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Classification of $L^p$ AF algebras. (arXiv:1707.09257v2 [math.OA] UPDATED)
来源于:arXiv
We define spatial $L^p$ AF algebras for $p \in [1, \infty) \setminus \{ 2
\}$, and prove the following analog of the Elliott AF algebra classification
theorem. If $A$ and $B$ are spatial $L^p$ AF algebras, then the following are
equivalent: 1) $A$ and $B$ have isomorphic scaled preordered $K_0$-groups. 2)
$A \cong B$ as rings. 3) $A \cong B$ (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 $L^p$ operator algebras,
and show that there is a unique way to make a spatial $L^p$ AF algebra into a
matrix normed $L^p$ operator algebra. We also show that any countable scaled
Riesz group can be realized as the scaled preordered $K_0$-group of a spatial
$L^p$ AF algebra. 查看全文>>