HE NONCOMMUTATIVE ℓ₁ - ℓ₂ INEQUALITY FOR HILBERT C^*-MODULES AND THE EXACT CONSTANT

Let 𝓐 be a unital C^*-algebra. Then it follows that $\sum\limits_{i=1}^{n}(a_ia^*_i)^{\frac{1}{2}}{\leq}\sqrt{n}\(\sum\limits_{i=1}^{n}a_ia^*_i\)^{\frac{1}{2}}$, ∀n ∈ ℕ, ∀a₁, …, a_n ∈ 𝓐. By modifications of arguments of Botelho-Andrade, Casazza, Cheng, and Tran given in 2019, for certain n-tuple x = (a₁, …, a_n) ∈ 𝓐ⁿ, we give a method to compute a positive element c_x in the C^*-algebra 𝓐 such that the equality $$\sum\limits_{i=1}^{n}(a_ia^*_i)^{\frac{1}{2}}=c_x\sqrt{n}\(\sum\limits_{i=1}^{n}a_ia^*_i\)^{\frac{1}{2}}$$ holds. We give an application for the integral of Kasparov. We also derive a formula for the exact constant for the continuous ℓ₁ - ℓ₂ inequality.