{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:W2ODE4TGFSHU6SJ6D5V7TTG6QW","short_pith_number":"pith:W2ODE4TG","canonical_record":{"source":{"id":"2606.27230","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-25T16:17:02Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"e16abcf293a46fe2efdd66d03a3973def01ce6f92803f63bb72186b3a48c443b","abstract_canon_sha256":"4cdc3762e49637e3d7065fe0e4956e5045f17912a701d0f6c5656ecffe9f40fc"},"schema_version":"1.0"},"canonical_sha256":"b69c3272662c8f4f493e1f6bf9ccde85adada4dbbee2ad3f7211a7fe50c0e108","source":{"kind":"arxiv","id":"2606.27230","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27230","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27230v1","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27230","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"pith_short_12","alias_value":"W2ODE4TGFSHU","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"pith_short_16","alias_value":"W2ODE4TGFSHU6SJ6","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"pith_short_8","alias_value":"W2ODE4TG","created_at":"2026-06-26T01:16:15Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:W2ODE4TGFSHU6SJ6D5V7TTG6QW","target":"record","payload":{"canonical_record":{"source":{"id":"2606.27230","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-25T16:17:02Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"e16abcf293a46fe2efdd66d03a3973def01ce6f92803f63bb72186b3a48c443b","abstract_canon_sha256":"4cdc3762e49637e3d7065fe0e4956e5045f17912a701d0f6c5656ecffe9f40fc"},"schema_version":"1.0"},"canonical_sha256":"b69c3272662c8f4f493e1f6bf9ccde85adada4dbbee2ad3f7211a7fe50c0e108","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T01:16:15.762275Z","signature_b64":"gR6d3UKQlCua5mTktGiaSg2/ZRLrzlyufc+PaD9DbTTxnXbxGDZEQACf2540rBiRnEBmVlxgoFBs0jQNg1kPAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b69c3272662c8f4f493e1f6bf9ccde85adada4dbbee2ad3f7211a7fe50c0e108","last_reissued_at":"2026-06-26T01:16:15.761930Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T01:16:15.761930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.27230","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-26T01:16:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"StWH6ZRSEJ5e/+TwFkYZMs9NcNhf4drk+bzurH4srl7TDiC7+Vwts7Hke84S4M2PAwXZBOeUSeljb7X0wZV4Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T17:50:50.012836Z"},"content_sha256":"96105eb2e4beeb4772c94021f22ce417930f0ae3d2db5bd3cccf4c444c55c821","schema_version":"1.0","event_id":"sha256:96105eb2e4beeb4772c94021f22ce417930f0ae3d2db5bd3cccf4c444c55c821"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:W2ODE4TGFSHU6SJ6D5V7TTG6QW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On some $p$-approximation properties of exact discrete groups and $\\ell^p$ uniform Roe algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.FA","authors_text":"Yeong Chyuan Chung","submitted_at":"2026-06-25T16:17:02Z","abstract_excerpt":"We study $p$-approximation properties of $\\ell^p$ uniform Roe algebras and their connections to coarse geometry and group theory. For a discrete metric space $X$ with bounded geometry, we prove that property A implies $p$-nuclearity of the $\\ell^p$ uniform Roe algebra $B^p_u(X)$ for every $p\\in(1,\\infty)$, while $B^1_u(X)$ is always 1-nuclear. We introduce the $p$-invariant translation approximation property ($p$-ITAP) for discrete groups, generalizing the 2-ITAP of Roe. We also introduce the $p$-operator ITAP. For exact groups, we show that the $p$-operator ITAP is equivalent to the $p$-appro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27230","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.27230/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-26T01:16:15Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"haJcG3F35iWnK1R1YMSXZ9OsuQMgZm1MrtgTDRPGEMIhDkpQX9TRtp2pyTlK715yCxvA/GUPAjTgwLxw4w9dBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T17:50:50.013386Z"},"content_sha256":"ebd797dd39ca047190c283c25d1a386255f4cdf1592e08a0a40c521ff4b8fbd7","schema_version":"1.0","event_id":"sha256:ebd797dd39ca047190c283c25d1a386255f4cdf1592e08a0a40c521ff4b8fbd7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W2ODE4TGFSHU6SJ6D5V7TTG6QW/bundle.json","state_url":"https://pith.science/pith/W2ODE4TGFSHU6SJ6D5V7TTG6QW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W2ODE4TGFSHU6SJ6D5V7TTG6QW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-28T17:50:50Z","links":{"resolver":"https://pith.science/pith/W2ODE4TGFSHU6SJ6D5V7TTG6QW","bundle":"https://pith.science/pith/W2ODE4TGFSHU6SJ6D5V7TTG6QW/bundle.json","state":"https://pith.science/pith/W2ODE4TGFSHU6SJ6D5V7TTG6QW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W2ODE4TGFSHU6SJ6D5V7TTG6QW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:W2ODE4TGFSHU6SJ6D5V7TTG6QW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4cdc3762e49637e3d7065fe0e4956e5045f17912a701d0f6c5656ecffe9f40fc","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-25T16:17:02Z","title_canon_sha256":"e16abcf293a46fe2efdd66d03a3973def01ce6f92803f63bb72186b3a48c443b"},"schema_version":"1.0","source":{"id":"2606.27230","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.27230","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"arxiv_version","alias_value":"2606.27230v1","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.27230","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"pith_short_12","alias_value":"W2ODE4TGFSHU","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"pith_short_16","alias_value":"W2ODE4TGFSHU6SJ6","created_at":"2026-06-26T01:16:15Z"},{"alias_kind":"pith_short_8","alias_value":"W2ODE4TG","created_at":"2026-06-26T01:16:15Z"}],"graph_snapshots":[{"event_id":"sha256:ebd797dd39ca047190c283c25d1a386255f4cdf1592e08a0a40c521ff4b8fbd7","target":"graph","created_at":"2026-06-26T01:16:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.27230/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study $p$-approximation properties of $\\ell^p$ uniform Roe algebras and their connections to coarse geometry and group theory. For a discrete metric space $X$ with bounded geometry, we prove that property A implies $p$-nuclearity of the $\\ell^p$ uniform Roe algebra $B^p_u(X)$ for every $p\\in(1,\\infty)$, while $B^1_u(X)$ is always 1-nuclear. We introduce the $p$-invariant translation approximation property ($p$-ITAP) for discrete groups, generalizing the 2-ITAP of Roe. We also introduce the $p$-operator ITAP. For exact groups, we show that the $p$-operator ITAP is equivalent to the $p$-appro","authors_text":"Yeong Chyuan Chung","cross_cats":["math.OA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-25T16:17:02Z","title":"On some $p$-approximation properties of exact discrete groups and $\\ell^p$ uniform Roe algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27230","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:96105eb2e4beeb4772c94021f22ce417930f0ae3d2db5bd3cccf4c444c55c821","target":"record","created_at":"2026-06-26T01:16:15Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4cdc3762e49637e3d7065fe0e4956e5045f17912a701d0f6c5656ecffe9f40fc","cross_cats_sorted":["math.OA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2026-06-25T16:17:02Z","title_canon_sha256":"e16abcf293a46fe2efdd66d03a3973def01ce6f92803f63bb72186b3a48c443b"},"schema_version":"1.0","source":{"id":"2606.27230","kind":"arxiv","version":1}},"canonical_sha256":"b69c3272662c8f4f493e1f6bf9ccde85adada4dbbee2ad3f7211a7fe50c0e108","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b69c3272662c8f4f493e1f6bf9ccde85adada4dbbee2ad3f7211a7fe50c0e108","first_computed_at":"2026-06-26T01:16:15.761930Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-26T01:16:15.761930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gR6d3UKQlCua5mTktGiaSg2/ZRLrzlyufc+PaD9DbTTxnXbxGDZEQACf2540rBiRnEBmVlxgoFBs0jQNg1kPAA==","signature_status":"signed_v1","signed_at":"2026-06-26T01:16:15.762275Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.27230","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96105eb2e4beeb4772c94021f22ce417930f0ae3d2db5bd3cccf4c444c55c821","sha256:ebd797dd39ca047190c283c25d1a386255f4cdf1592e08a0a40c521ff4b8fbd7"],"state_sha256":"3ac5ccc9e52552a438a219f37f3f8463cb1460d1424550efec3c192ef920caf8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qf/lcZMwaDagTcvqamVy0UsJgWEoXwG14B9yypvMQSAGO+0HKYCiP8T0/ZM6BbPq+PIN0B71MVT1PnXjGJ/UBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T17:50:50.015417Z","bundle_sha256":"a1c200abb5f8dfd336e25e1c580ef73c00d7178cfa06a351c7b6d5d9eece00a2"}}