Transformations Preserving Norms of Means of Positive Operators and Nonnegative Functions

Abstract

Abstract. Motivated by recent investigations on norm-additive and spectrally multiplicative maps on various spaces of functions, in this paper we determine all bijective transformations between the positive cones of standard operator algebras over a Hilbert space which preserve a given symmetric norm of a given mean of elements. A result of similar spirit is also presented concerning transformations between cones of nonnegative elements of certain algebras of continuous functions.Mathematics Subject Classification (2010). Primary: 47B49. Secondary: 47A64, 26E60.

Maps on positive operators preserving Rényi type relative entropies and maximal f-divergences

“…By the last displayed equality, the trace preserving property implies that φ leaves the quantity Tr Aσ h f B invariant. Then it is straightforward to see that φ fulfills the conditions of [17,Theorem 2] and the application of that result yields the statement of Theorem 2. Apparently, if f (1) = 0, then we have…”

On some isometries and other preservers

“…Determine the structure of transformations which preserve norms of quasi-arithmetic means. This question has not been answered yet for general quasi-arithmetic means, but for the corresponding results on Kubo-Ando means we refer to the papers [66,71]. In this chapter, we solve Problem B under certain quite general conditions.…”

“…Namely, in [69] Nagy completely described the structure of norm-additive maps on positive Schatten p-class operators with respect to the Schatten p-norm. In [66] among others Molnár and Szokol managed to determine the structure of maps of the same kind on the positive cone of a standard operator algebra (by what we mean a subalgebra of the algebra of all bounded linear operators containing the finite rank elements). Motivated by the aforementioned results, we consider this problem in the setting of C * -algebra equipped with a faithful normalized trace τ.…”

“…In the forthcoming lemma, we present a characterization of the usual order on A ++ what we shall also need (cf. [66,Lemma]…”

“…Conversely, if the latter inequality holds for all X ∈ P d , then we deduce that it is also satisfied by any element X ∈ H + d and, in particular, by each matrix of the form tQ with some rank-one projection Q and scalar t ≥ 0. According to [66,Lemma] we have A ≤ B.…”