{
  "annotation_id": "39ca9187-592c-485b-b80c-1b666af793f6",
  "date_created": "2026-02-17T05:53:08.617000Z",
  "date_modified": "2026-02-17T05:53:08.617000Z",
  "file_hash": "bf25ebfa08a137e5e153932c20ecdfddb8b63fd5f383d2bb8b02f44bb0dc4cdc",
  "private": false,
  "record": {
    "abstract": "We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family $(A_i)_{i\\in I}$ of Banach algebras and a Banach sequence lattice $E$ on~$I$, the $E$-sum $\\bigl(\\bigoplus_{i\\in I} A_i\\bigr)_{\\!E}$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that $C_E := \\sup\\{\\|\\chi_F\\|_E : F \\subseteq I \\text{ finite}\\} \u003c \\infty$, we prove that this $E$-sum is amenable if and only if the amenability constants of the summands are uniformly bounded, and we establish the two-sided estimate \\[ \\sup_{i\\in I}\\operatorname{AM}(A_i) \\;\\le\\; \\operatorname{AM}\\Bigl(\\bigl(\\textstyle\\bigoplus_{i\\in I} A_i\\bigr)_{\\!E}\\Bigr) \\;\\le\\; C_E^2 \\sup_{i\\in I}\\operatorname{AM}(A_i). \\] We show that the factor $C_E^2$ is sharp by exhibiting finite-dimensional examples where equality holds. We further prove that finiteness of $C_E$ is necessary whenever infinitely many summands are non-zero and the sum admits a bounded approximate identity.\n  Finally, we investigate weak amenability of $E$-sums. We prove that weak amenability passes to summands, that $E$-sums of commutative weakly amenable algebras are weakly amenable, and contrasting sharply with the Johnson amenability picture that for $1 \u003c p \u003c \\infty$, the $\\ell_p$-sum of infinitely many copies of a non-commutative weakly amenable algebra fails to be weakly amenable. In the $c_0$-type regime ($C_E \u003c \\infty$), we establish a two-sided estimate for weak amenability constants analogous to that for Johnson amenability.",
    "arxiv_id": "2601.06680",
    "authors": [
      "Tomasz Kania",
      "Jerzy K\u0105kol"
    ],
    "categories": [
      "math.FA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Amenability constants for unconditional sums of Banach algebras",
    "url": "https://arxiv.org/abs/2601.06680",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "972904fe-5784-4677-94e1-b46293dc75e6",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}