{
  "version": 1,
  "title": "Stable ranges: the full algebra and the correct signs",
  "scope": "An exact local roof example and algebraic generator maps; not a picture of a finite ergodic base.",
  "deck": {
    "roof_at_omega": "3/2",
    "roof_at_Tomega": "2",
    "height": "9/4",
    "height_after_deck": "3/4",
    "cover_before": 1,
    "cover_after": 0,
    "proof": "../../OA-FLOW-SR.html#sr-cover; SR8-SR10; Diagnostic2"
  },
  "sign_test": {
    "u": "13/100",
    "s": "1/5",
    "t": "2/5",
    "wrong_lhs": 1,
    "wrong_rhs": 0,
    "correct_lhs": 1,
    "correct_rhs": 1,
    "wrong_exponent": "floor(u+2t)-floor(u+t)",
    "correct_exponent": "-floor(u-t)",
    "proof": "../../OA-FLOW-SR.html#sr-cross; AC10; Diagnostic3"
  },
  "crossed_product_maps": {
    "integer_generator": "d_1 -> D_1; D_1 delta_m = delta_(m-1)",
    "real_perturbed_generator": "lambda_t^gamma -> U_t lambda_t^0",
    "final_coordinate": "U_t(z)=S_(n(t,S_(-t)z))",
    "full_center": "Z(R)=Z(Q_0)=L-infinity(Omega); chi restricts to f -> f composed with T^(-1)",
    "coarse_type": "P, Q, Q_0 and R have the same homogeneous coarse type I, II or III; this assertion does not include R crossed_chi Z. Finer subtypes are not asserted.",
    "normal_inverse": "Explicit regular coordinate unitaries, identity-multiplicity compression, and cocycle multiplication",
    "proof": "../../OA-FLOW-SR.html; Sections 3, 5, 6 and 7 (sr-iw, sr-cross, sr-center, sr-type), AC3, AC6-AC7, SR19-SR33."
  },
  "haar": {
    "group": "positive affine group (a,b)(a',b')=(aa',b+ab')",
    "left_measure": "a^(-2) da db",
    "modular_function": "Delta(a,b)=a^(-1)",
    "right_translate": "(a,b)(exp(t),0)=(a exp(t),b)",
    "squared_norm_before_correction": "exp(t)",
    "unitary_multiplier": "exp(-t/2)",
    "algebra_multiplier": "1",
    "proof": "../../OA-FLOW-SR.html#sr-models; SR39-SR41; Diagnostic4"
  },
  "human_source": "Masamichi Takesaki, Theory of Operator Algebras II, XII.3.4-3.5, printed pp.389-392; https://doi.org/10.1007/978-3-662-10451-4.",
  "terms": "Original diagram, data and renderer: CC0-1.0. DejaVu and STIX font components: see FONT-LICENSE.txt."
}