Transformations preserving norms of means of positive operators and nonnegative functions
Let \(H\) be a complex Hilbert space and \(B(H)\) the algebra of all bounded linear operators on \(H\). Denote by \(\mathcal A_+\) the cone of all positive operators in a subalgebra \(\mathcal{A}\) of \(B(H)\) which contains all finite rank operators, i.e., \(\mathcal A\) is a standard operator algebra on \(H\). Let \(1\leq p<\infty\) be a given number and denote by Tr the usual trace functional. The symbol \(C_p(H)\) stands for the set of all operators \(A\in B(H)\) for which \(\operatorname{Tr}| A|^p<\infty\). As for the case where \(p\) is infinite, \(C_\infty(H)\) denotes \(B(H)\) equipped with the usual operator norm \(\| \cdot\|\). In the paper under review, the authors first describe the structure of all bijective transformations between the positive cones of standard operator algebras on \(H\) which preserve a given symmetric norm of a given (Kubo-Ando) mean of their elements. They present their result in two theorems, one of them is as follows. Let \(\dim H>1\) and \(1<p\leq\infty\). Consider standard operator algebras \(\mathcal A\) and \(\mathcal B\) on \(H\) which are contained in \(C_p(H)\). Let \(\phi:\mathcal A_+\rightarrow\mathcal B_+\) be a bijective transformation such that, for a given \(0<\lambda<1\), \[ \|\lambda\phi(A)+(1-\lambda)\phi(B)\|_p=\|\lambda A+(1-\lambda)B\|_p \] holds for all \(A,\) \(B\in\mathcal A_+\). Then there exists a unitary or antiunitary operator \(U:H\rightarrow H\) such that \[ \phi(A)=UAU^\ast,\quad A\in\mathcal A_+. \] After presenting their result concerning means on operator algebras, the authors turn to the discussion of the problem concerning means on algebras of continuous functions. With the third and final theorem, which is of a similar spirit as the above result, the authors describe the structure of bijective transformations between cones of nonnegative elements of certain algebras of continuous functions.
scientific article; zbMATH DE number 6499542
scientific article