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Abstract. Corrected trapezoidal rules are proved for b a f (x) dx under the assumption that f ∈ L p ([a,b]) for some 1 p ∞ . Such quadrature rules involve the trapezoidal rule modified by the addition of a term k[ f (a) − f (b)] . The coefficient k in the quadrature formula is found that minimizes the error estimates. It is shown that when f is merely assumed to be continuous then the optimal rule is the trapezoidal rule itself. In this case error estimates are in terms of the Alexiewicz norm. This includes the case when f is integrable in the Henstock-Kurzweil sense or as a distribution. All error estimates are shown to be sharp for the given assumptions on f . It is shown how to make these formulas exact for all cubic polynomials f . Composite formulas are computed for uniform partitions.Mathematics subject classification (2010): Primary 26D15, 41A55, 65D30. Secondary 26A39, 46F10.

Let Γ be a discrete group. We show that if Γ is nonamenable, then the algebraic tensor products C * r (Γ) ⊗ C * r (Γ) and C * (Γ) ⊗ C * r (Γ) do not admit unique C * -norms. Moreover, when Γ1 and Γ2 are discrete groups containing copies of noncommutative free groups, then C * r (Γ1) ⊗ C * r (Γ2) and C * (Γ1) ⊗ C * r (Γ2) admit 2 ℵ 0 C * -norms. Analogues of these results continue to hold when these familiar group C * -algebras are replaced by appropriate intermediate group C * -algebras.

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