The Bishop-Phelps-Bollobás property for operators on \(\mathcal{C}(K)\)
This paper starts with an overview of known results that lead to the Bishop-Phelps-Bollobás property for operators. Consider \(X\) and \(Y\), Banach spaces, and \(\mathcal{L}(X,Y)\), the space of bounded operators from \(X\) into \(Y\). The unit sphere in \(X\) is denoted by \(S_X\) and the set of all norm 1 operators in \(\mathcal{L}(X,Y)\) is denoted by \(\mathcal{S}(X,Y)\). The pair \((X,Y)\) has the BPBp if for every \(\epsilon >0,\) there exists \(\eta (\epsilon)>0\) such that for every \(T\in \mathcal{S}(X,Y),\) and if \(x_0 \in S_X\) is such that \(\|Tx_0\|>1-\eta (\epsilon)\), then there exists a \(u_0 \in S_X\) and \(S\in \mathcal{S}(X,Y)\) with \(\|S(u_0)\|=1\), \(\|T-S\|<\epsilon\) and \(\|u_0-x_0\|<\epsilon\). In the introduction, the author also gives a list of pairs of Banach spaces that satisfy the Bishop-Phelps-Bollobás property for operators which motivates well the main theorem in the paper. Section 2 starts with the definition of \(\mathbb{C}\)-modulus of convexity of a complex Banach space \(Y\), given \(\epsilon>0\), \[ \delta(\epsilon) =\inf \{ \sup \{\|x+\lambda \epsilon y\|-1: \, \lambda \in \mathbb{C}, \, |\lambda|=1\}: x,y \in S_X\}. \] The Banach space \(Y\) is \(\mathbb{C}\)-uniformly convex if \(\delta (\epsilon)>0\) for every \(\epsilon >0\). This section is devoted to the proof of the following theorem. Theorem (cf. Theorem 2.4). The pair \((C_0(L), Y)\) satisfies the Bishop-Phelps-Bollobás property for operators for any locally compact Hausdorff topological space \(L\) and any \(\mathbb{C}\)-uniformly convex space \(Y\). Moreover, the function \(\eta\) appearing in the definition of BPBp depends only on the \(\mathbb{C}\)-modulus of convexity of \(Y\). Since the complex spaces \(L_p (\mu)\) (\(1\leq p<\infty\)) are \(\mathbb{C}\)-uniformly convex the author obtains the following result. Corollary (cf. Corollary 2.5). In the complex case, the pair \((C_0(L), L_p (\mu))\) (\(1\leq p<\infty\)) satisfies the Bishop-Phelps-Bollobás property for operators for any locally compact Hausdorff topological space \(L\) and every positive measure \(\mu\). The paper ends with an open problem. It is not known whether the pair \((c_0, \ell_1)\) has the BPBp for operators.
scientific article; zbMATH DE number 6575513
scientific article