Young's Inequality in Semifinite von Neumann Algebras

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This paper formulates Young-type inequalities for singular values (or s-numbers) and traces in the context of von Neumann algebras. In particular, it shown that if (·) is a faithful semifinite normal trace on a semifinite von Neumann algebra M and if p and q are positive real numbers for which p⁻¹+q⁻¹=1, then, for all positive operators a,b∈ M, (|ab|)≤ p⁻¹(aᵖ)+ q⁻¹(bq), with equality holding (in the cases where p⁻¹(aᵖ)+ q⁻¹(bq)<∞) if and only if bq=aᵖ.