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 𝒮 r ( ℬ ) of all matrices A = [ a j k ] over ℬ such that ϕ [ A [ r ] ] : = [ ϕ ( | a j k | r ) ] defines a bounded operator from ℓ p to ℓ q 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 𝒮 r ( ℬ ) setting.