\(K\)-theory for the Banach algebra of operators on Jame's quasi-reflexive Banach spaces (Q5951574)

Let \({\mathcal B}({\mathcal X})\) denote a Banach algebra of all continuous linear operators on a Banach space \({\mathcal X}.\) It is known that the \(K_0({\mathcal B}({\mathcal X}))\) vanishes for most ``classical'' Banach spaces, including \(l_p,\) \(L_p(X),\) \(c_o,\) and \(C(Y),\) where \(Y\) is an infinite compact metric spase. On the other hand, for each natural number \(m,\) there is a Banach space \({\mathcal X}\) such that \(K_0({\mathcal B}({\mathcal X})) \cong {\mathbb Z}^{m},\) and there are examples of Banach spaces \({\mathcal X}\) such that \(K_0({\mathcal B}({\mathcal X}))\) has torsion [see \textit{N. J. Laustsen}, J. Lond. Math. Soc., II. Ser. 59, 715-728 (1999; Zbl 0922.47037)]. The main purpose of this paper is to prove that \(K_0({\mathcal B}(J_p))\cong {\mathbb Z}\) and \(K_1({\mathcal B}(J_p))=0\) for \(p\)-th James spaces \(J_p\) (\(1< p < \infty \)). This enables to calculate the \(K\)-groups of \({\mathcal B}({\mathcal X})\) for each Banach space \({\mathcal X}\) which is a direct sum of finitely many James spaces and \(l_p\)-spaces: \(K_0({\mathcal B}({\mathcal X})\cong {\mathbb Z}^{m}\) and \(K_1({\mathcal B}({\mathcal X})\cong {\mathbb Z}^{m+n-1}\) for the Banach space \[ {\mathcal X}=J_{p_1} \oplus J_{p_2}\oplus \dots \oplus J_{p_m} \oplus l_{p_{m+1}} \oplus l_{p_{m+2}} \oplus \dots \oplus l_{p_{m+n}} \] where \(m\) and \(n\) are are natural numbers and \(p_{1},p_{2},\dots,p_{m+n}\) are distinct real numbers with \(p_1,\dots,p_{m}>1\) and \(p_{m+1}\dots, p_{m+n}\geq 1.\)