full_collection/Junk-Test.jsonl

{"id":"lem: wstar_of_masa","label":"wstar_of_masa","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A maximal abelian \\(*\\)–subalgebra of a \\(W^{*}\\)–algebra \\(M\\) is also a \\(W^{*}\\)–algebra.","doc_url":null,"latex_link":"sect0001.html#lem: wstar_of_masa","lean_link":null,"lean_decl_name":"wstar_of_masa","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"sigma_cts_of_normal","label":"sigma_cts_of_normal","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":["lem: wstar_of_masa","lem: exists_uw_ge_normal","lem: msr_th_lemma","lem: fin_lin_approx_of_stonean","lem: cut_down_of_wstar","lem: spec_masa_wstar_stonean","lem: zorn_base","lem: bdd_sigma_cts_iff_bdd_s_cts"],"nlp_text":"Every positive normal linear functional \\(\\varphi \\) on \\(M\\) is \\(\\sigma (M,M_{*})\\)–continuous.","doc_url":null,"latex_link":"sect0001.html#sigma_cts_of_normal","lean_link":null,"lean_decl_name":"sigma_cts_of_normal","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: selfadjoint_le_norm","label":"selfadjoint_le_norm","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all self-adjoint \\(x\\in M\\), \\(\\| x\\| 1-x \\ge 0\\).","doc_url":null,"latex_link":"sect0001.html#lem: selfadjoint_le_norm","lean_link":null,"lean_decl_name":"selfadjoint_le_norm","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: proj_sub_of_subproj","label":"proj_sub_of_subproj","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all \\(p,q\\in \\mathcal{P}(M)\\) such that \\(q\\le p\\), \\(p-q\\in \\mathcal{P}(M)\\).","doc_url":null,"latex_link":"sect0001.html#lem: proj_sub_of_subproj","lean_link":null,"lean_decl_name":"proj_sub_of_subproj","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: uw_pos_sep_pts","label":"uw_pos_sep_pts","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every nonzero element \\(a\\in M\\), there is a \\(\\sigma (M,M_{*})\\)–continuous positive linear functional \\(\\psi \\) on \\(M\\) such that \\(\\psi (a)\\ne 0\\).","doc_url":null,"latex_link":"sect0001.html#lem: uw_pos_sep_pts","lean_link":null,"lean_decl_name":"uw_pos_sep_pts","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: exists_uw_ge_normal","label":"exists_uw_ge_normal","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every positive normal linear functional \\(\\varphi \\) and nonzero \\(p\\in \\mathcal{P}(M)\\) there exists a positive \\(\\sigma (M,M_{*})\\)–continuous linear functional \\(\\psi \\) such that \\(\\varphi (p){\\lt}\\psi (p)\\).","doc_url":null,"latex_link":"sect0001.html#lem: exists_uw_ge_normal","lean_link":null,"lean_decl_name":"exists_uw_ge_normal","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: msr_th_lemma","label":"msr_th_lemma","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For all positive linear functionals \\(\\varphi ,\\psi \\) with \\(\\varphi \\) normal and \\(\\psi \\) \\(\\sigma (M,M_{*})\\)–continuous and every nonzero \\(p\\in \\mathcal{P}(M)\\) such that \\(\\varphi (p){\\lt}\\psi (p)\\), there exists a nonzero \\(p_1 \\in \\mathcal{P}(M)\\) such that \\(p_1\\le p\\) and for all nonzero \\(q\\in \\mathcal{P}(M)\\) with \\(q\\le p_1\\), we have \\(\\varphi (q){\\lt}\\psi (q)\\).","doc_url":null,"latex_link":"sect0001.html#lem: msr_th_lemma","lean_link":null,"lean_decl_name":"msr_th_lemma","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: proj_of_sup_proj","label":"proj_of_sup_proj","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For every increasing bounded net \\((p_{\\alpha })\\) of projections in \\(M\\), the supremum \\(p\\) is a projection in \\(M\\)","doc_url":null,"latex_link":"sect0001.html#lem: proj_of_sup_proj","lean_link":null,"lean_decl_name":"proj_of_sup_proj","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: fin_lin_approx_of_stonean","label":"fin_lin_approx_of_stonean","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K\\) be a Stonean space. Then every element \\(a\\) in \\(C(K)\\) can be uniformly approximated by finite linear combinations of projections in \\(C(K)\\). If \\(a\\ge 0\\) then the coefficients of the approximating linear combinations may be chosen nonnegative.","doc_url":null,"latex_link":"sect0001.html#lem: fin_lin_approx_of_stonean","lean_link":null,"lean_decl_name":"fin_lin_approx_of_stonean","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: cut_down_of_wstar","label":"cut_down_of_wstar","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(p\\in M\\) is a projection, \\(pMp\\) is also a \\(W^{*}\\) algebra with identity \\(p\\).","doc_url":null,"latex_link":"sect0001.html#lem: cut_down_of_wstar","lean_link":null,"lean_decl_name":"cut_down_of_wstar","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: spec_masa_wstar_stonean","label":"spec_masa_wstar_stonean","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(C\\) is any maximal commutative \\(C^*\\)–subalgebra of the \\(W^{*}\\)–algebra \\(M\\), its spectrum space (maximal ideal space) is Stonean.","doc_url":null,"latex_link":"sect0001.html#lem: spec_masa_wstar_stonean","lean_link":null,"lean_decl_name":"spec_masa_wstar_stonean","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: zorn_base","label":"zorn_base","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\varphi \\) be a normal positive linear functional on \\(M\\) and consider the predicate \\(P:\\mathcal{P}(M)\\to \\text{Prop}\\) defined, for \\(p\\in \\mathcal{P}(M)\\), by “\\(M\\ni x \\mapsto \\varphi (xp)\\) is \\(\\sigma (M,M_{*})\\)–continuous”. If \\((p_{\\alpha })\\) is a chain of projections in \\(M\\) such that \\(P(p_{\\alpha })\\) is true for each \\(\\alpha \\), then \\(P(\\sup (p_{\\alpha }))\\) is true. Hence by Zorn’s Lemma there is a maximal \\(p_0\\in \\mathcal{P}(M)\\) such that \\(P(p_0)\\) is true.","doc_url":null,"latex_link":"sect0001.html#lem: zorn_base","lean_link":null,"lean_decl_name":"zorn_base","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem: bdd_sigma_cts_iff_bdd_s_cts","label":"bdd_sigma_cts_iff_bdd_s_cts","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://JonBannon.github.io/Junk-Test/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A linear functional \\(\\rho \\) on \\(M\\) is \\(\\sigma \\) continuous on the unit sphere (hence \\(\\sigma \\)–continuous) if and only if it is \\(s\\)–continuous on the unit sphere (hence \\(s\\)–continuous).","doc_url":null,"latex_link":"sect0001.html#lem: bdd_sigma_cts_iff_bdd_s_cts","lean_link":null,"lean_decl_name":"bdd_sigma_cts_iff_bdd_s_cts","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}