
and let \(\Theta :\Gamma (M,SM) \rightarrow {\mathfrak {D}}(C^{1}(M),C(M))\) be the map defined by
A standard argument shows that the image \({\mathrm {Im}}\Theta \) is non-zero and finitely generated as a \(C^{1}(M)\)-module. The map \(\Theta \) yields the map \(\theta \) in Lemma 3.2 when localized at a point p: To be more precise, let \(\epsilon _{p}:\Gamma (M,SM) \rightarrow S_{p}M\) and \(e_{p}:{\mathfrak {D}}(C^{1}(M),C(M)) \rightarrow {\mathfrak {D}}_{p}(M)\) be the evaluation maps defined by
Then we have
Here the similarity between the spaces \({\hat{M}}\) and SM breaks down: every continuous map \(\sigma :M \rightarrow {\hat{M}}\) of a path-connected compact metric space M must be a constant map, because the space \({{\hat{M}}} = \beta {\tilde{M}} {\setminus } {\tilde{M}}\), being a remainder of the Stone–Čech compactification of a non-psuedo-compact Lindelöf space \({\tilde{M}}\), contains no metrizable compact connected subsets which are not singletons [9] and hence the image \(\sigma (M)\) must be a singleton. In particular there exists no continuous map \(\sigma :M \rightarrow {\hat{M}}\) such that \(\pi \circ \sigma = {\mathrm {id}}_{M}\) for such a space. This prevents us from defining a map which corresponds to \(\Theta \) (3.6) to obtain elements of \({\mathfrak {D}}({\mathrm {Lip}}M,C(M))\). More strongly, Theorem 1.3 states that there exists no non-zero derivations \({\mathrm {Lip}}M \rightarrow C(M)\) when M is a compact Lipschitz manifold. Combining [8, Theorem 3.5] we see that the map \(e_{p}:{\mathfrak {D}}({\mathrm {Lip}}M,C(M)) \rightarrow {\mathfrak {D}}_{p}(M)\) that corresponds to (3.7) reduces to the trivial map \(0 \rightarrow \) (an \(\infty \)-dimensional space).
The rest of this section is devoted to the proof of Theorem 1.3. First the theorem is proved for \(M = [0,1]^{m} \subset {\mathbb {R}}^{m}\) and the result is combined with Theorem 1.5 to prove the general case. We start with several lemmas. For simplicity let \(I = [0,1]\). For \(i\in \{1,\ldots ,n\}\) and \(a \in {\mathrm {int}}I\), the subspace \(H =\{(t_{1},\ldots ,t_{m}) \in I^{m}~|~ t_{i} = a\}\) is called a coordinate section. For two points \(x,y \in I^{m}\), \({\overline{xy}}\) denotes the segment joining x with y. For a subset S of \(I^{m}\), \({\mathrm {int}}S\) denotes the interior of S in \(I^m\). Notice that for each derivation \(D:{\mathrm {Lip}}M \rightarrow C(M)\) we have
for each constant function \(c \in {\mathrm {Lip}}M\).
Lemma 3.3
For \(a \in {\mathrm {int}}I\) and \(i=1,\ldots , m\), let \(H^{a,i} =\{(t_{1},\ldots ,t_{m}) \in I^{m}~|~ t_{i} = a\}\) be a coordinate section of \(I^m\) and let
For a Lipschitz function \(f \in {\mathrm {Lip}}I^{m}\) with \(f|H^{a,i} \equiv 0\), let
and
Then \(f_{+}\) and \(f_{-}\) are Lipschitz functions such that \(f = f_{+} + f_{-}\) and \(f_{+}\cdot f_{-} = 0\).
Proof
Let \(x \in H_{+}^{a,i}\) and \(y \in H_{-}^{a,i}\) and take the point \(m \in {\overline{xy}}\cap H^{a,i}\). We have
Thus \(f_{+}\) is a Lipschitz function. Similarly \(f_{-}\) is a Lipschitz function. The last equalities follow directly from the definition. \(\square \)
Lemma 3.4
Let \(D:{\mathrm {Lip}}I^{m} \rightarrow C(I^{m})\) be a derivation and let B be a convex body in \(I^{m}\). For each \(f \in {\mathrm {Lip}}I^{m}\) with \(f|B \equiv 0\), we have \(Df|B \equiv 0\).
Proof
Let \(g(x) = d(x,\overline{I^{m}{\setminus } B}),~x \in I^{m}\). It is straightforward to see
Restricting the equality \(0 = D(fg) = f\cdot Dg + g \cdot Df\) to \({\mathrm {int}}B\), we obtain
and hence \(Df|{\mathrm {int}}B = 0\). By the continuity of Df we have \(Df|B = 0\). \(\square \)
Lemma 3.5
Let H be a coordinate section of \(I^{m}\) and let \(D:{\mathrm {Lip}}I^{m} \rightarrow C(I^{m})\) be a derivation. For each function \(f \in {\mathrm {Lip}}M\) with \(f|H \equiv 0\), we have \(Df|H \equiv 0\).
Proof
We may assume \(H = H^{a,m}= \{(t_{1},\ldots ,t_{m-1},a)~|~t_{i} \in I,1\le i\le m-1\}\) for some \(a \in {\mathrm {int}}I\). Let \(H_{+} = H_{+}^{a,i}, H_{-} = H_{-}^{a,i}\), and let \(f \in {\mathrm {Lip}}M\) with \(f|H \equiv 0\). We see that the functions \(f_{+}\) and \(f_{-}\) defined by (3.9) and (3.10) are Lipschitz such that \(f = f_{+}+f_{-}, ~f_{+}f_{-} = 0\) due to Lemma 3.3. From the equality \(0 = D(f_{+}f_{-}) = f_{+}Df_{-}+f_{-}Df_{+}\) we see
We take an arbitrary \(p \in H\) and prove \(Df(p) = 0\) by considering two cases.
Case 1. There exists an \(\varepsilon >0\) such that, for the rectangular neighbourhood \(B_{\varepsilon } = \prod _{i=1}^{m}[p_{i}-\varepsilon ,p_{i}+\varepsilon ]\), we have either
Applying Lemma 3.4 to the convex body \(B_{\varepsilon }\cap H_{+}\) or \(B_{\varepsilon }\cap H_{-}\), we conclude \(Df|B_{\varepsilon }\cap H_{+}\equiv 0\) or \(Df|B_{\varepsilon }\cap H_{-}\equiv 0\). In particular we have \(Df(p) = 0\).
Case 2. There exist two sequences \((x_{k})_{k\ge 1}\) and \((y_{k})_{k\ge 1}\) such that
- (i)
\(x_{k} \in H_{+}, y_{k} \in H_{-}\) and \(f(x_{k}) \ne 0 \ne f(y_{k})\) for each \(k\ge 1\),
- (ii)
\(\lim _{k}x_{k} = \lim _{k}y_{k} = p\).
By (3.12), (3.13) and (i) above, we have \(Df_{-}(x_{k}) = Df_{+}(y_{k}) = 0\) for each k and hence by continuity of \(Df_{\pm }\) we see \(Df_{-}(p) = Df_{+}(p) = 0\). Then we see \(Df(p) = Df_{+}(p) + Df_{-}(p) = 0\).
Since p is an arbitrary point of H we have \(Df|H \equiv 0\). \(\square \)
Remark 3.6
The above lemma holds also for \(m=1\) in which case H is a singleton in \({\mathrm {int}}I\).
Proof of Theorem 1.3
Step 1. As before let \(I = [0,1]\). First we prove the theorem for \(M = I^{m}\) by induction on m.
- (i)
\(m=1\). Let \(D:{\mathrm {Lip}}I \rightarrow C(I)\) be a derivation, let \(f \in {\mathrm {Lip}}I\), and take a point \(p \in {\mathrm {int}}I\). Let \(f_{p}:I \rightarrow {\mathbb {C}}\) be the function defined by \(f_{p}(t) = f(t) - f(p), ~t \in I\). By (3.8) we have \(Df_{p} = Df\). Since \(f_{p}(p) = 0\), we have \((Df_{p})(p) = 0\) by Lemma 3.5 and Remark 3.6. Thus we obtain \(Df(p) = Df_{p}(p) = 0\). Since p is an arbitrary point of \({\mathrm {int}}I\) we see by continuity that \(Df \equiv 0\) on I.
- (ii)
Assume that theorem holds for m and let \(D:{\mathrm {Lip}}I^{m+1}\rightarrow C(I^{m+1})\) be a derivation. Take a point \(a = (a_{1},\ldots , a_{m+1}) \in {\mathrm {int}}I^{m+1}\) and take the coordinate section \(H=\{(t_{1},\ldots ,t_{m},a_{m+1})~|~t_{i} \in I,1\le i \le m\}\). The space H is isometric to \(I^m\) and the inclusion of H into \(I^{m+1}\) is denoted by \(\iota :H \rightarrow I^{m+1}\). Let \(R:I^{m+1} \rightarrow H\) be the projection defined by
$$\begin{aligned} R(t_{1},\ldots ,t_{m+1}) = (t_{1},\ldots ,t_{m},a_{m+1}),\quad (t_{1},\ldots ,t_{m+1}) \in I^{m+1}. \end{aligned}$$The map R is a Lipschitz map. We define an operator \(d:{\mathrm {Lip}}H \rightarrow C(H)\) by \(d = \iota ^{*}\circ D \circ R^{*}\) which is explicitly given by
$$\begin{aligned} df = D(f\circ R)|H,\quad f \in {\mathrm {Lip}}H. \end{aligned}$$We show that d is a derivation. Indeed using \(R|H = {\mathrm {id}}_{H}\) we have
$$\begin{aligned} d(fg)= & {} D(fg\circ R)|H = D((f\circ R)\cdot (g\circ R))|H\\= & {} (f\circ R|H)\cdot D(g \circ R)|H +(g\circ R|H)\cdot (f \circ R)|H\\= & {} f D(g \circ R)|H + g D(f \circ R)|H = fdg+gdf. \end{aligned}$$By the inductive hypothesis and the isometry \(H \equiv I^{m}\) we see \(d = 0\). Thus for each \(h \in {\mathrm {Lip}}H\), we have
$$\begin{aligned} dh = D(h\circ R)|H = 0. \end{aligned}$$(3.14)For an arbitrary \(f \in {\mathrm {Lip}}I^{m+1}\), consider the function \(g_{f}\) given by
$$\begin{aligned} g_{f} = f - (f|H)\circ R \end{aligned}$$which is a Lipschitz function on \(I^{m+1}\) such that \(g_{f}|H\equiv 0\). By Lemma 3.5 we see \((Dg_{f})|H \equiv 0\) and thus by (3.14) we have
$$\begin{aligned} Df|H= D((f|H)\circ R)|H =0. \end{aligned}$$In particular \(Df(a) = 0\). Since a is an arbitrary point of \({\mathrm {int}}I\) we see by continuity of Df that \(Df \equiv 0\) on \(I^{m+1}\). This finishes the inductive step and Step 1 is completed.
Step 2. For a proof of general M, we use the next lemma. The standard Euclidean metric on \(I^m\) is denoted by \(\rho \).
Lemma 3.7
Let \(D:{\mathrm {Lip}}M \rightarrow C(M)\) be a derivation. Let \(\alpha :I^{m} \rightarrow M\) be a bi-Lipschitz embedding of \(I^m\) into a compact metric space (M, d) such that \(\alpha ({\mathrm {int}}I^{m})\) is open in M. For each \(f \in {\mathrm {Lip}}M\) with \(f|\alpha (I^{m}) \equiv 0\), we have \(Df|\alpha (I^{m}) \equiv 0\).
Proof
For an \(\epsilon \in (0,1)\), let \(\epsilon I^{m} = [\epsilon ,1-\epsilon ]^{m}\). We define a function \(g:M \rightarrow [0,\infty )\) by
Notice that
and hence the above function is well-defined.
In order to see that g is a Lipschitz function, first notice that \( t \mapsto d(t,\overline{I^{m}{\setminus } \epsilon I^{m}}) \) is a Lipschitz function on \(I^{m}\). Since \(\alpha \) is a bi-Lipschitz embedding we see that \(g|\alpha (I^{m})\) is a Lipschitz function. This and (3.15) imply that g is a locally Lipschitz function. By the compactness of M we conclude that \(g \in {\mathrm {Lip}}M\) (see [11, p. 85]). Also by the definition \(g(q) \ne 0\) for each \(q \in \alpha ({\mathrm {int}}(\epsilon I^{m}))\).
For each \(f \in {\mathrm {Lip}}M\) with \(f|\alpha (I^{m}) \equiv 0\), we have \(fg \equiv 0\) and thus
which implies \(Df|\alpha ({\mathrm {int}}(\epsilon I^{m}))=0\). Since \(\epsilon \) is an arbitrary number in (0, 1) we see that \(Df|\alpha (I^{m}) \equiv 0\). \(\square \)
In order to finish the proof of Theorem, let M be a compact metric space as in the hypothesis and let \(D:{\mathrm {Lip}}M \rightarrow C(M)\) be a continuous derivation. Fix a point \(p \in M\). Take a bi-Lipschitz embedding \(\alpha :I^{m}\rightarrow M\) such that \(p \in \alpha (I^{m})\) and \(\alpha ({\mathrm {int}}I^{m})\) is open in M. First we show that there exists a Lipschitz map \(R:M \rightarrow \alpha (I^{m})\) such that \(R|\alpha (I^{m})= {\mathrm {id}}_{\alpha (I^{m})}\).
To show the above, let \({\mathrm {proj}}_{j}:I^{m}\rightarrow I\) be the projection to the j-th factor (\(1\le j \le m\)). The map \({\mathrm {proj}}_{j}\circ \alpha ^{-1}:\alpha (I^{m}) \rightarrow I\) is a Lipschitz function and we apply Theorem 1.5 to obtain a Lipschitz function \(r_{j}:M\rightarrow I\) such that \(r_{j}|\alpha (I^{m}) = {\mathrm {proj}}_{j}\circ \alpha ^{-1}\). Define \(r:M \rightarrow I^{m}\) by \(r(x) = (r_{j}(x))_{1\le j \le m}\) and let
Then the map R is the desired Lipschitz map (see [11, Lemma 5.6]).
Take a function \(f \in {\mathrm {Lip}}M\) and let \(g_{f}\) be the function given by
which is a Lipschitz function such that \(g_{f}|\alpha (I^{m}) \equiv 0\). By Lemma 3.7 we see \(Dg_{f}|\alpha (I^{m}) \equiv 0\). Thus we see
We notice that the Lipschitz homeomorphism \(\alpha :I^{m}\rightarrow \alpha (I^{m})\) induces algebraic isomorphisms \(\alpha ^{*}:{\mathrm {Lip}}({\mathrm {Im}}\alpha ) \rightarrow {\mathrm {Lip}}(I^{m})\) and \(\alpha ^{*}:C(\alpha (I^{m})) \rightarrow C(I^{m})\). It follows from this and Step 1 that the derivation \(d:{\mathrm {Lip}}(\alpha (I^{m})) \rightarrow C({\mathrm {Im}}\alpha )\) defined by
is the zero-homomorphism. It implies \(D(f|\alpha (I^{m}) \circ R)|\alpha (I^{m}) = 0\) for each \(f \in {\mathrm {Lip}}M\). Combining this with (3.16) we have \(Df|\alpha (I^{m}) = 0\) and thus \(Df(p) = 0\), as required.
This completes the proof of theorem. \(\square \)
For a compact metric space M as in Theorem 1.3 and \(n\ge 2\), take an alternating n-cochain \(F \in Z^{n}_{\mathrm {alt}}({\mathrm {Lip}}(M), C(M))\). By (1) of Theorem 1.6, F is an n-derivation. Fixing arbitrary Lipschitz functions \(f_{1}, \ldots , f_{n-1} \in {\mathrm {Lip}}(M)\), we have the linear operator \(f \mapsto F(f_{1},\ldots , f_{n-1},f)\) that is a derivation due to (1.12). It follows from the proof of Theorem 1.3 that the operator is zero and we conclude:
Corollary 3.8
Let M be a compact metric space as in Theorem 1.3. Then we have \(Z_{\mathrm {alt}}^{n}({\mathrm {Lip}}(M),C(M)) = 0\) for each \(n \ge 2\).
It is not known to the author whether the cohomology \({\mathrm {H}}^{n}({\mathrm {Lip}}(M), C(M)) \) is trivial for each \(n \ge 2\) and for each compact metric space M in Theorem 1.3.
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Acknowledgements
The author is grateful to the referees for their comments which were helpful to improve the exposition of the paper. Kazuhiro Kawamura is supported by JSPS KAKENHI Grant number 17K05241.
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Communicated by Armando R. Villena.
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Kawamura, K. Derivations and cohomologies of Lipschitz algebras. Banach J. Math. Anal. 14, 140–162 (2020). https://doi.org/10.1007/s43037-019-00025-1
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DOI: https://doi.org/10.1007/s43037-019-00025-1