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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ff8a7d72593335f3a37ff2c40f7dc1d5c017a095 | subsection | 103 | 224 | Combinatorial expression for | For this purpose, let {B}\subset \lbrace 1,\cdots ,p-1\rbrace be defined by
{B}={A}\backslash \lbrace p\rbrace .If p \in {A}, then n_{{A}} = n_{{B}}, and if p \notin {A} then n_{{A}} = n_{{B}} + |{B}|.Now we consider separately the cases p \in {A} and p\notin {A}.First, if p \in {A}, then,\mathcal {R}_{{A}}(c) &= \math... | {
"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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... | |
ab188f97ef8bfe91b04f482d244c91a0a91c786f | subsection | 104 | 224 | Combinatorial expression for | Then,\mathcal {R}_{{A}}(c) &= \mathcal {R}_{{B}}(c^{\prime })L(a_p)\\
\mathcal {P}_{{A}}(c) &= \mathcal {P}_{{B}}(c^{\prime })L(a_p).Focusing on \mathcal {P}_{{A}}(c):\mathcal {P}_{{A}}(c)F^{|{A}|} &= \mathcal {P}_{{B}}(c^{\prime })L(a_p)F^{|{B}|}.So:(\mathcal {R}_{{A}}(c)-(-1)^{n_{{A}}}\mathcal {P}_{{A}}(c)F^{|{A}|})|... | {
"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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-... | |
ada676db68b303bc44ac012414b552b6b2d398a8 | subsection | 105 | 224 | Combinatorial expression for | For all c\in (\mathcal {A}+\mathbb {C})^{\otimes (p+1)} and for all {A}\subseteq \lbrace 1,\ldots ,p\rbrace the operator\left(\mathcal {P}_{{A}}(c)-\mathcal {W}_{{A}}(c)\cdot |D|^{p-|{A}|}\right)\cdot |D|.has bounded extension.This proof is again similar to the proofs of Lemmas REF and REF .Once more, it suffices to pr... | {
"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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... | |
417608f20a6595f892ff6468e740ab3673f6ab2a | subsection | 106 | 224 | Combinatorial expression for | Then,\mathcal {P}_{{A}}(c) &= \mathcal {P}_{{B}}(c^{\prime })\delta (a_p),\\
\mathcal {W}_{{A}}(c)|D|^{p-|{A}|} &= \mathcal {W}_{{B}}(c^{\prime })\delta (a_p)|D|^{p-1-|{B}|}.So,\mathcal {P}_{{A}}(c)|D| &= \mathcal {P}_{{B}}(c^{\prime })\delta (a_p)|D|\\
&= \mathcal {P}_{{B}}(c^{\prime })|D|-\mathcal {P}_{{B}}(c^{\prime... | {
"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... | |
b20d9ca37e20eb9765e149cdaa050a55bf675c67 | subsection | 107 | 224 | Combinatorial expression for | In this case, we have:\mathcal {P}_{{A}}(c) &= \mathcal {P}_{{B}}(c^{\prime })L(a_p)\\
\mathcal {W}_{{A}}(c) &= \mathcal {W}_{{B}}(c^{\prime })[F,a_p].Multiplying by |D|, we have\mathcal {P}_{{A}}(c)|D| = \mathcal {P}_{{B}}(c^{\prime })|D|L(a_p)-\mathcal {P}_{{B}}(c^{\prime })L(\delta (a_p)).Note that \mathcal {P}_{{B}... | {
"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.... | |
7ecc77765e51cce7c3f39dc722e5c228aafad107 | subsection | 108 | 224 | Combinatorial expression for | So by the inductive hypothesis, it follows that (\mathcal {P}_{{A}}(c)-\mathcal {W}_{{A}}(c)|D|^{p-|{A}|})|D| has bounded extension
in the case p \notin {A}.The main idea used in the proof of Lemma REF is the algebraic identity,\prod _{k=1}^p (x_k+y_k) = \sum _{{A} \subseteq \lbrace 1,\cdots ,p\rbrace } z_{{{A}}}where ... | {
"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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... | |
f8812350df0ff7d97d799f76e49d515bcdc39bac | subsection | 109 | 224 | Combinatorial expression for | For all c\in \mathcal {A}^{\otimes (p+1)}, we have\mathrm {Tr}(\Omega (c)|D|^{2-p}e^{-s^2D^2}) = \sum _{{A} \subseteq \lbrace 1,2,\ldots ,p\rbrace } (-1)^{n_{{A}}}\mathrm {Tr}(\mathcal {W}_{{A}}(c)D^{2-|{A}|}e^{-s^2D^2})+O(s^{-1})as s\rightarrow 0.As in the preceding lemmas it suffices to prove the result for an elemen... | {
"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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89becbbc41ba4131a78bb8119b8f00c2d755cbb7 | subsection | 110 | 224 | Combinatorial expression for | By the cyclicity of the trace and the fact that \mathcal {A} commutes with \Gamma , we have:\mathrm {Tr}(\Omega (c)|D|^{2-p}e^{-s^2D^2}) = \mathrm {Tr}(\Omega (c^{\prime })|D|^{2-p}e^{-s^2D^2}a_0)and for all {A} \subseteq \lbrace 1,2,\ldots ,p\rbrace :\mathrm {Tr}(\mathcal {W}_{{A}}(c)D^{2-|{A}|}e^{-s^2D^2}) = \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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-0... | |
f3f20d554e298ed6587173d4c38e812876a704a2 | subsection | 111 | 224 | Auxiliary commutator estimates | This section is a slight detour from the main task of this chapter. Here we establish bounds on the \mathcal {L}_1-norm of commutators
of the form [f(s|D|),x], where x \in \mathcal {B}, s > 0 and f is the square of a Schwartz class function on \mathbb {R}. These bounds are used everywhere in the subsequent sections of ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.jfa.2003.11.016",
"end": 555,
"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. 21... | 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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0.043538790196180344,
-0.02... | |
0f413513b3a5fb66162afaa715d0aed7a03ed20f | subsection | 112 | 224 | Auxiliary commutator estimates | Then for all x \in \mathcal {B}, we have\left\Vert [f(s|D|),x]-\frac{s}{2}\lbrace f^{\prime }(s|D|),\delta (x)\rbrace \right\Vert _1 \le \frac{1}{2}s^2\Vert \widehat{h^{\prime \prime }}\Vert _1\cdot \left(\Vert \delta ^2(x)h(s|D|)\Vert _1+\Vert h(s|D|)\delta ^2(x)\Vert _1\right)Here, \lbrace \cdot ,\cdot \rbrace denote... | {
"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... | |
946b37a0ecf2925bff4b8a57449fed06f881ae76 | subsection | 113 | 224 | Auxiliary commutator estimates | Let h be a Schwartz function on \mathbb {R} and let f=h^2. Then for every x\in \mathcal {B}, we have\left\Vert |D|^m[f(s|D|),x]\right\Vert _1 \le s\Vert \widehat{h^{\prime }}\Vert _1\left(\left\Vert |D|^mh(s|D|)\delta (x)\right\Vert _1+\left\Vert |D|^m\delta (x)h(s|D|)\right\Vert _1\right).Since f = h^2, by the Leibniz... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.044709041714668274,
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0.033661480993032455,
-0.0166... | |
32245f515965c83b6fb65d7a4ae0822dc73dd956 | subsection | 114 | 224 | Auxiliary commutator estimates | For all x\in \mathcal {B} and for all integers m>-p, we have\Vert |D|^m[e^{-s^2D^2},x]\Vert _1=O(s^{1-p-m}),\quad s\downarrow 0.Let h(t)=e^{-\frac{1}{2}t^2}, t\in \mathbb {R}. By Lemma REF , we have\Vert |D|^m[e^{-s^2D^2},x]\Vert _1 \le s\Vert \widehat{h^{\prime }}\Vert _1(\Vert |D|^me^{-\frac{1}{2}s^2D^2}\delta (x)\Ve... | {
"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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-... | |
9ec5f4e8e77d637caf38499fb4f9444a81290d86 | subsection | 115 | 224 | Auxiliary commutator estimates | Let f(t)=e^{-t^2}, t\in \mathbb {R}.for every a\in \mathcal {A}, we have
\Big \Vert [f(s|D|),a]-s\delta (a)f^{\prime }(s|D|)\Big \Vert _{\infty }=O(s^2),\quad s\downarrow 0.
for every a\in \mathcal {A}, we have
\Big \Vert [f(s|D|),a]-s\delta (a)f^{\prime }(s|D|)\Big \Vert _1=O(s^{2-p}),\quad s\downarrow 0.
for ever... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.07306341826915741,
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-0.0... | |
b1404cba9a4f0aebcb67d9ed5c66b1a41b6f36d6 | subsection | 116 | 224 | Auxiliary commutator estimates | By Lemma REF , for all a \in \mathcal {A} we have\left\Vert [f(s|D|),a]-\frac{s}{2}\lbrace f^{\prime }(s|D|),\delta (a)\rbrace \right\Vert _1 \le \frac{1}{2}s^2\Vert \hat{h^{\prime \prime }}\Vert _1(\Vert \delta ^2(a)h(s|D|)\Vert _1+\Vert h(s|D|)\delta ^2(a)\Vert _1).Using Lemma REF , we have\Vert \delta ^2(a)h(s|D|)\V... | {
"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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... | |
0a4da46052b61331229481a4be07c7fc7b044567 | subsection | 117 | 224 | Auxiliary commutator estimates | Therefore,\Vert [f^{\prime }(s|D|),\delta (a)]\Vert _1 = O(s^{1-p}),\quad s \downarrow 0.By combining (REF ) and (REF ), we obtain (REF ).Finally, to prove (REF ), we use the inequality\Vert T\Vert _{p,1}\le \Vert T\Vert _1^{\frac{1}{p}}\Vert T\Vert _{\infty }^{1-\frac{1}{p}}and write\Vert [f(s|D|),a]-s\delta (a)f^{\pr... | {
"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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-0.011... | |
b4800a6bf48b6fde68ce1112f09af8cda3dcd820 | subsection | 118 | 224 | Auxiliary commutator estimates | First, for the case when m > 2 we apply the Leibniz rule[D^{2-m},x] &= -D^{2-m}[D^{m-2},x]D^{2-m}\\
&= -\sum _{k+l=m-3} D^{k+2-m}\partial (x)D^{l+2-m}.Now using the triangle inequality:\Vert D^{m-p}[D^{2-m},x]e^{-s^2D^2}\Vert _1 \le \sum _{k+l=m-3}\Vert |D|^{k+2-p}\partial (x)|D|^{l+2-m}e^{-s^2|D|^2}\Vert _1.Applying L... | {
"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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... | |
57e4b125bcad91c97c9f6ce9d0a8aba1d53bd713 | subsection | 119 | 224 | Exploiting Hochschild homology | Recall the multilinear mapping \mathcal {W}_p from Definition REF . In this section, we prove the following:Theorem 4.3.1
Let (\mathcal {A},H,D) be a spectral triple satisfying Hypothesis REF and Hypothesis REF . For every Hochschild cycle c\in \mathcal {A}^{\otimes (p+1)} we have:\mathrm {Tr}(\Omega (c)|D|^{2-p}e^{-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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... | |
58db8684c8341533edc90a634bdde7300458b9c0 | subsection | 120 | 224 | Exploiting Hochschild homology | Then, from the computation in Appendix , we have that the Hochschild coboundary is:(b\theta _s)(a_0\otimes &\cdots \otimes a_p)\\
&= (-1)^p\mathrm {Tr}\left(\Gamma a_0\left(\prod _{k=1}^{m-2}[b_k,a_k]\right)\delta ^2(a_{m-1})\left(\prod _{k=m}^{p-1}[b_{k+1},a_k]\right)[D^{2-|{A}|}e^{-s^2D^2},a_p]\right)\\
&+ 2(-1)^{m-1... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06909225136041641,
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-... | |
62015746bb1d062d5fe2fbb84ab105dc0f5db273 | subsection | 121 | 224 | Exploiting Hochschild homology | Let {A}_1,{A}_2 \subseteq \lbrace 1,\ldots ,p\rbrace , with |{A}_1|=|{A}_2| and that the symmetric difference {A}_1\Delta {A}_2=\lbrace m-1,m\rbrace for some m.
Then for every Hochschild cycle c\in \mathcal {A}^{\otimes (p+1)}, we have\mathrm {Tr}(\mathcal {W}_{{A}_1}(c)D^{2-|{A}_1|}e^{-s^2D^2})+\mathrm {Tr}(\mathcal {... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06463310122489929,
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0e6d9ba7f550f53bc3343ec0bb8ca1cf20c86bf3 | subsection | 122 | 224 | Exploiting Hochschild homology | Indeed,\Big \Vert \Gamma a_0\left(\prod _{k=1}^{m-2}[b_k,a_k]\right)&[F,\delta (a_{m-1})]\left(\prod _{k=m}^{p-1}[b_{k+1},a_k]\right)[D^{2-|{A}_1|}e^{-s^2D^2},a_p]\Big \Vert _1\\
&\le \left\Vert D^{|{A}_1|-p}[D^{2-|{A}_1|}e^{-s^2D^2},a_p]\right\Vert _1\\
&\times \left\Vert \Gamma a_0\prod _{k=1}^{m-2}[b_k,a_k][F,\delta... | {
"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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04b8c978fe4f270c11311c9427cfb6261b57b085 | subsection | 123 | 224 | Exploiting Hochschild homology | That is:{A}_j := ({A}\backslash \lbrace n\rbrace )\cup \lbrace j+n\rbrace ,\quad 0 \le j < m-n.Then by construction:|{A}_j|=|{A}| and {A}_j\Delta {A}_{j-1} = \lbrace n+j,n+j-1\rbrace for all 1 \le j < m-n.
{A}_0={A} and m-1,m\in {A}_{m-n-1}.Hence if 1 \le j < m-n the subsets {A}_j and {A}_{j-1} satisfy the conditions ... | {
"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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0.015760697424411774,
-0.0... | |
a0be9e41157679135c3f9c10fbe5d92907553c6a | subsection | 124 | 224 | Exploiting Hochschild homology | So on H_\infty :[F,a_{p-1}][F,a_p]|D|^2 &= [F,a_{p-1}]\cdot L(a_p)\cdot |D|\\
&= [F,a_{p-1}]\cdot |D|L(a_p)-[F,a_{p-1}]\cdot [|D|,L(a_p)]\\
&= [F,a_{p-1}]|D|\cdot L(a_p)-[F,a_{p-1}]\cdot \delta (L(a_p))\\
&= L(a_{p-1})\cdot L(a_p)-[F,a_{p-1}]\cdot L(\delta (a_p)).So for c=a_0\otimes \cdots \otimes a_p \in \mathcal {A}^... | {
"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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0.03287016600370407,
-0.... | |
62ca5d7266eafc280dd00569c7271ab4110e51fc | subsection | 125 | 224 | Exploiting Hochschild homology | Recall that\mathcal {W}_{\emptyset }(c) = \Gamma a_0\prod _{k=1}^p [F,a_k].Using the fact that F anticommutes with [F,a_k] for all k, we have:\mathrm {ch}(c) &= \Gamma F[F,a_0]\prod _{k=1}^p [F,a_k]\\
&= \Gamma a_0\prod _{k=1}^p [F,a_k]-\Gamma Fa_0F\prod _{k=1}^p [F,a_k].\\
&= \mathcal {W}_{\emptyset }(c)+(-1)^{p+1}\Ga... | {
"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... | |
6df438755855e9533d3c54f1e00df47b0a887171 | subsection | 126 | 224 | Exploiting Hochschild homology | Then using Theorem REF :\mathrm {Tr}(\Omega (c)|D|^{2-p}e^{-s^2D^2}) = \sum _{{A} \subseteq \lbrace 1,\ldots ,p\rbrace }(-1)^{n_{{A}}}\mathrm {Tr}(\mathcal {W}_{{A}}(c)D^{2-|{A}|}e^{-s^2D^2})+O(s^{-1}),\quad s\downarrow 0.Applying Lemma REF to every summand with |{A}| \ge 2, and Lemma REF to the summand {A} = \emptyset... | {
"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... | |
f37f821405aa52845b71ff7e17d44c643fbce89d | subsection | 127 | 224 | Preliminary heat semigroup asymptotic | In this section, we move closer to proving Theorem REF .
We will show that if (\mathcal {A},H,D) satisfies Hypothesis REF and Hypothesis REF (in particular, D has a spectral gap at 0), then for a Hochschild cycle c \in \mathcal {A}^{\otimes (p+1)}
we have\mathrm {Tr}(\mathcal {W}_p(c)|D|^{2-p}e^{-s^2D^2}) = \frac{1}{4}... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.051778923720121384,
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0.036065224558115005... | |
40e86ecdf40d18e922c36e204ed74555605867d4 | subsection | 128 | 224 | Preliminary heat semigroup asymptotic | Then using [F,a_p] = |D|^{-1}\partial (a_p)-|D|^{-1}\delta (a_p)F, we have (on H_\infty )\mathrm {ch}(c)(1-e^{-s^2D^2}) &= \Gamma F \left(\prod _{k=0}^{p-1}[F,a_k]\right)(|D|^p|D|^{-p})[F,a_p](1-e^{-s^2D^2})\\
&= \Gamma F \left(\prod _{k=0}^{p-1}[F,a_k]\right)|D|^p\cdot |D|^{-p-1}\partial (a_p)(1-e^{-s^2D^2})\\
&\quad ... | {
"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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... | |
c0f69e5b0d325af9220dc0f9f23a0c4dfcf860bf | subsection | 129 | 224 | Preliminary heat semigroup asymptotic | By Lemma REF , we have\mathrm {Tr}(\mathrm {ch}(c)e^{-s^2D^2}) = \mathrm {Tr}(\mathrm {ch}(c))+O(s),\quad s\downarrow 0.Since the spectral triple and p both have the same parity, we may apply Lemma REF to get:2\mathrm {Tr}(\mathcal {W}_{\emptyset }(c)e^{-s^2D^2}) = \mathrm {Tr}(\mathrm {ch}(c))+O(s),\quad s\downarrow 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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-... | |
81c6d92915f313113ef2e340ae579091287b6e5e | subsection | 130 | 224 | Preliminary heat semigroup asymptotic | By the Hölder inequality in the form (REF ):|(\mathcal {H}_s+2s^2\mathcal {V}_s)&(a_0\otimes \cdots \otimes a_p)| \\
&= \left|\mathrm {Tr}\Big (\Gamma a_0\Big (\prod _{k=1}^{p-1}[F,a_k]\Big )F\cdot \Big ([e^{-s^2D^2},a_p]+2s^2\delta (a_p)|D|e^{-s^2D^2}\Big )\Big )\right|\\
&\le \Big \Vert \Gamma a_0\prod _{k=1}^{p-1}[F... | {
"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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0.0050823581404984,
-0.02182... | |
292083ec9d81fa6180717cb973c00d3423791398 | subsection | 131 | 224 | Preliminary heat semigroup asymptotic | Therefore, we have(\mathcal {H}_s+2s^2\mathcal {V}_s)(c) = O(s),\quad s\downarrow 0,Combining (REF ) and (REF ), we arrive at4s^2\mathcal {V}_s(c) = \mathrm {Tr}(\mathrm {ch}(c))+O(s),\quad s\downarrow 0,for all Hochschild cycles c\in \mathcal {A}^{\otimes (p+1)}.From the definition of \mathcal {W}_p, if a_0\otimes \cd... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.027111971750855446,
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-0.0... | |
fafd93a1aa9f0108c2779abeb6f18ec7cba4bc92 | subsection | 132 | 224 | Preliminary heat semigroup asymptotic | Dividing by 4s^2,\mathrm {Tr}(\mathcal {W}_p(c)De^{-s^2D^2}) = \frac{1}{4}s^{-2}\mathrm {Tr}(\mathrm {ch}(c))+O(s^{-1}).Since \mathrm {Ch}(c) = \frac{1}{2}\mathrm {Tr}(\mathrm {ch}(c)), this formula coincides with (REF ).We remark that (REF ) follows as a simple combination of Theorems REF and REF . | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.041506607085466385,
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... | |
25f94a43a5de49c65120c708a24fda1feb40faa6 | subsection | 133 | 224 | Heat semigroup asymptotic: the proof of the first main result | In this section, we finally complete the proof of Theorem REF . We start by removing the assumption of Hypothesis REF .The following two lemmas show that if the parity of p does not match (\mathcal {A},H,D), then the statement of (REF ) becomes trivial.Lemma 4.5.1
Let (\mathcal {A},H,D) be a spectral triple satisfying... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03526703268289566,
0.029379025101661682,
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0.0018018677365034819,
0... | |
467e755932473f75529d66806008a0e7f26458c4 | subsection | 134 | 224 | Heat semigroup asymptotic: the proof of the first main result | Thus since p+1 is even:\mathrm {ch}(c) &= \Gamma F\prod _{k=0}^p[F,a_k]\\
&= -F\Gamma \prod _{k=0}^p[F,a_k]\\
&= -F\cdot \prod _{k=0}^p[F,a_k]\Gamma .Thus,\mathrm {Tr}(\mathrm {ch}(c)) &= -\mathrm {Tr}(\Gamma F\prod _{k=0}^p[F,a_k])\\
&= -\mathrm {Tr}(\mathrm {ch}(c)).This proves the second assertion.Now, we deal with ... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03461148589849472,
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... | |
66f1db3073a05d7b06eebb34f63a13fda6be1db0 | subsection | 135 | 224 | Heat semigroup asymptotic: the proof of the first main result | Consider the multilinear mapping \theta _s:\mathcal {A}^{\otimes p}\rightarrow \mathbb {C} defined by the formula\theta _s(a_0\otimes \cdots \otimes a_{p-1}) = \mathrm {Tr}(\left(\prod _{k=0}^{p-1}\partial (a_k)\right) |D|^{2-p}e^{-s^2D^2}).The Hochschild coboundary b\theta _s is computed in Section by the formula:(b\t... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05195587873458862,
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0.022667773067951202,
-0.... | |
1956ea56eda8fcddd7a9416b2063bd472c6d83ec | subsection | 136 | 224 | Heat semigroup asymptotic: the proof of the first main result | Thus,\mathrm {Tr}(\Omega (c)|D|^{2-p}e^{-s^2D^2}) = O(s^{-1}),\quad s\downarrow 0.for every Hochschild cycle c.
This completes the proof of (REF ).The proof of (REF ) is similar to Lemma REF .(REF ).
For all a \in \mathcal {A}, we have F[F,a]=-[F,a]F. Since p+1 is odd,F\cdot \prod _{k=0}^p[F,a_k]=-\left(\prod _{k=0}^p[... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.jfa.2003.11.016",
"end": 704,
"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. 21... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.07353037595748901,
-0.024484699591994286,
-0.01981658861041069,
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0.0112660126760602,
0.02768830582499504,
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0.003951113671064377,
0.04118921607732773,
-0.00... | |
35070263aedd2a2038202eae8aacf91998a5b047 | subsection | 137 | 224 | Heat semigroup asymptotic: the proof of the first main result | If a\in \mathcal {A}, then
as \mu \downarrow 0 we have:[{\rm sgn}(D_{\mu }),\pi (a)]\rightarrow [F_0,\pi (a)]in \mathcal {L}_{p+1}.We have\mathrm {sgn}(D_{\mu }) = \begin{pmatrix}
\frac{D}{(D^2+\mu ^2)^{1/2}} & \frac{\mu }{(D^2+\mu ^2)^{1/2}} \\
\frac{\mu }{(D^2+\mu ^2)^{1/2}} & -\frac{D}{(D^2+\mu ^2)^{1/2}}
\end{pmatr... | {
"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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0.042326346039772034,
0.01235154177993536,
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0.03924418240785599,
0.012900838628411293,
-0.0... | |
46bbc0d9ed808ab09c9ce1ce318505ad2eec5704 | subsection | 138 | 224 | Heat semigroup asymptotic: the proof of the first main result | \end{pmatrix}On the other hand, we have[F_0,\pi (a)] = \begin{pmatrix}
[\mathrm {sgn}(D),a] & -aP\\ Pa & 0
\end{pmatrix}.Therefore:\Big \Vert [{\rm sgn}(D_{\mu }),\pi (a)&]-[F_0,\pi (a)]\Big \Vert _{p+1}\\
&\le \Big \Vert \Big ({\rm sgn}(D)-\frac{D}{(D^2+\mu ^2)^{\frac{1}{2}}}\Big )a\Big \Vert _{p+1}+\Big \Vert a\Big (... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.055965688079595566,
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-0.... | |
71d509a9c608a4c05fc383a9aaaf8a092fccb808 | subsection | 139 | 224 | Heat semigroup asymptotic: the proof of the first main result | Hence,a^*\Big ({\rm sgn}(D)-\frac{D}{(D^2+\mu ^2)^{\frac{1}{2}}}\Big )^2a\downarrow 0,\quad a\Big ({\rm sgn}(D)-\frac{D}{(D^2+\mu ^2)^{\frac{1}{2}}}\Big )^2a^*\downarrow 0,\quad \mu \downarrow 0,a^*\Big (P-\frac{\mu }{(D^2+\mu ^2)^{\frac{1}{2}}}\Big )^2a\downarrow 0,\quad a\Big (P-\frac{\mu }{(D^2+\mu ^2)^{\frac{1}{2}}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1315,
"openalex_id": "",
"raw": "Simon B. Trace ideals and their applications. Second edition. Mathematical Surveys and Monographs, 120. American Mathematical Society, Providence, RI, 2005.",
"source_ref_id": "d249ca4eeaedfc... | 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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6e3633af393298535051b8974575cb9e809dd55a | subsection | 140 | 224 | Heat semigroup asymptotic: the proof of the first main result | Since for each k we have that [\mathrm {sgn}(D_\mu ),\pi (a_k)] \rightarrow [F_0,\pi (a_k)]
in \mathcal {L}_{p+1}, it follows that the second summand converges to 0 in \mathcal {L}_1.Lemma 4.5.5
Let (\mathcal {A},H,D) be a spectral triple satisfying Hypothesis REF .
For every c\in \mathcal {A}^{\otimes (p+1)}, we have... | {
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} | 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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bf6a6a727d9da0ad7d0cfebc63041bebd1e2a6f2 | subsection | 141 | 224 | Heat semigroup asymptotic: the proof of the first main result | We have(\mathrm {Tr}_2\otimes \mathrm {Tr})(\Omega _{\mu }(\pi (c))(1\otimes (1+D^2)^{1-\frac{p}{2}}e^{-s^2D^2}))= \frac{p}{2}\mathrm {Ch}_{\mu }(\pi (c))s^{-2} + O(s^{-1}),\quad s\downarrow 0.For \mu >0, the spectral triple (\pi (\mathcal {A}),H_0,D_{\mu }) satisfies Hypothesis REF and the spectral gap assumption. | {
"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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9169408285c6a43dcd51ac53bcc3a9c6e7793965 | subsection | 142 | 224 | Heat semigroup asymptotic: the proof of the first main result | This allows us to apply Theorems REF and REF (or Lemmas REF and REF ) to the Hochschild cycle \pi (c)\in (\pi (\mathcal {A}))^{\otimes (p+1)}.A combination of Theorems REF and REF (if the parities of p and (\pi (\mathcal {A}),H_0,D_{\mu }) match) or one of Lemmas REF and REF (if parities of p and
(\pi (A),H_0,D_{\mu })... | {
"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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5ad9ad6839533deb7c3f71d9c941d3f27768b102 | subsection | 143 | 224 | Heat semigroup asymptotic: the proof of the first main result | \end{array}\right.}In particular,\mathcal {T}_{\emptyset }(a_0\otimes \cdots a_p) = \begin{pmatrix}
\Omega (a_0\otimes \cdots \otimes a_p) & 0\\
0 & 0
\end{pmatrix}.For c\in \mathcal {A}^{\otimes (p+1)}, we apply (REF ) to get\Omega _{\mu }(\pi (c)) = \sum _{{A}\subseteq \lbrace 1,\cdots ,p\rbrace }\mu ^{|{A}|}\mathcal... | {
"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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13e59061d30a400a80926a9b709a65579609f7c7 | subsection | 144 | 224 | Residue of the | In this chapter we complete the proofs of Theorem REF and Theorem REF .For a spectral triple (\mathcal {A},H,D) satisfying Hypothesis REF , we define the zeta function of a Hochschild cycle c \in \mathcal {A}^{\otimes (p+1)} by the formula\zeta _{c,D}(z) := \mathrm {Tr}(\Omega (c)(1+D^2)^{-z/2}),\quad \Re (z) > p+1.Ind... | {
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"source_ref_id": "044f93993... | 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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ac340d84e8983efad45f529d72447dc5b79fc924 | subsection | 145 | 224 | Analyticity of the | This section contains the proof of Theorem REF . The proof is relatively short, since we are able to use Theorem REF .Lemma 5.1.1
Let h\in L_{\infty }(0,1) and u \in L_\infty (1,\infty ). Then,the function
F(z) := \int _0^1s^{z-1}h(s)ds,\quad \Re (z)>0,
is analytic.
the function
G(z) := \int _1^{\infty }s^{z-1}u(s)e... | {
"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.... | |
07f2a27d0fa7974fc9c9fbcba7a2a6ed1affdf1b | subsection | 146 | 224 | Analyticity of the | Thus, F is holomorphic on this half-plane.To prove (REF ), we consider the functionsG_n(z) := \int _1^ns^{z-1}u(s)e^{-s}ds,\quad z\in \mathbb {C}Exactly the same argument as above shows that each G_n is entire.
For all n\ge 1, we have:|G(z)-G_n(z)| &\le \int _n^\infty s^{\Re (z)-1}|u(s)|e^{-s}\,ds\\
&\le \Vert u\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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9f8bae4972a94423120a6c81f17d14b4a37c218d | subsection | 147 | 224 | Analyticity of the | Then for all x>0, we have\int _0^{\infty }s^{z-1}e^{-s^2x^2}\,ds &= x^{-z}\int _0^\infty t^{z-1}e^{-t^2}\,dt\\
&= x^{-z}\int _0^\infty u^{\frac{z-1}{2}}e^{-u}\frac{u^{-\frac{1}{2}}}{2}\,du\\
&= \frac{x^{-z}}{2}\Gamma \left(\frac{z}{2}\right).Thus,x^{2-z} = \frac{2}{\Gamma \left(\frac{z}{2}\right)}\int _0^\infty s^{z-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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d0c997e47327ee3c5f939096e4bcffdcb3605e61 | subsection | 148 | 224 | Analyticity of the | Now applying Lemma REF with the function h_p:\Vert s^{z-1}\Omega (c)(1+D^2)^{1-\frac{p}{2}}e^{-s^2(1+D^2)}\Vert _1 &\le \left\Vert \left(\frac{1+D^2}{1+s^2D^2}\right)^{1-\frac{p}{2}}\right\Vert _{\infty }\Vert s^{z-1}\Omega (c)h_p(s|D|)\Vert _1\\
&= s^{\Re (z)+p-3}\Vert \Omega (c)h_p(s|D|)\Vert _1\\
&= O(s^{\Re (z)-3})... | {
"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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804d4d7f9a8cc94569b90710909b819e56c4030b | subsection | 149 | 224 | Analyticity of the | First we define a function h on (0,\infty ) by:h(s) := {\left\lbrace \begin{array}{ll}
e^s\mathrm {Tr}(\Omega (c)(1+D^2)^{1-\frac{p}{2}}e^{-s^2(1+D^2)}), \quad s \ge 1\\
s\mathrm {Tr}(\Omega (c)(1+D^2)^{1-\frac{p}{2}}e^{-s^2(1+D^2)})-\frac{p}{2}\mathrm {Ch}(c)s^{-1}, \quad 0 < s < 1.
\end{array}\right.}By Theorem REF ,... | {
"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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89b46665fb3e027a7349705ad5929b1e713c8b2b | subsection | 150 | 224 | Analyticity of the | Since the function \frac{1}{\Gamma \left(\frac{z}{2}\right)} is entire, and \Gamma (1) = 1, we may equivalently say that
the functionz\mapsto \zeta _{c,D}(z+p-2) - p\mathrm {Ch}(c)(z-2)^{-1},\quad \Re (z) > 2has analytic continuation to the set \Re (z) > 1.
In other words, for \Re (z) > p\zeta _{c,D}(z) - p\mathrm {Ch}... | {
"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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4bf9eb763ab06b3df1b0d442ebcabdf15a0177ff | subsection | 151 | 224 | An Integral Representation for | Integral representationIn this section, we follow the convention that for all s \in \mathbb {R} we have 0^{is}=0, so in particular we have the unusual convention that 0^{i0} = 0.
This section is devoted to the proof of Theorem REF (stated below). Theorem REF is a strengthening of (which corresponds to the special
case ... | {
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"source_ref_id": "044f93993f... | 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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718d5f60ce21284d2ab5ea7ac8d58ab30d9b4bd2 | subsection | 152 | 224 | An Integral Representation for | Since \lim _{s\rightarrow 0} \frac{\tanh (s)}{s} = 1, it is evident that f_w is continuous at 0
and that f_w is smooth in [-1,1].It suffices now to show that the function \tanh (ws)-\tanh (s) is Schwartz, since for |s| > 1 the function \frac{1}{\tanh (s)} is
smooth and bounded with all derivatives bounded. For s > 1, w... | {
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"raw": "Bratteli O., Robinson D. Operator algebras and quantum statistical mechanics. 1. C^*- and W^*-algebras, symmetry groups, decomposition of states. Texts and Monographs in Physics. Springer-Verlag, 1... | 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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df52bd88d0cdff8595915ef77e4d0c136332de98 | subsection | 153 | 224 | An Integral Representation for | \end{array}\right.}Recalling our convention stated at the start of this section, that 0^{is} = 0 for all t \ge 0, we have:t^{is} = \exp (is\log _0(t))(1-\chi _{\lbrace 0\rbrace }(t)).Let P_k be the support projection of X_k (i.e., the projection onto the orthogonal complement of the kernel of X_k). Then since P_k = 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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5f3870a2084fa8fe5a962d563c82293946add783 | subsection | 154 | 224 | An Integral Representation for | If we rewrite the definition of g_z in terms of exponentials,
then for t \ne 0 we getg_z(t) = 1-\frac{e^{\frac{z}{2}t}-e^{-\frac{z}{2}t}}{(e^{\frac{t}{2}}-e^{-\frac{t}{2}})(e^{\left(\frac{z-1}{2}\right)t}+e^{-\left(\frac{z-1}{2}\right)t})},and therefore:\phi _{1,z}(\lambda ,\mu ) = 1-\frac{\lambda ^z-\mu ^z}{(\lambda -... | {
"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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43b0baeed391ada34a1d5e5f5d93587d9033f8ac | subsection | 155 | 224 | An Integral Representation for | First,\phi _{2,z}(\lambda ,\mu ) = (\lambda ^{z-1}+\mu ^{z-1})(\lambda -\mu ),\quad \lambda ,\mu \ge 0and secondly,\phi _{3,z}(\lambda ,\mu ) = (\lambda ^{z-1}+\mu ^{z-1})(\lambda -\mu )-(\lambda ^z-\mu ^z),\quad \lambda ,\mu \ge 0.Both functions are bounded on compact subsets of [0,\infty )^2, and so in particular on ... | {
"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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... | |
70b2c078d2b92c8aad2563f1db6421149aee1091 | subsection | 156 | 224 | An Integral Representation for | If \lambda =0 or \mu =0 one has \phi _{1,z}(\lambda ,\mu )=0 and \phi _{3,z}(\lambda ,\mu )=0.Using formulae (REF ) and (REF ), we obtain that T^{X,Y}_{\phi _{2,z}}:\mathcal {L}_{\infty }\rightarrow \mathcal {L}_{\infty } andT^{X,Y}_{\phi _{2,z}}(A) = X^zA-X^{z-1}AY+XAY^{z-1}-AY^z.Since \phi _{3,z} bounded on \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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3139eab4f711880a420dbcd02d56e8a892be3b6b | subsection | 157 | 224 | An Integral Representation for | We have,B^{z} = \sum _{k=1}^n \lambda _k^z P_k.Therefore,B^zA^z-Y^z &= \sum _{k=1}^n (P_k\lambda _k^zA^z-P_kY^z)\\
&= \sum _{k=1}^n P_k((\lambda _kA)^z-Y^z).Applying Lemma REF to each term in the above sum, with X = \lambda _kA, ifV_{k,z} = (\lambda _kA)^{z-1}(\lambda _kA-Y)+(\lambda _k A-Y)Y^{z-1}then(\lambda _kA)^z-Y... | {
"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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5e2bfa76cfcf9dad0d49fbcd5439497c6f5b638d | subsection | 158 | 224 | An Integral Representation for | To do this we will select a sequence \lbrace B_n\rbrace _{n=1}^\infty with B_n\rightarrow B in the uniform norm and such that the spectrum of each B_n is finite.The following lemma shows that under certain conditions, if B_n\rightarrow B in the uniform norm, then B_n^{is} \rightarrow B^{is} in the weak operator topolog... | {
"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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3911c4f96f5f8eeb9c737ece9ccdfd76d3550b58 | subsection | 159 | 224 | An Integral Representation for | These conditions are sufficient for \phi and \psi to be operator Lipschitz (see ): i.e.,
there are constants C_{\phi } and C_{\psi } such that\Vert \phi (C_n)-\phi (C)\Vert _\infty &\le C_{\phi }\Vert C_n-C\Vert _\infty ,\\
\Vert \psi (C_n)-\psi (C)\Vert _\infty &\le C_{\psi }\Vert C_n-C\Vert _\infty .Select N> 0 such ... | {
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"source_re... | 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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31f98e3ce0b4a1af45c1f8554829185f128829b2 | subsection | 160 | 224 | An Integral Representation for | In this case we have \mathrm {Spec}(B) \subseteq [0,1] and 1 \in \mathrm {Spec}(B).
For every n\ge 1 , setB_n=\sum _{m=1}^n\frac{m}{n}\chi _{(\frac{m-1}{n},\frac{m}{n}]}(B).Recall that Y := A^{1/2}BA^{1/2}, and let Y_n := A^{1/2}B_nA^{1/2}, and let T_{n,z}(s) be defined as T_z(s)
with the occurances of B replaced with ... | {
"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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6dc43bcd3650d3c9d1163d7487c5796fbc4ab2cf | subsection | 161 | 224 | An Integral Representation for | One can also see that \sup _{s \in \mathbb {R}} \sup _{n\ge 1} \Vert T_{n,z}(s)\Vert _\infty < \infty .In other words, for every \xi ,\eta \in H, we have\langle T_{n,z}(s)\xi ,\eta \rangle \rightarrow \langle T_z(s)\xi ,\eta \rangle .Since |\langle T_{n,z}(s)\xi ,\eta \rangle | \le \sup _{n\ge 1} \Vert T_{n,z}(s)\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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1223bec628f50dbb096f68f7b307acc5ed7793d8 | subsection | 162 | 224 | Analyticity of the mapping | So far we have considered the function,g_z(t) := 1-\frac{e^{\frac{z}{2}t}-e^{-\frac{z}{2}t}}{(e^{\frac{t}{2}}-e^{-\frac{t}{2}})(e^{\left(\frac{z-1}{2}\right)t}+e^{-\left(\frac{z-1}{2}\right)t})}, t\ne 0with g_z(0) := 1-\frac{z}{2} as a Schwartz function of t with a fixed parameter z \in \mathbb {C}, with \Re (z) > 1.We... | {
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"raw": "Rudin W. Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991.",
"source_ref_id": "1ed1... | 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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edddd9ef19930b1eb2e645e429d4eafd7a5295f5 | subsection | 163 | 224 | Analyticity of the mapping | Since H^2(\mathbb {R}) is a Hilbert space, for any continuous linear functional \varpi
on H^2(\mathbb {R}) there exists h \in H^2(\mathbb {R}) such that:\varpi (g_z) = \int _{\mathbb {R}} g_z(t)h(t)\,dt + \int _{\mathbb {R}} g_z^{\prime }(t)h^{\prime }(t)\,dt + \int _{\mathbb {R}}g_z^{\prime \prime }(t)h^{\prime \prim... | {
"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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f269a4b986cb7c8d5e2683770d222883935ba457 | subsection | 164 | 224 | Analyticity of the mapping | Then G is continuous on its domain.It suffices to prove that the mappings G:\lbrace z \in \mathbb {C}\;:\; \Re (z) > 1\rbrace \rightarrow H^2(-1,1)\rbrace , G:\lbrace z \in \mathbb {C}\;:\; \Re (z) > 1\rbrace \rightarrow H^2(1,\infty )\rbrace and G:\lbrace z \in \mathbb {C}\;:\; \Re (z) > 1\rbrace \rightarrow H^2(-\inf... | {
"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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99f795c6cc1d3110d3b8ca3071989312702bc651 | subsection | 165 | 224 | Analyticity of the mapping | Then G is holomorphic on its domain.To show that G is holomorphic with values in H^2(\mathbb {R}), it suffices to show for all continuous linear functionals \varpi on H^2(\mathbb {R}) that z\mapsto \varpi (G(z)) is holomorphic.Since H^2(\mathbb {R}) is a Hilbert space, it suffices to show for all h \in H^2(\mathbb {R})... | {
"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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c526bc42874e8bb9a098f663fc445d450c114c28 | subsection | 166 | 224 | The function | The difference of two \zeta -functions admits analytic continuationAs indicated in the title, in this section we prove (under certain assumptions on A and B) that for all X \in \mathcal {L}_\infty the mappingz \mapsto \mathrm {Tr}(XB^zA^z)-\mathrm {Tr}(X(A^{\frac{1}{2}}BA^{\frac{1}{2}})^z)defined initially on \lbrace 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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7400b0c07b26c0a8862a63759c134f156c8fb9d7 | subsection | 167 | 224 | The function | Then by the Araki-Lieb-Thirring inequality:|B^{p/r}A|^r \prec \prec _{\log } B^pA^r.Now using (REF ):\Vert B^{p/r}A\Vert _{r,\infty } &= \Vert |B^{p/r}A|^r\Vert _{1,\infty }\\
&\le e\Vert B^pA^r\Vert _{1,\infty }\\
&\le e\Vert A\Vert _\infty ^{r-1}\Vert B^pA\Vert _{1,\infty }.The next lemma provides a sufficient condit... | {
"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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5276f804b181799fc3dbab47f3385b132f390dca | subsection | 168 | 224 | The function | The proof will be completed upon showing that
G is indeed the derivative of F considered as an \mathcal {I}-valued mapping.For every z and z_0 in the interior of \gamma , we have:\frac{F(z)-F(z_0)}{z-z_0}-G(z_0)=\frac{z-z_0}{2\pi i}\int _{\gamma }\frac{F(w)}{(w-z)(w-z_0)^2}\,dwAgain, this integral is an \mathcal {I}-va... | {
"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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44bb8a791d8a1c5f5998ff7e6ccff701013d9f38 | subsection | 169 | 224 | The function | The mappingz\rightarrow [B,A^z],\quad \Re (z)>1,is a holomorphic \mathcal {L}_{\frac{p}{2},\infty }-valued function.We take care to note that since p > 2, the ideal \mathcal {L}_{p/2,\infty } can be equipped with a norm generating the same topology as
that of the canonical quasi-norm (see ). Denote such a norm as \Vert... | {
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{
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"raw": "Bennett C., Sharpley R. Interpolation of operators. volume 129 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1988.",... | 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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de7628445fa8e5fc4a4900b51be7a3b34c388ea2 | subsection | 170 | 224 | The function | Thus,\Vert F(z)\Vert _{\mathcal {L}_{p/2,\infty }}^{\prime } \le c_{p/2}\Vert \phi _z^{\prime }\Vert _{L_\infty (\mathbb {R})}\Vert [B,A]\Vert _{\mathcal {L}_{p/2,\infty }}^{\prime }and similarly taking f = \phi _{z_1}-\phi _{z_2},\Vert F(z_1)-F(z_2)\Vert _{\mathcal {L}_{p/2,\infty }}^{\prime } \le c_{p/2}\Vert \phi _{... | {
"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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b96353a980fc8ed31d063c352b6867f5f0c68fe5 | subsection | 171 | 224 | The function | If \Re (z)>q-1, thenB^{z-1}A^{z-1}=B^{z-q+1}B^{q-2}A\cdot A^{z-2}.By Lemma REF , we have thatB^{q-2}A\in \mathcal {L}_{\frac{p}{q-2},\infty }\subset \mathcal {L}_{\frac{p}{p-2},1}and correspondingly by Lemma REF the mapping z\mapsto B^{z-1}A^{z-1} is continuous in the \mathcal {L}_{p/(p-2),1} norm.Moreover Lemma REF im... | {
"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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e6f8b1064f816e5e425c06e1eb5f4ff9767ec0ee | subsection | 172 | 224 | The function | Hence, the first summand is \mathcal {L}_1-valued holomorphic for \Re (z)>q-1.The second summand in (REF ) can be written asz\rightarrow B^{z-q+1}\cdot B^{q-2}[B,A]\cdot A^{z-1}.By our assumption of Condition REF , the operator B^{q-2}[B,A] belongs to \mathcal {L}_1 and accordingly the map z\rightarrow B^{z-1}\cdot [B,... | {
"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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fd615e53b4e6435e315bfe0b6420da9d789aceb7 | subsection | 173 | 224 | The function | We write F_0(z) as:F_0(z) &= B^{z-1}[B,A^{z-1}A^{1/2}]A^{1/2}+[B,A^{1/2}]A^{1/2}Y^{z-1}\\
&= F_2(z)+B^{z-1}A^{z-1}[B,A^{1/2}]A^{1/2}+[B,A^{1/2}]A^{1/2}F_3(z)\\
&= F_2(z)+F_1(z)[B,A^{1/2}]A^{1/2}+[B,A^{1/2}]F_3(z).Hence, by (REF ), (REF ) and (REF ) and Condition REF .(REF ),F_0(z) \in \mathcal {L}_1+\mathcal {L}_{p/(p-... | {
"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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658173fde0ac6a08f3420017df0f1d8b3be3b98f | subsection | 174 | 224 | The function | Then we have\frac{G(z)-G(z_0)}{z-z_0}-G_1(z_0) = \int _{\mathbb {R}}H(s)\Big (\frac{\hat{g}_z(s)-\hat{g}_{z_0}(s)}{z-z_0}-\hat{g}_{1,z_0}(s)\Big )ds.So by the triangle inequality,\Big \Vert \frac{G(z)-G(z_0)}{z-z_0}-G^{\prime }(z_0)\Big \Vert _{\infty }\le \operatornamewithlimits{ess\,sup}_{s \in \mathbb {R}}\Vert H(s)... | {
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"source_re... | 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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8d9baf189ded04705cad4dcd588330e7f084e719 | subsection | 175 | 224 | The function | Indeed, by Lemma REF , we have\Vert G(z)\Vert _{r,\infty } &\le \frac{r}{r-1}\int _{\mathbb {R}}(1+|s|)|\hat{g}_z(s)|ds,\\
\Vert G^{\prime }(z)\Vert _{r,\infty } &\le \frac{r}{r-1}\int _{\mathbb {R}}(1+|s|)|\hat{g}_{1,z}(s)|ds,\\
\Big \Vert \frac{G(z)-G(z_0)}{z-z_0}-G^{\prime }(z_0)\Big \Vert _{r,\infty } &\le \frac{r}... | {
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"source_re... | 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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8adc29d74da03def2b581e059fef833f86a329e7 | subsection | 176 | 224 | The function | If X\in \mathcal {L}_{\infty }, thenthe mapping
G_1(z) := \int _{\mathbb {R}}[BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]Y^{-is}XB^{is}\hat{g}_z(s)ds,
is an \mathcal {L}_{\frac{p}{2},\infty }-valued holomorphic function for \Re (z)>1.
the mapping
G_2(z) := \int _{\mathbb {R}}A^{is}Y^{-is}XB^{is}\hat{g}_z(s)ds,
is \mathcal {... | {
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... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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] | [
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6cdc47f662208a44ad86b93902f65b125ff6d374 | subsection | 177 | 224 | The function | Since p > 2 and \phi _s is a Lipschitz function, we have\Vert [BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]\Vert _{\frac{p}{2},\infty } &\le c_p\Vert \phi _s^{\prime }\Vert _{\infty }\Vert [BA,A^{\frac{1}{2}}]\Vert _{\frac{p}{2},\infty }\\
&\le c_p(1+|s|)\Vert [BA,A^{\frac{1}{2}}]\Vert _{\frac{p}{2},\infty }.Hence for j = 1,2 ... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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e9be7f6487bbfac2e650f8710c2ad616a08e545b | subsection | 178 | 224 | The function | Using the Leibniz rule:B^{z-1+is}[BA^{\frac{1}{2}},A^{z-\frac{1}{2}+is}]Y^{-is} &= B^{z-1+is}[BA^{\frac{1}{2}},A^{z-1} A^{\frac{1}{2}+is}]Y^{-is}\\
&= B^{is} B^{z-1}A^{z-1}[BA^{\frac{1}{2}},A^{\frac{1}{2}+is}] Y^{-is}\\
&\quad +B^{is} B^{z-1}[BA,A^{z-1}] A^{is}Y^{-is}.By the \mathcal {L}_1-triangle inequality, we have\... | {
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} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
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a1f676c1ea236530f5a74fb772609f7135c705c7 | subsection | 179 | 224 | The function | Since p > 2, we may apply the result of to obtain:\Vert [BA^{1/2},\phi (A^{1/2})]\Vert _{p/2,\infty } \le C_p(1+|s|)\Vert [BA^{1/2},A^{1/2}]\Vert _{p/2,\infty }.Therefore,\Vert [BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]\Vert _{\frac{p}{2},\infty } \le C_p(1+|s|)\Vert [BA^{\frac{1}{2}},A^{\frac{1}{2}}]\Vert _{\frac{p}{2},\in... | {
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... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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] | [
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6d3eadd08c30a69c68f4024ed25191efebcdffb6 | subsection | 180 | 224 | The function | Using the Hölder inequality in the form of (REF ), we obtain\Vert B^{is}[BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]Y^{z-1-is}\Vert _1&\le \Vert [BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]Y^{z-1}\Vert _1\\
&\le \Vert [BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]\Vert _{\frac{p}{2},\infty }\Vert Y^{z-1}\Vert _{\frac{p}{p-2},1}.Now combining... | {
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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_r... | 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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4c03f9d7decbfc822c03dca2b28e741d4faa57f5 | subsection | 181 | 224 | The function | Since p > 2, we therefore have a weak operator topology integral:B^zA^z-(A^{1/2}BA^{1/2})^z = T_z(0)-\int _{\mathbb {R}} T_z(s)\widehat{g}_z(s)\,ds.From Lemma REF , we have that\int _{\mathbb {R}}\Vert T_z(s)\Vert _1\cdot |\hat{g}_z(s)|ds<\infty .Thus by Lemma REF , we have that \int _{\mathbb {R}} T_z(s)\widehat{g}_z(... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
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688c4a1760f4327b04fe3a14026456820d08ffdc | subsection | 182 | 224 | The function | If X\in \mathcal {L}_{\infty }, then for \Re (z) > p:{\rm Tr}\Big (X\Big (B^zA^z-(A^{\frac{1}{2}}BA^{\frac{1}{2}})^z\Big )\Big )={\rm Tr}(XF_0(z))-\sum _{k=1}^3{\rm Tr}(G_k(z)F_k(z))Here, the functions F_k are as in Lemma REF and the functions G_k are as in Lemma REF .By Lemma REF ,\int _{\mathbb {R}} \Vert T_z(s)\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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6d1c325f84073494ed8fc2f4249f00deb8c6cca3 | subsection | 183 | 224 | The function | Therefore,{\rm Tr}(X\cdot T_z(s)) &= {\rm Tr}([BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]Y^{-is}XB^{is} B^{z-1}A^{z-1})\\
&+ {\rm Tr}(A^{is}Y^{-is}XB^{is} B^{z-1}[BA,A^{z-1}])+{\rm Tr}(Y^{-is}XB^{is}[BA^{\frac{1}{2}},A^{\frac{1}{2}+is}]Y^{z-1}).Thus,\int _{\mathbb {R}}{\rm Tr}&(XT_z(s))\hat{g}_z(s)ds\\
&={\rm Tr}\Big (\int _... | {
"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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d1c165e0de1e89f7d77a2d8458fd14b6e9f922d2 | subsection | 184 | 224 | The function | Finally, for the G_3F_3 term, we use Lemma REF .
(REF ) and Lemma REF .(REF ) to see that mapping z\rightarrow G_3(z)F_3(z) is \mathcal {L}_1-valued analytic for \Re (z)>p-1.Hence, A is holomorphic in the set \Re (z) > p-1. By Lemma REF ,A(z) = {\rm Tr}\Big (X\Big (B^zA^z-(A^{\frac{1}{2}}BA^{\frac{1}{2}})^z\Big )\Big )... | {
"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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89943d4f16f2e233aadb3bb3396590d20d700779 | subsection | 185 | 224 | Criterion for universal measurability in terms of a | In this section we provide a sufficient condition for universal measurability of operators in \mathcal {L}_{1,\infty }.We recall that a linear functional \varphi on the weak Schatten ideal \mathcal {L}_{1,\infty } is called
a trace if for all unitary operators U and T \in \mathcal {L}_{1,\infty }, we have \varphi (U^*T... | {
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"raw": "Sukochev F., Usachev A., Zanin D. Singular traces and residues of the \\zeta -function. Indiana U. Math. J., 66 (2017), no. 4, 1107–1144.",
"source_ref_id": "a140a7c977344d3598eb63fbd5d15a44a... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
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] | [
"math.OA"
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d0b646d0b1905631c1db9353c7f59d352a77dbf1 | subsection | 186 | 224 | Criterion for universal measurability in terms of a | For \Re (z)>\varepsilon , we have|f(z)-f_n(z)|=|\int _n^{\infty }e^{-tz}db(t)|\le \sum _{k\ge n}e^{-k\varepsilon }\mathrm {Var}_{[k,k+1]}(b)\le c_b\frac{e^{-n\varepsilon }}{1-e^{-\varepsilon }}.Therefore, f_n\rightarrow f uniformly on the half-plane \lbrace \Re (z)>\varepsilon \rbrace . Since \varepsilon >0 is arbitrar... | {
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"end": 689,
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"raw": "Subhankulov M. Tauberian theorems with remainders. Nauka, Moscow, 1976.",
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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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f4ad65f88a059c558bdd988fa5b76bb1f65f14c6 | subsection | 187 | 224 | Criterion for universal measurability in terms of a | By definition we have:\int _{-1}^1\frac{(1-t^2)^2}{\frac{1}{x}+it}f(\frac{1}{x}+it)e^{(\frac{1}{x}+it)x}dt = \int _{-1}^1\int _0^{\infty }\frac{(1-t^2)^2}{\frac{1}{x}+it}e^{(\frac{1}{x}+it)(x-s)}db(s)dt.Examining the integrand, we see that the function(s,t) \mapsto \frac{(1-t^2)^2}{x^{-1}+t}e^{(\frac{1}{x}+it)(x-s)}is ... | {
"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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5a6341ffa1d66dc1fbf2e0819171f57a66b119b8 | subsection | 188 | 224 | Criterion for universal measurability in terms of a | Then by the triangle inequality, there is a positive constant c_{\mathrm {abs}} such that\Big |\int _0^{\infty }\min \lbrace 1,(x-s)^{-2}\rbrace \cdot O(1)\cdot db(s)\Big |
&\le c_{abs}\cdot \int _0^{\infty }\min \lbrace 1,(x-s)^{-2}\rbrace dh(s)\\
&=c_{abs}\cdot \sum _{k\ge 0}\int _k^{k+1}\min \lbrace 1,(x-s)^{-2}\rbr... | {
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"raw": "Shubin M. Pseudodifferential operators and spectral theory. Springer-Verlag, Berlin, second edition, 2001.",
"source_ref_id": "4f9e8ea... | 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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4395846c462b3b746c475e6d016e304a4a3a2ae0 | subsection | 189 | 224 | Criterion for universal measurability in terms of a | In particular, since the (closure of the) set\Big \lbrace \frac{1}{x}+it:\ x\ge 1,\ t\in [-1,1]\Big \rbrace ,is a compact subset in \lbrace \Re (z)>-\epsilon \rbrace , it follows that\sup _{x\ge 1}\sup _{t\in [-1,1]}|f_0(\frac{1}{x}+it)|<\infty .The assertion of Lemma REF is now written as follows.\frac{1}{2\pi }\int _... | {
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"raw": "Kalton N., Lord S., Potapov D., Sukochev F. Traces of compact operators and the noncommutative residue. Adv. Math. 235 (2013), 1–55.",
... | 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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242cb8f3153ae769c9df464791005c775811f0dd | subsection | 190 | 224 | Criterion for universal measurability in terms of a | It is proved in that V is V-modulated, and therefore that AV is V-modulated.The relevance of the notion of a V-modulated operator to measurability comes from , which states
that if V \ge 0 is in \mathcal {L}_{1,\infty }, \ker (V) = 0, T is V-modulated and \lbrace e_n\rbrace _{n\ge 0} is an eigenbasis for V ordered
so t... | {
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"source_ref_id": "a82390d35c6878... | 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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63015e9bffbe80151fae3f1d194a634118d41fc5 | subsection | 191 | 224 | Criterion for universal measurability in terms of a | We have\sum _{k=0}^n\lambda (k,AV) = \sum _{\mu (k,V)>\frac{\Vert V\Vert _{1,\infty }}{n}}\langle Ae_k,e_k\rangle \mu (k,V)+O(1).Here, e_k is the eigenvector of V corresponding to the eigenvalue \mu (k,V).We have that AV is V-modulated. now states that as n\rightarrow \infty\sum _{k=0}^n\lambda (k,AV) &= \sum _{k=0}^n\... | {
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"source_ref_id": "a82390d35c687... | 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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df3a3f26ceb7cd4f968545c910e548023529b3c8 | subsection | 192 | 224 | Criterion for universal measurability in terms of a | Thus,\Big |\sum _{k=m(n)+1}^n\langle Ae_k,e_k\rangle \mu (k,V)\Big | &\le \Vert A\Vert _{\infty } \sum _{k=m(n)+1}^n\mu (k,V)\\
&\le \frac{\Vert A\Vert _{\infty }\Vert V\Vert _{1,\infty }}{n}\sum _{k=m(n)+1}^n1\\
&= O(1).Finally, we have\sum _{k=0}^n\lambda (k,AV) &= \sum _{k=0}^n\langle Ae_k,e_k\rangle \mu (k,V)+O(1)\... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.013621091842651367,
0.03836798295378685,
-0.027516894042491913,
-0.027379538863897324,
0.027730558067560196,
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0.024495072662830353,
0.0004898060578852892,
0.002413260517641902,
0.018039362505078316,
0.03687233477830887,
... | |
a7a96707c0875ed721cedeae51de0a5ee34ea3a7 | subsection | 193 | 224 | Criterion for universal measurability in terms of a | If x \ge 0, then:\mathrm {Var}_{[x,x+1]}b &\le |\mathrm {Res}_{w=0}\zeta _{A,V}(w)|+\sum _{\mu (k,V)\in (e^{-1-x},e^{-x}]}|\langle Ae_k,e_k\rangle |\mu (k,V)\\
&\le |\mathrm {Res}_{w=0}\zeta _{A,V}(w)|+2\Vert A\Vert _{\infty }e^{-x}\sum _{\mu (k,V)\in (e^{-1-x},e^{-x}]}1\\
&\le |\mathrm {Res}_{w=0}\zeta _{A,V}(w)|+2\Ve... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.03127877041697502,
0.027967797592282295,
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-0.03237733989953995,
-0.0023420932702720165,
-0.007545049302279949,
-0.0032423113007098436,
... | |
0c7913df617fe8c9a3ff9e6635cbbe61b15a3915 | subsection | 194 | 224 | Criterion for universal measurability in terms of a | Using the identity e^{-\alpha _k z} = \mu (k,V)^z, we have:\int _0^{\infty }e^{-zt}db(t) &= \sum _{k\ge 0}e^{-\alpha _k z}\cdot b_k-\mathrm {Res}_{w=0}\zeta _{A,V}(w)\int _0^{\infty }e^{-zt}dt\\
&= \sum _{k\ge 0}\mu (k,V)^z\cdot \langle Ae_k,e_k\rangle \mu (k,V)-\mathrm {Res}_{w=0}\zeta _{A,V}(w)\int _0^{\infty }e^{-zt... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.020464660599827766,
0.029148023575544357,
-0.041265055537223816,
0.0014469093875959516,
0.002745026722550392,
0.02623322233557701,
0.02925485000014305,
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0.003591998014599085,
0.0073938313871622086,
0.001806490821763873,
0.01681734248995781,
-0.0022204385604709387,
... | |
d70b543d63db5fbbafc4c0a5c5712c73525b0337 | subsection | 195 | 224 | Properties of the algebra | For this section, (\mathcal {A},H,D) is a smooth spectral triple.
Recall from Definition REF that \mathcal {B} is the *-algebra generated by all elements of the form \delta ^k(a) or \partial (\delta ^k(a)), k\ge 0, a\in \mathcal {A}.
Recall that we define H_\infty := \bigcap _{k\ge 1} \mathrm {dom}(D^k), and that for a... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 493,
"openalex_id": "",
"raw": "Carey A., Phillips J., Rennie A., Sukochev F., The local index formula in semifinite von Neumann algebras. I. Spectral flow. Adv. Math. 202 (2006), no. 2, 451–516.",
"source_ref_id": "97ff2bed... | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04114235192537308,
0.029010241851210594,
-0.019792890176177025,
-0.06983212381601334,
-0.0069435350596904755,
-0.0010691899806261063,
0.06040112301707268,
0.03439720347523689,
0.04221059009432793,
0.024722035974264145,
-0.030551554635167122,
0.014734028838574886,
0.06366687268018723,
-0... | |
c320b6a7c9db79b570b017ea87b8235111f0e4e7 | subsection | 196 | 224 | Properties of the algebra | Then for all x \in \mathcal {B} and m\ge 0 we haveThe operators |D|^{-m}x|D|^m,|D|^mx|D|^{-m}:H_\infty \rightarrow H_\infty have bounded extension
|D|^{1-m}[|D|^m,x]:H_\infty \rightarrow H_\infty has bounded extension.By Lemma REF , on H_\infty we have:|D|^mx|D|^{-m} &= \sum _{k=0}^m\binom{m}{k}\delta ^{m-k}(x)|D|^{k... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.055051039904356,
0.03329306095838547,
-0.03741273656487465,
-0.01641765981912613,
-0.03454422205686569,
0.0225361380726099,
0.04250892251729965,
0.055203620344400406,
0.029844744130969048,
-0.024565458297729492,
-0.03207241743803024,
-0.015593726187944412,
0.01750098168849945,
-0.019682... | |
f29261c0e300b2c9b6bf862a0fd13a5ea38e2011 | subsection | 197 | 224 | Properties of the algebra | Clearly,(1+s^2D^2)^{-\frac{p+1}{2}}=|(1-isD)^{-p-1}|.Setting \lambda =\frac{1}{s}, we obtain from Hypothesis REF that:\Vert x(1+s^2D^2)^{-\frac{p+1}{2}}\Vert _1 &= s^{-p-1}\Vert x(D+\frac{i}{s})^{-p-1}\Vert _1\\
&= s^{-p-1}\cdot O(s)\\
&= O(s^{-p}),\quad s\downarrow 0.Since the operator (1+s^2D^2)^{\frac{p+1}{2}}h(sD) ... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.06449788808822632,
0.005957054439932108,
-0.043354541063308716,
0.021951859816908836,
-0.019373774528503418,
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0.04542921483516693,
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0.02831317111849785,
-0.015590549446642399,
-0.02381296269595623,
-0.011753931641578674,
0.022394252941012383,
-0... | |
7e9bb81a2ae1cfb0102421f51f9e0acaab55172e | subsection | 198 | 224 | Properties of the algebra | Using the triangle inequality:\Vert |D|^{-m_1}x|D|^{-m_2}e^{-s^2D^2}\Vert _1 &\le \sum _{k,l=0}^{\infty }\Vert \chi _{[2^k,2^{k+1}]}(|D|)|D|^{-m_1}x|D|^{-m_2}e^{-s^2D^2}\chi _{[2^l,2^{l+1}]}(|D|)\Vert _1\\
&\le \sum _{k,l=0}^{\infty }2^{-km_1-lm_2}\cdot e^{-2^{2l}s^2}\cdot \Vert \chi _{[2^k,2^{k+1}]}(|D|)x\chi _{[2^l,2... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05461001396179199,
0.03992727771401405,
-0.029167059808969498,
-0.028266558423638344,
-0.03431059792637825,
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0.03513478487730026,
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-0.0019164254190400243,
-0.03843153268098831,
-0.0008280026377178729,
0.01095863338559866,
0.... | |
73a61b0bf0102f09aad4bf551d5f4a29cdea7a5a | subsection | 199 | 224 | Properties of the algebra | We have\Vert |D|^{-m_1}x|D|^{-m_2}e^{-s^2D^2}\Vert _1 &\le \sum _{l=0}^{\infty }\Big \Vert x|D|^{-m_2}e^{-s^2D^2}\chi _{[2^l,2^{l+1}]}(|D|)\Big \Vert _1\\
&\le \sum _{l=0}^{\infty }2^{-lm_2}\cdot e^{-2^{2l}s^2}\cdot \Big \Vert x\chi _{[2^l,2^{l+1}]}(|D|)\Big \Vert _1\\
&\le \sum _{l=0}^{\infty }2^{-lm_2}\cdot e^{-2^{2l... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0402153879404068,
0.026927223429083824,
-0.029413986951112747,
-0.0632217675447464,
-0.03188549354672432,
0.02558467723429203,
0.042869970202445984,
0.026530561968684196,
0.0010469579137861729,
-0.0021129860542714596,
-0.019421163946390152,
-0.012349908240139484,
0.03045140951871872,
-0... | |
7333a88aa95907c15b09532df2b3408d7be21d4b | subsection | 200 | 224 | Properties of the algebra | Since for t > 1 we have:t^{-p-1}(1-e^{-t^2})\le t^{-p-1}\le 2^{\frac{p+1}{2}}\cdot (t^2+1)^{-\frac{p+1}{2}},it follows that|D|^{-p-1}(1-e^{-s^2D^2})\chi _{(\frac{1}{s},\infty )}(|D|) \le s^{p+1}\cdot 2^{\frac{p+1}{2}}\cdot (1+s^2D^2)^{-\frac{p+1}{2}}.So we may estimate the first summand by:\Vert x|D|^{p-1}(1-e^{-s^2D^2... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.04837454855442047,
0.015305889770388603,
-0.036807384341955185,
-0.027758140116930008,
-0.023729471489787102,
-0.0014001150848343968,
0.05566887930035591,
0.011323001235723495,
-0.002397744683548808,
0.012459879741072655,
-0.021165771409869194,
0.006462656427174807,
-0.0024854904040694237... | |
01aa81caa4737f9c386e016cd1811672d64e98d7 | subsection | 201 | 224 | Properties of the algebra | For every x\in \mathcal {B}, we have\Vert |D|^{-p-1}x(1-e^{-s^2D^2})\Vert _1 = O(s),\quad s\downarrow 0.On the subspace H_\infty , we have:|D|^{-p-1}x &= x|D|^{-p-1}+[|D|^{-p-1},x]\\
&= x|D|^{-p-1}-|D|^{-p-1}[|D|^{p+1},x]|D|^{-p-1}.By Lemma REF , we have (again on H_\infty ):[|D|^{p+1},x] &= |D|^{p+1}x-x|D|^{p+1}\\
&= ... | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.05141127109527588,
0.03499627858400345,
-0.03984754905104637,
-0.020274652168154716,
-0.03270794078707695,
0.021983277052640915,
0.08622448146343231,
0.03267743065953255,
0.010793316178023815,
0.00025862958864308894,
-0.04039674997329712,
-0.011708649806678295,
0.010419554077088833,
-0.... | |
149b71c26c6ff09db88410e7558d628ab9415569 | subsection | 202 | 224 | Properties of the algebra | Thus,\Vert |D|^{-p-1}\delta ^{p+1-k}(x)|D|^{k-p-1}(1-e^{-s^2D^2})\Vert _1=O(s),\quad s\downarrow 0.Hence, each summand in (REF ) extends to a trace class operator with norm bounded by O(s), s\downarrow 0. By the triangle
inequality, this completes the proof. | {
"cite_spans": []
} | 1803.01551 | The Connes character formula for locally compact spectral triples | [
"Fedor Sukochev",
"Dmitriy Zanin"
] | [
"math.OA"
] | 2,018 | en | Mathematics | [
-0.0682164654135704,
0.0027667174581438303,
-0.04750135540962219,
-0.009236374869942665,
-0.020089687779545784,
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-0.006185548845678568,
-0.044359005987644196,
0.024071015417575836,
0.03355908393859863,
-0.0... |
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