{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:5TQTDP4CC4DAW43VLAW3JF2HRH","short_pith_number":"pith:5TQTDP4C","canonical_record":{"source":{"id":"math/9402205","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1994-02-08T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"2da333f8e668da0383e7885eec354d373761e7a0dff8223866bb4eabf3995847","abstract_canon_sha256":"efccc5945599d9618c2dcae40a0760f73e621fe78a03fd684194f18d27bb303d"},"schema_version":"1.0"},"canonical_sha256":"ece131bf8217060b7375582db4974789dda0aa6c90a58857a41dc0522930a464","source":{"kind":"arxiv","id":"math/9402205","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9402205","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9402205v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9402205","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"5TQTDP4CC4DA","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"5TQTDP4CC4DAW43V","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"5TQTDP4C","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:5TQTDP4CC4DAW43VLAW3JF2HRH","target":"record","payload":{"canonical_record":{"source":{"id":"math/9402205","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1994-02-08T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"2da333f8e668da0383e7885eec354d373761e7a0dff8223866bb4eabf3995847","abstract_canon_sha256":"efccc5945599d9618c2dcae40a0760f73e621fe78a03fd684194f18d27bb303d"},"schema_version":"1.0"},"canonical_sha256":"ece131bf8217060b7375582db4974789dda0aa6c90a58857a41dc0522930a464","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:51.663079Z","signature_b64":"d69ngwqU0WwWZsAMT5zZtFpwhqJxXUkLOVzecHLIEA5KNTDuTXXDcS8bp/AmfwO1GXnyGXaGxEBUzYkvMPl0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ece131bf8217060b7375582db4974789dda0aa6c90a58857a41dc0522930a464","last_reissued_at":"2026-05-18T01:05:51.662618Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:51.662618Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9402205","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":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lc9TLlIfS4IR1Y3ZWnRO+dTRgU8cCiT9D0xQ36wQdmh7BNBungO21rUvU3WUQi39rTQZwFC5gT2yNE73EeGABg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T03:01:18.943725Z"},"content_sha256":"23ba5eedc0e768d916e1449ec2f7061c8f4128cd9e3c4375cee0469fe9425070","schema_version":"1.0","event_id":"sha256:23ba5eedc0e768d916e1449ec2f7061c8f4128cd9e3c4375cee0469fe9425070"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:5TQTDP4CC4DAW43VLAW3JF2HRH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the structure of tensor products of l_p spaces","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alvaro Arias, Jeff Farmer","submitted_at":"1994-02-08T00:00:00Z","abstract_excerpt":"We examine some structural properties of (injective and projective) tensor products of $\\ell_p$-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals.\n  Our main results are: \\medbreak \\item{(1)} If $1<p<\\infty$, then $B(\\ell_p)\\approx B(L_p)$ ($B(X)$ consists of the bounded linear operators on $X$). \\medbreak \\item{(2)} If ${1\\over p_i}+{1\\over p_j}\\leq1$ for every $i\\neq j$, or if all of the $p_i$'s are equal, then $\\ell_{p_1}\\hat{\\otimes}\\cdots"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9402205","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":""},"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":"2026-05-18T01:05:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HINkWj7nVAKbpJEjyrmZ5uDxqjaLoSBtn3jxJb8loMaJXXSNeJ/kOqyqx5e5lAObyxs04p3TlYZKAe3VCDPcBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T03:01:18.944065Z"},"content_sha256":"76d574a0a5f559fb7c19fc8f2535a89ebaf0dd4c18beff2741af07b56df7910a","schema_version":"1.0","event_id":"sha256:76d574a0a5f559fb7c19fc8f2535a89ebaf0dd4c18beff2741af07b56df7910a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5TQTDP4CC4DAW43VLAW3JF2HRH/bundle.json","state_url":"https://pith.science/pith/5TQTDP4CC4DAW43VLAW3JF2HRH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5TQTDP4CC4DAW43VLAW3JF2HRH/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":"2026-06-29T03:01:18Z","links":{"resolver":"https://pith.science/pith/5TQTDP4CC4DAW43VLAW3JF2HRH","bundle":"https://pith.science/pith/5TQTDP4CC4DAW43VLAW3JF2HRH/bundle.json","state":"https://pith.science/pith/5TQTDP4CC4DAW43VLAW3JF2HRH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5TQTDP4CC4DAW43VLAW3JF2HRH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:5TQTDP4CC4DAW43VLAW3JF2HRH","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":"efccc5945599d9618c2dcae40a0760f73e621fe78a03fd684194f18d27bb303d","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1994-02-08T00:00:00Z","title_canon_sha256":"2da333f8e668da0383e7885eec354d373761e7a0dff8223866bb4eabf3995847"},"schema_version":"1.0","source":{"id":"math/9402205","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9402205","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"arxiv_version","alias_value":"math/9402205v1","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9402205","created_at":"2026-05-18T01:05:51Z"},{"alias_kind":"pith_short_12","alias_value":"5TQTDP4CC4DA","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"5TQTDP4CC4DAW43V","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"5TQTDP4C","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:76d574a0a5f559fb7c19fc8f2535a89ebaf0dd4c18beff2741af07b56df7910a","target":"graph","created_at":"2026-05-18T01:05:51Z","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"},"paper":{"abstract_excerpt":"We examine some structural properties of (injective and projective) tensor products of $\\ell_p$-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals.\n  Our main results are: \\medbreak \\item{(1)} If $1<p<\\infty$, then $B(\\ell_p)\\approx B(L_p)$ ($B(X)$ consists of the bounded linear operators on $X$). \\medbreak \\item{(2)} If ${1\\over p_i}+{1\\over p_j}\\leq1$ for every $i\\neq j$, or if all of the $p_i$'s are equal, then $\\ell_{p_1}\\hat{\\otimes}\\cdots","authors_text":"Alvaro Arias, Jeff Farmer","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1994-02-08T00:00:00Z","title":"On the structure of tensor products of l_p spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9402205","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":"sha256:23ba5eedc0e768d916e1449ec2f7061c8f4128cd9e3c4375cee0469fe9425070","target":"record","created_at":"2026-05-18T01:05:51Z","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":"efccc5945599d9618c2dcae40a0760f73e621fe78a03fd684194f18d27bb303d","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1994-02-08T00:00:00Z","title_canon_sha256":"2da333f8e668da0383e7885eec354d373761e7a0dff8223866bb4eabf3995847"},"schema_version":"1.0","source":{"id":"math/9402205","kind":"arxiv","version":1}},"canonical_sha256":"ece131bf8217060b7375582db4974789dda0aa6c90a58857a41dc0522930a464","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ece131bf8217060b7375582db4974789dda0aa6c90a58857a41dc0522930a464","first_computed_at":"2026-05-18T01:05:51.662618Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:51.662618Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d69ngwqU0WwWZsAMT5zZtFpwhqJxXUkLOVzecHLIEA5KNTDuTXXDcS8bp/AmfwO1GXnyGXaGxEBUzYkvMPl0CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:51.663079Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9402205","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:23ba5eedc0e768d916e1449ec2f7061c8f4128cd9e3c4375cee0469fe9425070","sha256:76d574a0a5f559fb7c19fc8f2535a89ebaf0dd4c18beff2741af07b56df7910a"],"state_sha256":"a9fcb227fa247cebfebd626cc76735a99e2d35b51512c0a7545cdfb8f3f99670"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3TH46iGdfIC001CzoOzOGBKSFUobg9KqEIgqx/GK6qrED0Asxik/8C9dsXdlXK4r3I/ynn8sIolm8s/RFDhpCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T03:01:18.945958Z","bundle_sha256":"fb7f48a7e50c7ca38baa6eabc76624d4f99c47c7a8a6e0535bcc86f09cf0e84d"}}