For each triple of positive numbers p,q,r≥1 and each
commutative C*-algebra ℬ with identity 1 and the
set s(ℬ) of states on ℬ, the set 𝒮r(ℬ) of all matrices A=[ajk] over ℬ such that ϕ[A[r]]:=[ϕ(|ajk|r)] defines a bounded operator from ℓp to
ℓq for all ϕ∈s(ℬ) is shown to be a Banach
algebra under the Schur product operation, and the norm ‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}.
Schatten's theorems about the dual of the compact
operators, the trace-class operators, and the decomposition of the
dual of the algebra of all bounded operators on a Hilbert space
are extended to the 𝒮r(ℬ) setting.