• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.249-259
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• 2022

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.

• Journal of the Korean Mathematical Society
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• v.59 no.4
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• pp.789-804
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• 2022

Striking result of Vybíral [51] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vybíral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C^*-algebras and derive the results of Schur and Vybíral. As an application, we state C^*-algebraic version of Novak's conjecture and solve it for commutative unital C^*-algebras. We formulate Pólya-Szegő-Rudin question for the C^*-algebraic Schur product of positive matrices.

• Journal of the Korean Mathematical Society
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• v.45 no.2
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• pp.575-585
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• 2008

Suppose that D is a $C^*$-discrete quantum group and $D_0$ a discrete quantum group associated with D. If there exists a continuous action of D on an operator algebra L(H) so that L(H) becomes a D-module algebra, and if the inner product on the Hilbert space H is D-invariant, there is a unique $C^*$-representation $\theta$ of D associated with the action. The fixed-point subspace under the action of D is a Von Neumann algebra, and furthermore, it is the commutant of $\theta$(D) in L(H).

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.401-415
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• 2020

It is well-known that if char𝕜 = 0, then an Artinian monomial complete intersection quotient 𝕜[x₁, …, x_n]/(x₁^a₁, …, x_n^a_n) has the strong Lefschetz property in the narrow sense, and it is decomposed by the direct sum of irreducible 𝖘𝖑₂-modules. For an Artinian ring A = 𝕜[x₁, x₂, x₃]/(x₁⁶, x₂⁶, x₃⁶), by the Schur-Weyl duality theorem, there exist 56 trivial representations, 70 standard representations, and 20 sign representations inside A. In this paper we find an explicit basis for A, which is compatible with the S₃-module structure.