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Banach spaces whose algebra of bounded operators has the integers as their \(K_0\)-group. (English) Zbl 1332.46026

Summary: Let \(X\) and \(Y\) be Banach spaces such that the ideal of operators which factor through \(Y\) has codimension one in the Banach algebra \(\mathcal{B}(X)\) of all bounded operators on \(X\), and suppose that \(Y\) contains a complemented subspace which is isomorphic to \(Y \oplus Y\) and that \(X\) is isomorphic to \(X \oplus Z\) for every complemented subspace \(Z\) of \(Y\). Then the \(K_0\)-group of \(\mathcal{B}(X)\) is isomorphic to the additive group \(\mathbb{Z}\) of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that \(K_0(\mathcal{B}(C([0, \omega_1]))) \cong \mathbb{Z}\), where \(C([0, \omega_1])\) denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal \(\omega_1\), endowed with the order topology.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
19K14 \(K_0\) as an ordered group, traces
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