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20fa507a9aefc290c512f58dff04a2c9db5c6479 | subsection | 3 | 224 | The main results | In this paper we prove three key theorems (Theorems REF , REF and REF ) and a new result concerning universal measurability (Theorem REF ).Essential to our approach is a certain set of assumptions on a spectral triple to be outlined below. The notion of a spectral triple, and all of the corresponding notations are expl... | {
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6dd6874e1482c32792a0e0cafbf26bd72a30c92c | subsection | 4 | 224 | The main results | The definition of dimension in REF .(REF ) is strictly stronger, and we discuss this issue in REF .Condition REF .(REF ) is new and specific to the locally compact situation.
Indeed, if \mathcal {A} is unital then REF .(REF ) is redundant, as it follows from REF .(REF ).In order to show that Hypothesis REF .(REF ) is r... | {
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b86a8d1aa85f4df893a2ce0714ecf65baac23244 | subsection | 5 | 224 | The main results | The use of locality in noncommutative geometry was pioneered by Rennie in .Definition 1.2.4 A Hochschild cycle c=\sum _{j=1}^m a_0^j\otimes \cdots \otimes a_p^j \in \mathcal {A}^{\otimes (p+1)} is said to be local if there exists a positive element \phi \in \mathcal {A} such that \phi a_0^j=a_0^j for all 1\le j\le m.Fo... | {
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c803ccf11eeed9f168984d63bd0983bb5b4102df | subsection | 6 | 224 | The main results | A new result of this paper, and a crucial component of our proof of Theorem REF , is the following:Theorem 1.2.7
Let 0\le V\in \mathcal {L}_{1,\infty } and let A\in \mathcal {L}_{\infty }.
Define the \zeta -function:\zeta _{A,V}(z) := \mathrm {Tr}(AV^{1+z}),\quad \Re (z) > 0.If there exists \varepsilon > 0 such that \... | {
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102b3298a7e754a4d09548af9a93aee075ac8218 | subsection | 7 | 224 | Context of this paper | Connes' Character Formula dates back to Connes' 1995 paper . There the character theorem was discovered in order to “compute by a local formula the cyclic cohomology Chern character of (\mathcal {A},H,D)”. Connes' work initiated a lengthy and ongoing program to strengthen, generalise and better understand the Character... | {
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8419d65339c5025e2f9cccdbe89193a706193431 | subsection | 8 | 224 | Context of this paper | To be precise: the proof
relied on an equality between\lim _{s\downarrow 0} \mathrm {Tr}(Z|D|^{-n-s})and\mathrm {Tr}_\omega (Z|D|^{-n})(in the notation of ). In the case where |D|^{-1} is compact this result can be attained using
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777710e2b4b5edee30e8254521c3ea1c4d3bd364 | subsection | 9 | 224 | Structure of the paper | This paper is structured as follows:Chapter is devoted to preliminary definitions and concepts: we introduce the relevant definitions for operator ideals, traces, spectral
triples, operator valued integrals and double operator integrals.
Chapter provides important technical properties of spectral triples. In Section ... | {
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261b142489430924c5802fca1f8e50eda3965fa3 | subsection | 10 | 224 | General notation | Fix throughout a separable, infinite dimensional complex Hilbert space H. We denote by \mathcal {L}_{\infty } the algebra of all bounded operators on H, with operator norm denoted \Vert \cdot \Vert _\infty . For a compact operator T on H,
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2beed6a6dabd1faae1dace82b3ba89bf2559165e | subsection | 11 | 224 | Ideals in | For p \in (0,\infty ), we let \mathcal {L}_{p} denote the Schatten-von Neumann ideal of \mathcal {L}_{\infty },\mathcal {L}_p := \lbrace T \in \mathcal {L}_{\infty }\;:\; \mu (T) \in \ell _p\rbracewhere \ell _p is the space of p-summable sequences. As usual, for p \ge 1 the ideal \mathcal {L}_{p} is equipped
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642e55c9834f7b488a4f919a70faf82939cd26d0 | subsection | 12 | 224 | Ideals in | If A_k \in \mathcal {L}_{p_k,\infty } for all k = 1,\ldots , n, then
A_1A_2\cdots A_n \in \mathcal {L}_{p,\infty }, with an inequality of norms:\Vert A_1A_2\cdots A_n\Vert _{p,\infty } \le c_{p_1,p_2,\ldots ,p_n}\Vert A_1\Vert _{p_{1},\infty }\Vert A_2\Vert _{p_2,\infty }\cdots \Vert A_n\Vert _{p_n,\infty }where c_{p_1... | {
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55b960124fb706a9fb3002b535b80eefd0d67cf3 | subsection | 13 | 224 | Ideals in | To be precise, we have that for all A,B \in \mathcal {L}_{1,\infty } if B \prec \prec _{\log } A
then\Vert B\Vert _{1,\infty } \le e\Vert A\Vert _{1,\infty }.Indeed, since the sequence \lbrace \mu (k,B)\rbrace _{k=0}^\infty is nonincreasing, for all n \ge 0 we have:\mu (n,B)^{n+1} \le \prod _{k=0}^n \mu (k,B).So if B \... | {
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3f84131e9a656cf9296d00470dd335e823244de8 | subsection | 14 | 224 | Traces on | Definition 2.1.1
If \mathcal {I} is an ideal in \mathcal {L}_{\infty }, then a unitarily invariant
linear functional \varphi :\mathcal {I}\rightarrow \mathbb {C} is said to be a trace.Here \varphi being “unitarily invariant" means that \varphi (U^*TU) = \varphi (T) for all T \in \mathcal {I} and unitary operators U.
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866a8bfb1f0a61cfd1d94c052ef0eca96a2fe1c2 | subsection | 15 | 224 | Traces on | Combining , for all n \ge 0 we have:\sum _{k=0}^n \mu (k,A+B) \le \sum _{k=0}^n \mu (k,A)+\mu (k,B) \le \sum _{k=0}^{2n+1} \mu (k,A+B).Hence,0 \le \sum _{k=0}^n \mu (k,A)+\mu (k,B)-\mu (k,A+B) \le \sum _{k=n+1}^{2n+1} \mu (k,A+B).However A+B \in \mathcal {L}_{1,\infty }, so there is a constant C > 0 such that for all k... | {
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497eff0203eef6ae08e11578f5539dd47da54575 | subsection | 16 | 224 | Traces on | There exist traces on \mathcal {L}_{1,\infty } which fail to be continuous (see ).
Every trace on \mathcal {L}_{1,\infty } vanishes on \mathcal {L}_1 (see ).We are mostly interested in normalised traces \varphi :\mathcal {L}_{1,\infty }\rightarrow \mathbb {C}, that is, satisfying \varphi ({\rm diag}(\lbrace \frac{1}{k... | {
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353ec4c2469268088d789ec6af87d3c0c6b61729 | subsection | 17 | 224 | Spectral triples | A spectral triple is an algebraic model for a Riemannian manifold, defined as follows:Definition 2.2.1
A spectral triple (\mathcal {A},H,D) consists of the following data:a separable Hilbert space H.
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228ed4399dc4513698dacfe92a0c25e771df4b2e | subsection | 18 | 224 | Properties of spectral triples | Smoothness of a spectral triple is defined in terms of boundedness of commutators with |D| (see Subsection REF for discussion of this issue).
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8d80cacd8f8d7eddd1a0cdb59c8b0e2f21eee11e | subsection | 19 | 224 | Properties of spectral triples | The kth iterated commutator \delta ^k(TS)|\mathrm {dom}(D^k) is given by:\delta ^k(TS) = \sum _{j=0}^k \binom{k}{j} \delta ^{k-j}(T)\delta ^j(S).Since for all j we have \delta ^j(S) \in \mathrm {dom}_\infty (\delta ) and \delta ^{k-j}(T) \in \mathrm {dom}_\infty (\delta ), the above expression
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29029df4aa0042eff96e09b6489ddbfee1d0d939 | subsection | 20 | 224 | Properties of spectral triples | Then,DT\xi = \partial (T)\xi + TD\xi .Since T:\mathrm {dom}(D)\rightarrow \mathrm {dom}(D) and \partial (T):\mathrm {dom}(D^2)\rightarrow \mathrm {dom}(D), it follows that DT\xi \in \mathrm {dom}(D) and therefore T:\mathrm {dom}(D^2)\rightarrow \mathrm {dom}(D^2).Now since the operators D and |D| commute on \mathrm {do... | {
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0b73e29e6c5de1bc642eb5696dd1fefe12b545c4 | subsection | 21 | 224 | Properties of spectral triples | For k=1, by the definition of smoothness we have \partial (a) \in \mathrm {dom}(\delta ) and a \in \mathrm {dom}(\delta )\cap \mathrm {dom}(\partial ), and by definition a:\mathrm {dom}(D)\rightarrow \mathrm {dom}(D). So by Lemma REF it follows that
\delta (a) \in \mathrm {dom}(\partial ) and\partial (\delta (a)) = \de... | {
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f9dd63fe33ae7b3a239e0f4d25d9972a974d9887 | subsection | 22 | 224 | Properties of spectral triples | If T \in \mathrm {dom}_\infty (\delta ), then T:H_\infty \rightarrow H_\infty since by definition if T \in \mathrm {dom}(\delta ^n) then T:\mathrm {dom}(D^k)\rightarrow \mathrm {dom}(D^k) for all 0 \le k \le n. Moreover since F:\mathrm {dom}(D^n)\rightarrow \mathrm {dom}(D^n) for all n, we also
have F:H_\infty \rightar... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
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97d77f974842021f4f92dc66e5230e3d74670cd6 | subsection | 23 | 224 | Properties of spectral triples | Then L(T) \in \mathrm {dom}(\delta ) and\delta (L(T)) = L(\delta (T)).Since \partial (T) \in \mathrm {dom}(\delta ), we have from Lemma REF that \delta (T) \in \mathrm {dom}(\partial )\cap \mathrm {dom}(\delta ) and hence L(\delta (T)) is defined and bounded.The required identity can be checked on \mathrm {dom}(D^2), s... | {
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"Fedor Sukochev",
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9a57f4234146748237d982c53ead99cc8a3605ce | subsection | 24 | 224 | Properties of spectral triples | If (\mathcal {A},H,D) is a p-dimensional spectral triple satisfying Hypothesis REF , then [F,a]\in \mathcal {L}_{p,\infty } for all a\in \mathcal {A}.Let \mathcal {A}^{\otimes (p+1)} denote the (p+1)-fold algebraic tensor power of \mathcal {A}. We now define the two important mappings \mathrm {ch} and \Omega .Definitio... | {
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bb75bcd9e2c4d7cb35debdefc809508e39e19e71 | subsection | 25 | 224 | Properties of spectral triples | Consider the following unitary self-adjoint operators on the Hilbert space H_0=\mathbb {C}^2\otimes H defined by:F_0 := \begin{pmatrix} F & P \\ P & -F\end{pmatrix}\\
\Gamma _0 := \begin{pmatrix} \Gamma & 0 \\ 0 & (-1)^{\rm deg}\Gamma \end{pmatrix}.Here, {\rm deg}=1 for even triples and {\rm deg}=0 for odd triples. The... | {
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"Fedor Sukochev",
"Dmitriy Zanin"
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c48f582e64c5bd8ba5456a7bd7352aa599821ecf | subsection | 26 | 224 | Discussion of smoothness | It is tempting to define smoothness only in terms of \partial , without reference to \delta . One might naively suggest that (\mathcal {A},H,D) is smooth if for all n \ge 0
we have a\cdot \mathrm {dom}(D^n)\rightarrow \mathrm {dom}(D^n) and the nth iterated commutator [D,[D,[\cdots ,[D,a]\cdots ] extends to a bounded o... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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121894fd466081b18714e64a1e2b3e93ac7bb28a | subsection | 27 | 224 | Discussion of dimension | As we have defined it, we say that a spectral triple (\mathcal {A},H,D) is p-dimensional if for all a \in \mathcal {A} the operators a(D+i)^{-p} and \partial (a)(D+i)^{-p} are in \mathcal {L}_{1,\infty }.An alternative definition, also used in the literature, is to say that (\mathcal {A},H,D) is p-dimensional if a(D+i)... | {
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"raw": "Gayral V., Gracia-Bondia J., Iochum B., Schücker T., Varilly J. Moyal planes are spectral triples. Comm. Math. Phys. 246 (2004), no. 3, 569–6... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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4e4efdecf0a0b4762b0e0a6a14aca52daf50b86a | subsection | 28 | 224 | Hochschild (co)homology | Hochschild homology and cohomology provide noncommutative generalisations of the notion of differential forms and de Rham currents
respectively. A detailed exposition of the theory of Hochschild (co)homology and its relationship with noncommutative geometry may be found in , .Let A be a (possibly non-unital) algebra. T... | {
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82186736e1e64d84cc32d6660bcadbaf4e3256ae | subsection | 29 | 224 | Hochschild (co)homology | The Hochschild cochain complex is,C_1(A)\xrightarrow{} C_2(A)\xrightarrow{} C_3(A) \xrightarrow{} \cdotswhere the Hochschild coboundary operator b is defined as follows: if \theta :A^{\otimes n}\rightarrow \mathbb {C}, then b\theta :A^{\otimes (n+1)}\rightarrow \mathbb {C}
is given on an elementary tensor a_0\otimes a_... | {
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"Fedor Sukochev",
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7e3c11fa0d93fadf25231e05fa4dd6d6b96ec856 | subsection | 30 | 224 | Weak integration in | This section concerns the theory of “weak operator topology integrals" of operator valued functions. The following definitions, and the subsequent construction of weak integrals, are folklore. We provide suitable references whenever they
exist, otherwise we supply a proof. For example, one can look at , and consider th... | {
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"Fedor Sukochev",
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36b9bb08dcbb080350e0ea87987b4a0658fae9da | subsection | 31 | 224 | Weak integration in | We say that f is integrable in the weak operator topology if\int _{\mathbb {R}}\Vert f(s)\Vert _{\infty }ds < \infty .In particular, for all \xi ,\eta \in H, we have\int _{\mathbb {R}} |\langle f(s)\xi ,\eta \rangle |\,ds < \infty .Hence for a function f satisfying (REF ), we may therefore define the sesquilinear form(... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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e84cb5bbf819e37cfdd951dad796f27eb37c95ee | subsection | 32 | 224 | Weak integration in | Let T be the unique bounded linear operator such that x_\xi = T\xi , we now define\int _{\mathbb {R}} f(s)\,ds := T.Due to the above computation, we have that\left\Vert \int _{\mathbb {R}} f(s)\,ds\right\Vert _\infty \le \int _{\mathbb {R}} \Vert f(s)\Vert _{\infty }\,ds.Furthermore, we have that if A \in \mathcal {L}_... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
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209caf149f51145b4d9f21693ea21d09d0d5e677 | subsection | 33 | 224 | Properties of the weak integral | The authors thank Professor Peter Dodds for his assistance with the arguments in this subsection.Lemma 2.3.2 Let s\rightarrow a(s), s\in \mathbb {R}, be continuous in the weak operator topology. If a(s)\in \mathcal {L}_1 for every s\in \mathbb {R} and if\int _{\mathbb {R}}\Vert a(s)\Vert _1ds<\infty ,then a(s) is integ... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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ef4d02bc7c150d00e2968a288fc3c47cddc15f95 | subsection | 34 | 224 | Properties of the weak integral | We have{\rm Tr}(p_nAp_n)=\int _{\mathbb {R}}{\rm Tr}(p_na(s)p_n)ds.Clearly, {\rm Tr}(p_na(s)p_n)\rightarrow {\rm Tr}(a(s)) as n\rightarrow \infty for every s\in \mathbb {R}. Since the function s \mapsto \mathrm {Tr}(a(s)) is integrable, we can apply the dominated convergence theorem to obtain\int _{\mathbb {R}}{\rm Tr}... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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"math.OA"
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c6c4a0a98dd6d0b55081a86c59e5c854e234ecc3 | subsection | 35 | 224 | Properties of the weak integral | Then,p|A|p = \int _{\mathbb {R}} pU^*a(s)p\,ds.Thus,\mathrm {Tr}(p|A|p) &\le \int _{\mathbb {R}} |\mathrm {Tr}(pU^*a(s)p)|\,ds\\
&\le \int _{\mathbb {R}} \Vert pU^*a(s)p\Vert _1\,ds.The latter integral converges because \Vert pU^*a(s)p\Vert _1 \le n\Vert a(s)\Vert _\infty . Now,\Vert pU^*a(s)p\Vert _1 &\le \sum _{k=0}^... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
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d43b7ed752b47a67c9609149b86599e75dd363e9 | subsection | 36 | 224 | Double operator integrals | Here, we state the definition and basic properties of double operator integrals. This theory was initiated by the work
of Birman and Solomyak , , , and more recent summaries
of the theory may be found in , .Heuristically, given self-adjoint operators X and Y with spectra \sigma (X) and \sigma (Y), spectral resolutions ... | {
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257bc6bab70db90c1f317edb1192ac964c5194dc | subsection | 37 | 224 | Double operator integrals | Namely,T_{\phi _1+\phi _2}^{X,Y}=T_{\phi _1}^{X,Y}+T_{\phi _2}^{X,Y},\quad T_{\phi _1\cdot \phi _2}^{X,Y}=T_{\phi _1}^{X,Y}\circ T_{\phi _2}^{X,Y}.If, in (REF ) we take \Omega to be a one-point set, then \phi (\lambda ,\mu ) = a(\lambda )b(\mu ) andT_{\phi }^{X,Y}(A)=a(X)Ab(Y). | {
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"Fedor Sukochev",
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ad7d426fce65e3b8f261a3450ebbdbe18ef03f58 | subsection | 38 | 224 | Fourier transform conventions | We follow the convention that the Fourier transform of a function g \in L_1(\mathbb {R}) is defined by the formula\mathcal {F}(g)(t) := (2\pi )^{-1/2}\int _{\mathbb {R}} g(s)e^{-its}\,dsSo that the inverse Fourier transform is given for h \in L_1(\mathbb {R}) by,\mathcal {F}^{-1}(h)(s) := (2\pi )^{-1/2}\int _{\mathbb {... | {
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"raw": "Potapov D., Sukochev F. Unbounded Fredholm modules and double operator integrals. J. Reine Angew. Math. 626 (2009), 159–185.",
"source_re... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
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e24b54358eb14bb88ed94a3cdc9e01699ba501d6 | subsection | 39 | 224 | Spectral Triples: Basic properties and examples | This chapter is primarily concerned with Hypothesis REF . We study the consequences of this hypothesis, and also show that it
is satisfied for two important classes of examples.We begin with the proof of Proposition REF , an important prerequisite to the definition
of the Chern character (Definition REF ).
Next, we sho... | {
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"raw": "Kordyukov Y. Differential operators on manifolds and their applications in geometry and topology. Proceedings of Crimean Autumn School, 2009.",
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df5bf5a21ad676d335b56fc1a9091a863984ab3b | subsection | 40 | 224 | A spectral triple defines a Fredholm module | This section is devoted to the proof of Proposition REF . We prove this in several steps, initially working with the assumption that D has a spectral
gap at 0 (i.e., that D has bounded inverse). We later show how this assumption can be removed.Note that if D has a spectral gap at 0, then F = D|D|^{-1} = |D|^{-1}D.Remar... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
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ea90ac59db976ca844dea839061424bdb64fa5c8 | subsection | 41 | 224 | A spectral triple defines a Fredholm module | Similarly,[|D|^{-1},\partial (\delta (a))] &= -|D|^{-1}[|D|,\partial (\delta (a))]|D|^{-1} \\
&= -|D|^{-1}\partial (\delta ^2(a))|D|^{-1}.Additionally, working with operators on H_\infty :|D|^{-1}[D^2,a] &= |D|^{-1}\cdot (D\partial (a)+\partial (a)D)\\
&= F\partial (a)+|D|^{-1}\partial (a)D\\
&= F\partial (a)+\partial ... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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fd3307ac245d542ee3d73f7ce7ce7563c67ba9ac | subsection | 42 | 224 | A spectral triple defines a Fredholm module | Hence, the operator |D|^{-1}[D^2,a]|D|^{-p} has extension to an operator in \mathcal {L}_{1,\infty }.On the other hand since |D|^2 = D^2, we have (again, as operators on H_\infty )|D|^{-1}[D^2,a] &= |D|^{-1}[|D|^2,a]\\
&= |D|^{-1}\cdot (|D|\delta (a)+\delta (a)|D|)\\
&= \delta (a)+|D|^{-1}\delta (a)|D|\\
&= \delta (a)+... | {
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"raw": "Carey A., Gayral V., Rennie A., Sukochev F. Index theory for locally compact noncommutative geometries. Mem. Amer. Math. Soc. 231 (2014), no.... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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78db4a33a322a475a05aa6561cfbccf91696656b | subsection | 43 | 224 | A spectral triple defines a Fredholm module | For every a\in \mathcal {A} and for every 0 < s\le p, we have
a|D|^{-s}\in \mathcal {L}_{\frac{p}{s},\infty }, \partial (a)|D|^{-s}\in \mathcal {L}_{\frac{p}{s},\infty }, and \delta (a)|D|^{-s}\in \mathcal {L}_{\frac{p}{s},\infty }.We prove here only the third statement: that \delta (a)|D|^{-s} \in \mathcal {L}_{p/s,\i... | {
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"source_ref_id": "97ff2be... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
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fcf7a576b55147934782b1ff4cedae0d70b4886f | subsection | 44 | 224 | A spectral triple defines a Fredholm module | Then (\mathcal {A},H,D_0) is spectral triple, and:(\mathcal {A},H,D_0) is p-dimensional if and only if (\mathcal {A},H,D) is p-dimensional;
Let \delta _0 denote the bounded extension of [|D_0|,T], and define \mathrm {dom}_\infty (\delta _0) identically to \mathrm {dom}_\infty (\delta ) with D_0 in place of D. Then we... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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218d53d2cc46e4091dd05cd675cb31a1c115cd87 | subsection | 45 | 224 | A spectral triple defines a Fredholm module | Similarly, let \delta _1(T) denote the commutator of the bounded extension of |D_0|-|D| with T, \delta _1(T) := [\frac{1}{|D_0|+|D|},T].Then we have the following identity on H_\infty :[D_0,a] = \partial _1(a)+\partial (a).Since \partial (a) and \partial _1(a) are bounded, it follows that [D_0,a] extends to a bounded l... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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f8b754ee243d007222f0776ef81ce05cb43f9daa | subsection | 46 | 224 | A spectral triple defines a Fredholm module | So if T \in \mathrm {dom}_\infty (\delta ) we can compute the kth iterated commutator of T with |D_0| as:[|D_0|,[|D_0|,[\cdots ,[|D_0|,T]\cdots ]]] &= (\delta +\delta _1)^k(T)\\
&= \sum _{j=0}^k \binom{k}{j}\delta _1^{k-j}(\delta ^{j}(T))Thus the kth iterated commutator of |D_0| and T has bounded extension, so T \in \m... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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5399c8959d941838e452fb094d59b81b41390c3a | subsection | 47 | 224 | A spectral triple defines a Fredholm module | From (REF ), we have that\delta _0^k(a)(D_0+i\lambda )^{-p-1} &= \delta _0^k(a)(D+i\lambda )^{-p-1}\left(\frac{D+i\lambda }{D_0+i\lambda }\right)^{p+1}\\
&= \left(\sum _{l=0}^k \binom{k}{l} \delta _1^{k-l}(\delta ^l(a))(D+i\lambda )^{-p-1}\right)\left(\frac{D+i\lambda }{D_0+i\lambda }\right)^{p+1}.However since |D_0|-|... | {
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"raw": "Carey A., Phillips J., Rennie A., Sukochev F. The Hochschild class of the Chern character for semifinite spectral triples. J. Funct. Anal. 2... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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3e9cfeeb1e91343fa58907df87e905c74dddc110 | subsection | 48 | 224 | A spectral triple defines a Fredholm module | As an equality of operators on H_\infty , we have:[F,a] &= [D_0|D_0|^{-1},a]\\
&= [D_0,a]|D_0|^{-1}+D_0[|D_0|^{-1},a].Using (REF ),[F,a] = [D_0,a]|D_0|^{-1}-F[|D_0|,a]|D_0|^{-1}.Since the spectral triple (\mathcal {A},H,D_0) satisfies Hypothesis REF and has a spectral gap at 0,
we may apply Lemma REF with s = p to conc... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
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1c4e24e83468dc9f4900006f81487cea4c69a7ae | subsection | 49 | 224 | Restatement of Hypothesis 1.2.1 | In this section, we introduce the operator \Lambda , formally defined by:\Lambda (T) = (1+D^2)^{-\frac{1}{2}}[D^2,T].Strictly speaking, \Lambda (T) will be defined to be the bounded extension of the above operator. What is here denoted \Lambda appeared in the unital settings of
(there denoted L), (there denoted L_1) a... | {
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"Fedor Sukochev",
"Dmitriy Zanin"
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83d61a53743bef8674269ab022f9c65e88dee720 | subsection | 50 | 224 | Restatement of Hypothesis 1.2.1 | We
provide a full proof here since to the best of our knowledge no published proof exists in the non-unital setting.The easiest direction to establish is that \mathrm {dom}_\infty (\delta _0)\subseteq \mathrm {dom}_\infty (\Lambda ), as the following Lemma shows:Lemma 3.2.3
We have \mathrm {dom}_\infty (\delta _0) \su... | {
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"Fedor Sukochev",
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a176e12c9480328fd958fb0b10ac57b2483e468d | subsection | 51 | 224 | Restatement of Hypothesis 1.2.1 | Then for all \xi \in H_\infty we have:[|D_0|,T]\xi = \frac{1}{2}\Lambda (T)\xi + \frac{1}{\pi }\int _0^\infty \lambda ^{1/2} \frac{D_0^2}{(\lambda +D_0^2)^2}\Lambda ^2(T)\frac{1}{\lambda +D_0^2}\xi \,d\lambda .The integral above may be understood as a weak operator topology integral.This is essentially a combination of... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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70c7c1705d62985b17bf78bfbab6fe816bf8d63f | subsection | 52 | 224 | Restatement of Hypothesis 1.2.1 | Multiplying (REF ) by (1+D^2)\xi , we get:(1+D^2)^{1/2}\xi = \frac{1}{\pi }\int _0^\infty \frac{1+D^2}{1+\lambda +D^2}\xi \frac{d\lambda }{\lambda ^{1/2}}.The above is a convergent Bochner integral in H, since\left\Vert \frac{1+D^2}{1+\lambda +D^2}\xi \right\Vert _H \le \frac{1}{1+\lambda }\Vert (1+D^2)\xi \Vert _H.Now... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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5a97aa712b74fb2df530d12d84c538daa2ae4a2f | subsection | 53 | 224 | Restatement of Hypothesis 1.2.1 | This will allow
us to relate \mathrm {dom}_\infty (\delta _0) to \mathrm {dom}_\infty (\Lambda ).
We need to take care to ensure that the relevant version of a Fubini's theorem applies.Lemma 3.2.5
For all m\ge 1, and T \in \mathrm {dom}_\infty (\Lambda ). Then for all \xi \in H_\infty the mth iterated commutator of |D... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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b5148cd6317b0ab49f2677b668a3ad7691f66e33 | subsection | 54 | 224 | Restatement of Hypothesis 1.2.1 | Hence,\delta _0^m = \frac{1}{2^m}\sum _{k=0}^m \binom{m}{k} \left(\frac{2}{\pi }\right)^k\Theta ^k\circ \Lambda ^{m-k}.By the Fubini theorem for Hilbert space valued functions (see ), for all \xi \in H_\infty we have:\Theta ^k(T)\xi = \int _{[0,\infty )^k} \prod _{l=1}^k \frac{\lambda _l^{1/2}D_0^2}{(\lambda _l+D_0^2)^... | {
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c2cfbe429b27a34228fc2871433268fb895b0ea1 | subsection | 55 | 224 | Restatement of Hypothesis 1.2.1 | (\mathcal {A},H,D) is p-dimensional, i.e., for every a\in \mathcal {A},
a(D+i)^{-p}\in \mathcal {L}_{1,\infty },\quad \partial (a)(D+i)^{-p}\in \mathcal {L}_{1,\infty }.
For every a\in \mathcal {A} and for all k\ge 0, we have
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"math.OA"
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d4c50fb2d67dfdc0227fddfef0c2a0c56813d53a | subsection | 56 | 224 | Restatement of Hypothesis 1.2.1 | The proof of the second assertion is similar.By the spectral theorem,\left\Vert \prod _{l=1}^k \frac{\lambda _l^{1/2}(1+D^2)}{(1+\lambda _l+D^2)^2}\right\Vert _\infty &\le \prod _{l=1}^k\left\Vert \frac{\lambda _l^{1/2}(1+D^2)}{(1+\lambda _l+D^2)^2}\right\Vert _\infty \\
&\le \prod _{l=1}^k \sup _{t_l\ge 1} \frac{\lamb... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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6411e0161c6c2757bf09bfb5625859a11de1eb60 | subsection | 57 | 224 | Restatement of Hypothesis 1.2.1 | Hence \Vert \delta _0^m(a)(D+i\lambda )^{-p-1}\Vert _1 = O(\lambda ^{-1}).Now using the fact that the operator \left(\frac{D+i\lambda }{D_0+i\lambda }\right)^{p+1} has bounded extension, and\left\Vert \left(\frac{D+i\lambda }{D_0+i\lambda }\right)^{p+1}\right\Vert _\infty &\le \sup _{t \in \mathbb {R}} \left(\frac{t^2+... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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abbc18a30acfe94d2babf9a45b19a0bad2afb918 | subsection | 58 | 224 | Restatement of Hypothesis 1.2.1 | For T \in \mathrm {dom}_\infty (\delta ), we define \alpha (T) and \beta (T) by:\alpha (T) := \frac{|D|}{(D^2+1)^{1/2}}\delta (T),\\
\beta (T) := \frac{1}{(D^2+1)^{1/2}}\delta ^2(T).We can express \Lambda in terms of \alpha and \beta , by applying the Leibniz rule as follows:\Lambda (T) &= (1+D^2)^{-1/2}[|D|^2,T]\\
&= ... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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2f29efb3b803c7c4d2840a4fca61c40916159c86 | subsection | 59 | 224 | Example: Noncommutative Euclidean space | We now discuss the most heavily studied example of a non-unital spectral triple: noncommutative Euclidean space. Subsection REF covers
the definitions of noncommutative Euclidean spaces and their associated spectral triples. Subsection REF is devoted to the proof that
these spectral triples satisfy Hypothesis REF .Nonc... | {
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"Fedor Sukochev",
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f140bdbc48d64a00ad0bcaf96d3c76ebf1c6ac24 | subsection | 60 | 224 | Definitions for Noncommutative Euclidean spaces | Our approach to noncommutative Euclidean space is to proceed from the Weyl commutation relations, in line with and . An alternative
approach is to use the Moyal product, as in and . We caution the reader that the approach considered here is the "Fourier dual" of the approach in .
We briefly cite the required facts need... | {
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842eacc405d183f77ef0bed17720b6449c5ef004 | subsection | 61 | 224 | Definitions for Noncommutative Euclidean spaces | This is proved in , where a spatial isomorphism is constructed.Theorem 3.3.3
If \det (\theta ) \ne 0, then there is a spatial isomorphismL_\infty (\mathbb {R}^p_\theta )\cong \mathcal {L}_\infty (L_2(\mathbb {R}^{p/2})).We now focus exclusively on the case where \det (\theta ) \ne 0.Definition 3.3.4 The semifinite tra... | {
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f8f8cb02257379842a21944e23686645dcb45e07 | subsection | 62 | 224 | Definitions for Noncommutative Euclidean spaces | For r \in [1,\infty ), the space L_r(\mathbb {R}^p_\theta ) is defined by:L_r(\mathbb {R}^p_\theta ) := \lbrace x \in L_\infty (\mathbb {R}^p_\theta )\;:\;\tau _\theta (|x|^r) < \infty \rbrace .The space L_{r}(\mathbb {R}^p_\theta ) is equipped with the norm \Vert x\Vert _{L_r} = \tau _\theta (|x|^r)^{1/r}.Note that L_... | {
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"Fedor Sukochev",
"Dmitriy Zanin"
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2dedc47fda6de61b586042d144edd2e22798081e | subsection | 63 | 224 | Definitions for Noncommutative Euclidean spaces | If x \in L_\infty (\mathbb {R}^p_\theta ), and the operator [D_k^\theta ,x] has bounded extension, then its extension is an element
of L_\infty (\mathbb {R}^d_\theta ).Definition 3.3.7 If x \in L_\infty (\mathbb {R}^d_\theta ) is such that [D_k^\theta ,x] has bounded extension, then we denote \partial _kx for the exten... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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3ac6d2ee7b8e312c4318b1be4cfb54b5af53ee3f | subsection | 64 | 224 | Definitions for Noncommutative Euclidean spaces | The space W^{\infty ,1}(\mathbb {R}^p_\theta )
is important because it forms a part of our spectral triple for noncommutative Euclidean space.The remainder of this section is devoted to showing that the triple(1_{2^{p/2}}\otimes W^{\infty ,1}(\mathbb {R}^p_\theta ),L_2(\mathbb {R}^{p},\mathbb {C}^{2^{p/2}}),D^\theta )i... | {
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"Fedor Sukochev",
"Dmitriy Zanin"
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83e3305a6da5920917c43d1d826955597cec1e29 | subsection | 65 | 224 | Verification of Hypothesis | Now we prove that the triple (REF ) is a spectral triple satisfying Hypothesis REF . In fact
it is easier to use Hypothesis REF .Our main reference for this section is . As in that reference, the spaces \ell _1(L_\infty ) and \ell _{1,\infty }(L_\infty ) are defined as follows:
Let K = [0,1]^p be the unit p-cube. Then ... | {
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"Fedor Sukochev",
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e336863a3523e424f190f0d2d4bf9d8c75241e13 | subsection | 66 | 224 | Verification of Hypothesis | Then there exist constants c_p > 0 and c_p^{\prime } > 0 such that for all x \in W^{p,1}(\mathbb {R}^p_\theta ) we have:(1\otimes x)(D^\theta +i\lambda )^{-p-1} \in \mathcal {L}_1 and
\Vert (1\otimes x)(D^\theta +i\lambda )^{-p-1}\Vert _1 \le c_p\frac{\Vert x\Vert _{W^{p,1}}}{\lambda },
(1\otimes x)(D^\theta +i)^{-p}... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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d713faf51d95b4f2770b4f445b1188ef82f89dac | subsection | 67 | 224 | Verification of Hypothesis | Since (D^\theta )^2 = 1\otimes \sum _{k=1}^p (D_k^\theta )^2, we have\Lambda (1\otimes x) = (1+(D^\theta )^2)^{-\frac{1}{2}}\cdot (\sum _{k=1}^p1\otimes [(D_k^\theta )^2,x]).Applying the Leibniz rule,[(D_k^\theta )^2,x] &= [D_k^\theta ,x]D_k^\theta +D_k^\theta [D_k^\theta ,x]\\
&= 2D_k^\theta [D_k^\theta ,x]-[D_k^\thet... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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e27d4cbe655fca8651f6d95a73c7a1da92083418 | subsection | 68 | 224 | Verification of Hypothesis | First we prove that we indeed have a spectral triple.By the definition of W^{\infty ,1}(\mathbb {R}^p), if x \in W^{\infty ,1}(\mathbb {R}^p) then [D^\theta ,1\otimes x]
has bounded extension, and therefore 1\otimes x:\mathrm {dom}(D^\theta )\rightarrow \mathrm {dom}(D^\theta ).If x \in W^{\infty ,1}(\mathbb {R}^p_\the... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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8866ff8be342d53766a4f9e536fff6897f5af4f8 | subsection | 69 | 224 | Verification of Hypothesis | Applying an identical argument to \partial (1\otimes x) yields \partial (1\otimes x) \in \mathrm {dom}_\infty (\Lambda ), and so (1\otimes W^{\infty ,1}(\mathbb {R}^p_\theta ),L_2(\mathbb {R}^d,\mathbb {C}^{2^{p/2}}),D^\theta ) is \Lambda -smooth.We now show that Hypothesis REF .(REF ) holds.
Let x \in W^{\infty ,1}(\m... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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3f1a30a2f6daed4efb3763e1eee9a582f6f8f861 | subsection | 70 | 224 | Verification of Hypothesis | Using the \mathcal {L}_1-norm triangle inequality,\Vert \partial (\Lambda ^m(1\otimes x))(D^\theta +i\lambda )^{-p-1}\Vert _1 \le C_p\sum _{k=1}^p \Vert \Lambda ^m(1\otimes \partial _k x)(D^\theta +i\lambda )^{-p-1}\Vert _1.By assumption, each \partial _k x is in W^{\infty ,1}(\mathbb {R}^p_\theta ), and so by Lemma RE... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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5cc3a8b38601259e4cdb7338420537bb2269d066 | subsection | 71 | 224 | Example: Riemannian manifolds | The authors wish to thank Professor Yuri Kordyukov for significant contributions to this section, including providing many of the proofs. | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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"math.OA"
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dbfdaf81b228a62f1414bc9992f26e9e2e3bc9d7 | subsection | 72 | 224 | Basic notions about manifolds | We briefly recall the relevant definitions for Riemannian manifolds. The material in this subsection is standard,
and may be found in for example or .
Let X be a second countable p-dimensional complete smooth Riemannian manifold with metric tensor g.
Recall that g defines a canonical measure \nu _g on X. The notation L... | {
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"Fedor Sukochev",
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f9b6568a6cbb652cf0d00c4332064e032f131478 | subsection | 73 | 224 | Basic notions about manifolds | The use of the Hodge-Dirac operator to define spectral triples for arbitrary Riemannian manifolds has previously been studied in , and for related work see .To the best of our knowledge, the main results of this paper, Theorems REF , REF and REF are new in the setting of the Hodge-Dirac operator on arbitrary complete m... | {
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"raw": "Rennie, A. Summability for nonunital spectral triples. K-Theory 31 (2004), no. 1, 71-–100.",
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9f9e7d43d993e1dc75d3386d785fb3b7237b22a3 | subsection | 74 | 224 | Proof of Theorem | The proof proceeds by showing the required Cwikel-type estimates for the case of a torus: X = \mathbb {T}^p with the flat metric. We then
deduce the general case by an argument involving local coordinates.We define the p-torus as \mathbb {T}^p := \mathbb {R}^p/\mathbb {Z}^p. The space \mathbb {T}^p is a smooth p-dimens... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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b281040d87b3eceed36094c5feaf922fe2f196d9 | subsection | 75 | 224 | Proof of Theorem | Hence,
the L_2-norms corresponding to \nu _g and \nu _{g_0} are equivalent.Sobolev spaces on \mathbb {T}^p – and more generally on a compact Riemannian manifold – are defined following :Definition 3.4.4 Let g be a metric on \mathbb {T}^p, and for j=1,\ldots ,p we let \frac{\partial }{\partial x_j} denote the differenti... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00209-004-0718-0",
"end": 379,
"openalex_id": "https://openalex.org/W2031482444",
"raw": "Lawson B., Michelsohn M., Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989.",
"source_... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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8880ece5fc1e65597e51f0ad34cc22589902718c | subsection | 76 | 224 | Proof of Theorem | Then the operatorD_0(D_g+i)^{-1}defined initially on \Omega (\mathbb {T}^p) has bounded extension to L_2\Omega (\mathbb {T}^p,g) (or equivalently L_2\Omega (\mathbb {T}^p,g_0)).By the definition of the Sobolev norm \Vert \cdot \Vert _{H^1\Omega (\mathbb {T}^p,g)}, for all v \in H^1\Omega (\mathbb {T}^p,g) we have:\Vert... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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a4a4fcfddf6fd52cad3d1a80dcd5f1ff733bc7d5 | subsection | 77 | 224 | Proof of Theorem | For \lambda \rightarrow \infty , we have
\Vert (D_g+i\lambda )^{-1}\Vert _{\mathcal {L}_{p+1}(L_2\Omega (\mathbb {T}^p,g))} = O(\lambda ^{-\frac{1}{p+1}}).Part (REF ) follows immediately from Lemma REF and Lemma REF .REF .Now we prove (REF ). First we compute (D_0+i\lambda )(D_g+i\lambda )^{-1} (working on the dense d... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1023/b:kthe.0000021311.27770.e8",
"end": 1103,
"openalex_id": "https://openalex.org/W2024484826",
"raw": "Rennie, A. Summability for nonunital spectral triples. K-Theory 31 (2004), no. 1, 71-–100.",
"source_ref_id": "de6f97c99706b9... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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910d25594b193f5730866ecc51afce2e55a5e3b6 | subsection | 78 | 224 | Proof of Theorem | By identifying the edges of [0,1]^p, we may view h as a continuous function h:U\rightarrow \mathbb {T}^p.We define three smooth “cut-off" functions \phi _1,\phi _2,\phi _3 compactly supported in h(U), defined so that for each j = 1,2,3 we have 0\le \phi _j \le 1,
andfor all x \in U, \phi _1(h(x))f(x) = f(x),
we have ... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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aed5f7c9281cf2eadd668b43812011fa7bd21b84 | subsection | 79 | 224 | Proof of Theorem | \end{array}\right.}By construction, V induces an isometry from L_2\Omega (\mathrm {supp}(\psi _2),g)\rightarrow L_2\Omega (\mathrm {supp}(\phi _2),g_1), andVV^* &= M_{\chi _{\mathrm {supp}{\phi _2}}},\\
V^*V &= M_{\chi _{\mathrm {supp}{\psi _2}}}.We also have that if j = 1,2 thenM_{\phi _j}V = VM_{\psi _{j}}.We use the... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.006... | |
7015a450e8b720a53296ea1daf58f2404685059b | subsection | 80 | 224 | Proof of Theorem | Hence (working on \Omega _c(X)):(D_g+i)M_{\psi _2}V^*PV &= V^*(D_{g_1}+i)M_{\phi _2}PV\\
&= V^*(D_{g_1}+i)M_{\phi _1}(D_{g_1}+i)^{-1}M_{\phi _1}V\\
&= V^*([D_{g_1},M_{\phi _1}](D_{g_1}+i)^{-1}M_{\phi _1}+M_{\phi _1}^2)V.Now recalling that D_{g_1} is a local operator, we have [D_{g_1},M_{\phi _1}] = [D_{g_1},M_{\phi _1}... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.0243... | |
be37fd7fbab93baed90fc8ac2e6710a4d3576b0a | subsection | 81 | 224 | Proof of Theorem | We
also use the same cut-off functions \phi _1, \phi _2 and \phi _3, and \psi _1,\psi _2,\psi _3.In place of the operators P and Q, we introduce P_{\lambda } and Q_{\lambda } given by:P_{\lambda } &= M_{\phi _1}(D_{g_1}+i\lambda )^{-1}M_{\phi _1}\\
Q_{\lambda } &= M_{\phi _2}(D_{g_1}+i\lambda )^{-1}M_{\phi _2}.Followin... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.02... | |
734dd654e0392fb8160206d188b533d6723cedaf | subsection | 82 | 224 | Proof of Theorem | Then,M_f(D_g+i)^{-j-1} &= M_{\phi }M_f(D_g+i)^{-1}\cdot (D_g+i)^{-j}\\
&= -M_{\phi }[(D_g+i)^{-1},M_f](D_g+i)^{-j}+M_{\phi }(D_g+i)^{-1}M_f(D_g+i)^{-j}\\
&= M_{\phi }(D_g+i)^{-1}[D,M_f](D_g+i)^{-j-1}+M_{\phi }(D_g+i)^{-1}M_f(D_g+i)^{-j}\\
&= M_{\phi }(D_g+i)^{-1}[D,M_f]M_{\phi }(D_g+i)^{-j-1}+M_{\phi }(D_g+i)^{-1}M_f(D... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03863096609711647,
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0.0004157558723818511,
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0.00... | |
b713004dd181ba3ea779279b689db9ea3c9f83df | subsection | 83 | 224 | Proof of Theorem | Then,M_f(D_g+i\lambda )^{-j-1} &= M_{\phi }\cdot M_f(D_g+i\lambda )^{-1}(D_g+i\lambda )^{-j}\\
&= -M_{\phi }[(D_g+i\lambda )^{-1},M_f](D_g+i\lambda )^{-j}\\
&\quad +M_{\phi }(D_g+i\lambda )^{-1}M_f(D_g+i\lambda )^{-j}\\
&= M_{\phi }(D_g+i\lambda )^{-1}[D,M_f]M_{\phi }(D_g+i\lambda )^{-j-1}\\
&\quad +M_{\phi }(D_g+i\lam... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.... | |
18e1d0ea9b99170d9d37442878ba6e2a5521c601 | subsection | 84 | 224 | Proof of Theorem | By Lemma REF , the spectral triple (\mathcal {A},H,D_0) satisfies Hypothesis REF .
Since |D_0| \ge 1, we have that \Vert |D_0|^{-1}\Vert _\infty \le 1.Let us establish Condition REF .(REF ).
We haveB^pA=|D_0|^{-p}a^2.Since (\mathcal {A},H,D) is p-dimensional, we have that |D_0|^{-p}a \in \mathcal {L}_{1,\infty }, and s... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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-0... | |
3051725bb7cf4444ac26640033636d5738e5d883 | subsection | 85 | 224 | Proof of Theorem | Using (REF ), we have:[B,A^{\frac{1}{2}}]=[|D_0|^{-1},a]=-|D_0|^{-1}\delta _0(a)|D_0|^{-1}.By Theorem 9 in , we have|D_0|^{-1}\delta _0(a)|D_0|^{-1}\prec \prec \delta _0(a)|D_0|^{-2}.By Lemma REF , we have\delta _0(a)|D_0|^{-2}\in \mathcal {L}_{\frac{p}{2},\infty }.Since the norm in the space \mathcal {L}_{\frac{p}{2},... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/pspum/087/01428",
"end": 183,
"openalex_id": "https://openalex.org/W2278675715",
"raw": "Sukochev F. On a conjecture of A. Bikchentaev. Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fr... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06492207944393158,
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c173893a0698a8ac4530ad497fc6fbb44f7a5ce1 | subsection | 86 | 224 | Proof of Theorem | Therefore, for all z \in \mathbb {C} with \Re (z) > 0:a^{2z}a_0^j=a^{2z}\chi _{\lbrace 1\rbrace }(a)a_0^j=1^{2z}a_0^j=a_0^j.Recall that \Omega (c) = \sum _{j=0}^m \Gamma a_0^j\partial (a_1^j)\cdots \partial (a_p^j). Since a commutes with \Gamma , we have for all \Re (z) > 0,a^{2z}\Omega (c) = \Omega (c).Let A=a^2 and B... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.09540902823209763,
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... | |
1349f0e2b4add57fbad4fad4b50689d1141ba51b | subsection | 87 | 224 | Proof of Theorem | It has just been demonstrated thatz\rightarrow {\rm Tr}(\Omega (c)V^z)admits an analytic continuation to the set \lbrace \Re (z)>1-\frac{1}{p}\rbrace \setminus \lbrace 1\rbrace , with a simple pole at z=1 and the corresponding residue being \mathrm {Ch}(c).By Theorem REF , we therefore have\varphi (\Omega (c)V)=\mathrm... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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909b35f9b827ee1fa651c51c586c3e19467ddccf | subsection | 88 | 224 | Proof of Theorem | For \varepsilon > 0, set f_{\varepsilon }(t)=f(t)e^{-\varepsilon ^2t^2},
t\in \mathbb {R}. Then,f_\varepsilon ^{\prime }(t) &= (f^{\prime }(t)-2\varepsilon ^2tf(t))e^{-\varepsilon ^2t^2},\\
f_{\varepsilon }^{\prime \prime }(t) &= (f^{\prime \prime }(t)-4\varepsilon ^2tf^{\prime }(t)+(4t\varepsilon ^4-2\varepsilon ^2)f(... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1515/crelle.2009.006",
"end": 1012,
"openalex_id": "https://openalex.org/W2964079016",
"raw": "Potapov D., Sukochev F. Unbounded Fredholm modules and double operator integrals. J. Reine Angew. Math. 626 (2009), 159–185.",
"source_r... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05006767064332962,
0.04817602410912514,
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0.00867259968072176,
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... | |
6dac2439e244093513b122182fac349d3db3d7a5 | subsection | 89 | 224 | Proof of Theorem | We have\Big \Vert \chi _{[0,N]}(|D|)[f_{\epsilon }(|D|),x]\chi _{[0,N]}(|D|)\Big \Vert _{r,\infty }\le \frac{c_{abs}r}{r-1}\Vert \delta (x)\Vert _{r,\infty }.Since as \varepsilon \rightarrow 0, we have that f_{\varepsilon } converges uniformly to f on the set [0,N], we have that:
As \epsilon \rightarrow 0, we have\chi ... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03475264087319374,
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0.00... | |
17bb02f08d0b92fae9959e08dd1946dbdaae970d | subsection | 90 | 224 | Proof of Theorem | Using (REF ), we have on H_\infty :B^{q-2}[B,A] &= |D_0|^{\frac{2-q}{3}}[|D_0|^{-\frac{1}{3}},a^4]\\
&= -|D_0|^{\frac{1-q}{3}}[|D_0|^{\frac{1}{3}},a^4]|D_0|^{-\frac{1}{3}}\\
&= -[|D_0|^{\frac{1}{3}},|D_0|^{\frac{1-q}{3}}a^4|D_0|^{-\frac{1}{3}}].By Lemma REF , we have\Vert B^{q-2}[B,A]\Vert _1\le c_{abs}\Vert [|D_0|,|D_... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05191536992788315,
0.031771838665008545,
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0.021410129964351654,
0.04629959911108017,
0.... | |
c3e8a35648eab0e2330e6f8620148d89adc23f58 | subsection | 91 | 224 | Proof of Theorem | We have:[B,A^{\frac{1}{2}}] &= [|D_0|^{-\frac{1}{3}},a^2]\\
&= -|D_0|^{-\frac{1}{3}}[|D_0|^{\frac{1}{3}},a^2]|D_0|^{-\frac{1}{3}}\\
&= -[|D_0|^{\frac{1}{3}},|D_0|^{-\frac{1}{3}}a^2|D_0|^{-\frac{1}{3}}].By Lemma REF , we have\Vert [B,A^{\frac{1}{2}}]\Vert _{\frac{3}{2},\infty }\le c_{\mathrm {abs}}\Vert [|D_0|,|D_0|^{-\... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.053167764097452164,
... | |
a1b75331cc66897e3b3d77a9ce8c694cb85f613c | subsection | 92 | 224 | Proof of Theorem | We haveB^{q-2}[B,A] &= |D_0|^{\frac{2-q}{3}}[|D_0|^{-\frac{1}{3}},a^4]\\
&= -|D_0|^{\frac{1-q}{3}}[|D_0|^{\frac{1}{3}},a^4]|D_0|^{-\frac{1}{3}}\\
&= -[|D_0|^{\frac{1}{3}},|D_0|^{\frac{1-q}{3}}a^4|D_0|^{-\frac{1}{3}}].By Lemma REF , we have\Vert B^{q-2}[B,A]\Vert _1\le c_{abs}\Vert [|D_0|,|D_0|^{\frac{1-q}{3}}a^4|D_0|^{... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04373931884765625,
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0.017443839460611343,
-0.0... | |
8a566afef0b581d98377556fbabb34811841ec54 | subsection | 93 | 224 | Proof of Theorem | We may compute on H_\infty :[B,A^{\frac{1}{2}}] &= [|D_0|^{-\frac{1}{3}},a^2]\\
&= -|D_0|^{-\frac{1}{3}}[|D_0|^{\frac{1}{3}},a^2]|D_0|^{-\frac{1}{3}}\\
&= -[|D_0|^{\frac{1}{3}},|D_0|^{-\frac{1}{3}}a^2|D_0|^{-\frac{1}{3}}].Therefore using Lemma REF , we have\Vert [B,A^{\frac{1}{2}}]\Vert _{3,\infty }\le c_{abs}\Vert [|D... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.04732082784175873,
-0.0... | |
c75c010f9563f23bc1873da99c046ea7c2fa5434 | subsection | 94 | 224 | Proof of Theorem | Since c is local, we may choose 0 \le a \in \mathcal {A} such that aa_0^j = a_0^j for all j.By exactly the same argument as in the p>2 case, we can show that for all \Re (z) > 0:a^{4z}\Omega (c)=\Omega (c).We let A=a^4 and B=(1+D^2)^{-\frac{1}{6}} as in Lemmas REF and REF .We have that:\mathrm {Tr}(\Omega (c)B^zA^z) &=... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.02935131825506687,
0.018352201208472252,
0.... | |
c0824903a2618752635ff3aeb45e77a188c703bf | subsection | 95 | 224 | Proof of Theorem | So by rescaling the argument,
we can equivalently say that the functionz\mapsto \mathrm {Tr}(\Omega (c)V_1^z)has analytic continuation to the set \lbrace z\;:\;\Re (z) > 2/3\rbrace \setminus \lbrace 1\rbrace with a simple
pole at 1 with corresponding residue \mathrm {Ch}(c).Thus by Theorem REF , for any continuous norm... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
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0.01916341669857502,
0... | |
790a43e1a302b4c09abb9777ff898a81d6dd4429 | subsection | 96 | 224 | Proof of Theorem | Thus,\varphi (\Omega (c)B^6A^6) = \mathrm {Ch}(c).So \varphi (a^{12}\Omega (c)(1+D^2)^{-1}) = \mathrm {Ch}(c). Since a^{12}\Omega (c) = \Omega (c), this completes the proof for the case p=2. | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.02464938350021839,
-0.003865298116579652,
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-0... | |
c83fb407085f718b051add975868c9631923dd4b | subsection | 97 | 224 | Asymptotic of the heat trace | In this chapter we complete the proof of Theorem REF . This will require some delicate computations
exploiting Hochschild homology.For the remainder of this chapter, we assume that (\mathcal {A},H,D) is a spectral triple satisfying Hypothesis REF .
Furthermore we will need the following auxiliary assumption:Hypothesis ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.jfa.2003.11.016",
"end": 1240,
"openalex_id": "https://openalex.org/W2034870188",
"raw": "Carey A., Phillips J., Rennie A., Sukochev F. The Hochschild class of the Chern character for semifinite spectral triples. J. Funct. Anal. 2... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05091509968042374,
0.008890300989151001,
-0.009500794112682343,
-0.030646737664937973,
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0.0141405388712883,
0.014941810630261898,
-0.0... | |
ce7743773c732439a84449aaf886ffb1cf0bdffb | subsection | 98 | 224 | Combinatorial expression for | Combinatorial expressionWe begin this section with the introduction of a new set of multilinear maps:Definition 4.1.1
Let {A} \subseteq \lbrace 1,2,\ldots ,p\rbrace . We define the multilinear map \mathcal {W}_{A}:\mathcal {A}^{\otimes (p+1)}\rightarrow \mathcal {L}_{\infty } by:\mathcal {W}_{{A}}(a_0\otimes a_1\otime... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04287000745534897,
0.0040657855570316315,
-0.04891147464513779,
-0.01777350902557373,
0.0007952310261316597,
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0.0015256230253726244,
0.027751082554459572,
0.01934489980340004,
0.01646147295832634,
-0.025371110066771507,
-0.001874609268270433,
0.011617619544267654,
... | |
be0aa008ce45eaeaac043cdb6138d9b54d22999e | subsection | 99 | 224 | Combinatorial expression for | However first we need a Lemma which constitutes the core of the proof of Theorem REF . Most of this section is devoted to the proof of the following lemma, which is split into various parts.Lemma 4.1.2
Let (\mathcal {A},H,D) be a smooth spectral triple where D has a spectral gap at 0. For all c\in (\mathcal {A}+\mathb... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04945281893014908,
0.019826916977763176,
-0.034250658005476,
-0.04032541811466217,
0.012843996286392212,
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-0.0038692252710461617,
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0.01904849335551262,
0.013294260948896408,
-0.... | |
e7abe442c4fffd9b35b3f6cd8cb9e9a4cd3b8e7b | subsection | 100 | 224 | Combinatorial expression for | Here we have |{B}| = |{A}|-1, and\mathcal {W}_{{A}}(c)|D|^{p-|{A}|} &= \mathcal {W}_{{B}}(c^{\prime })\delta (a_p)|D|^{p-1-|{B}|}\\
&= \left(\mathcal {W}_{{B}}(c^{\prime })|D|^{p-1-|{B}|}\right)\left(|D|^{-p+1+|{B}|}\delta (a_p)|D|^{p-1-|{B}|}\right)The first factor in the right hand side has bounded extension by the i... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06553883105516434,
0.012753831222653389,
-0.025995848700404167,
-0.001800181926228106,
-0.01110620703548193,
0.025675475597381592,
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0.04247209057211876,
0.0021358090452849865,
-0.036278244107961655,
0.0041381302289664745,
0.019847769290208817,
-... | |
bc7f25789f3d6615c9b8e929b2f0047e11aeb138 | subsection | 101 | 224 | Combinatorial expression for | \end{array}\right.}We also define the multilinear map \mathcal {P}_{{A}}:\mathcal {A}^{\otimes (p+1)}\rightarrow \mathcal {L}_{\infty } by\mathcal {P}_{{A}}(a_0\otimes \cdots \otimes a_p) := \Gamma a_0\prod _{k=1}^p y_k(a_k)where for each 1\le k \le p and a \in \mathcal {A},y_k(a) := {\left\lbrace \begin{array}{ll}
\de... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.048269595950841904,
0.01711709424853325,
-0.009161154739558697,
-0.042075708508491516,
0.011518187820911407,
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0.02233460545539856,
0.012456423602998257,
0.030389707535505295,
0.006857516709715128,
-0.022182045504450798,
0.021663345396518707,
0.012670006603002548,
... | |
657ebc4eedb7288a10b40fa116fe91c014ebce91 | subsection | 102 | 224 | Combinatorial expression for | If {A} = \lbrace 1\rbrace , then\mathcal {R}_{{A}}(c) &= \Gamma a_0F\delta (a_1)\\
\mathcal {P}_{{A}}(c)F^{|{A}|} &= \Gamma a_0\delta (a_1)F.So,\left(\mathcal {R}_{{A}}(c)-(-1)^{n_{{A}}}\mathcal {P}_{{A}}(c)\cdot F^{|{A}|}\right)\cdot |D| &= \Gamma a_0(F\delta (a_1)-\delta (a_1)F)|D|\\
&= \Gamma a_0[F,\delta (a_1)]|D|\... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.08976099640130997,
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-0.00443160580471158,
-0.019160402938723564,
0.017360301688313484,
-0.02372167631983757,
-0.0... |
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