Classification of Lᵖ AF algebras

الأصل · EN

We define spatial Lᵖ AF algebras for p ∈ [1, ∞) { 2 }, and prove the following analog of the Elliott AF algebra classification theorem. If A and B are spatial Lᵖ AF algebras, then the following are equivalent: 1) A and B have isomorphic scaled preordered K₀-groups. 2) A B as rings. 3) A B (not necessarily isometrically) as Banach algebras. 4) A is isometrically isomorphic to B as Banach algebras. 5) A is completely isometrically isomorphic to B as matrix normed Banach algebra. As background, we develop the theory of matrix normed Lᵖ operator algebras, and show that there is a unique way to make a spatial Lᵖ AF algebra into a matrix normed Lᵖ operator algebra. We also show that any countable scaled Riesz group can be realized as the scaled preordered K₀-group of a spatial Lᵖ AF algebra.