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Transformations preserving norms of means of positive operators and nonnegative functions. (English) Zbl 1354.47026
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.

##### MSC:
 47B49 Transformers, preservers (linear operators on spaces of linear operators) 47A64 Operator means involving linear operators, shorted linear operators, etc. 26E60 Means
##### Keywords:
preservers; operator means; means of functions; symmetric norms
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##### References:
  Abatzoglou, T.J., Norm derivatives on spaces of operators, Math. Ann., 239, 129-135, (1979) · Zbl 0398.47013  Busch, P.; Gudder, S.P., Effects as functions on projective Hilbert spaces, Lett. Math. Phys., 47, 329-337, (1999) · Zbl 0932.47055  Chan, J.T.; Li, C.K.; Tu, C.C.N., A class of unitarily invariant norms on $$B$$($$H$$), Proc. Am. Math. Soc., 129, 1065-1076, (2001) · Zbl 0963.47013  Hatori, O.; Jiménez-Vargas, A.; Villegas-Vallecillos, M., Maps which preserve norms of non-symmetrical quotients between groups of exponentials of Lipschitz functions, J. Math. Anal. Appl., 415, 825-845, (2014) · Zbl 1325.46011  Hatori, O.; Hirasawa, G.; Miura, T., Additively spectral-radius preserving surjections between unital semisimple commutative Banach algebras, Cent. Eur. J. Math., 8, 597-601, (2010) · Zbl 1211.46052  Hatori, O., Lambert, S., Luttman, A., Miura, T., Tonev, T., Yates, R.: Spectral preservers in commutative Banach algebras. In: Jarosz, K. (ed.) Function Spaces in Modern Analysis. Contemporary Mathematics, vol. 547, pp. 103-124. American Mathematical Society, Providence (2011) · Zbl 1239.46036  Hosseini, M.; Font, J.J., Norm-additive in modulus maps between function algebras, Banach J. Math. Anal., 8, 79-92, (2014) · Zbl 1450.46035  Hosseini, M.; Sady, F., Maps between Banach function algebras satisfying certain norm conditions, Cent. Eur. J. Math., 11, 1020-1033, (2013) · Zbl 1275.46035  Jiménez-Vargas, A.; Lee, K.; Luttman, A.; Villegas-Vallecillos, M., Generalized weak peripheral multiplicativity in algebras of Lipschitz functions, Cent. Eur. J. Math., 11, 1197-1211, (2013) · Zbl 1298.46043  Kubo, F.; Ando, T., Means of positive linear operators, Math. Ann., 246, 205-224, (1980) · Zbl 0412.47013  Lee, K., Characterizations of peripherally multiplicative mappings between real function algebras, Publ. Math. Debr., 84, 379-397, (2014) · Zbl 1340.46041  Molnár, L., Order-automorphisms of the set of bounded observables, J. Math. Phys., 42, 5904-5909, (2001) · Zbl 1019.81005  Molnár, L., Some characterizations of the automorphisms of $$B$$($$H$$) and $$C$$($$X$$), Proc. Am. Math. Soc., 130, 111-120, (2002) · Zbl 0983.47024  Molnár L.: Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces. Lecture Notes in Mathematics, vol. 1895. Springer, Berlin (2007) · Zbl 1119.47001  Molnár, L., Maps preserving the geometric Mean of positive operators, Proc. Am. Math. Soc., 137, 1763-1770, (2009) · Zbl 1183.47032  Molnár, L., Maps preserving the harmonic Mean or the parallel sum of positive operators, Linear Algebra Appl., 430, 3058-3065, (2009) · Zbl 1182.47035  Molnár, L., Maps preserving general means of positive operators, Electron. J. Linear Algebra, 22, 864-874, (2011) · Zbl 1276.47049  Molnár, L.: General Mazur-Ulam type theorems and some applications. Operator Theory: Advances and Applications (to appear) · Zbl 1325.46011  Nagy, G.: Preservers for the $$p$$-norm of linear combinations of positive operators. Abstr. Appl. Anal. 2014, 434121-1-434121-9 (2014) · Zbl 07022383  Rao, N.V., Tonev, T.V., Toneva, E.T.: Uniform algebra isomorphisms and peripheral spectra. In: Malios A., Haralampidou, M. (eds.) Topological Algebras. Contemporary Mathematics, vol. 427, pp. 401-416 (2007) · Zbl 1123.46035  Šemrl, P., Comparability preserving maps on bounded observables, Integral Equ. Oper. Theory, 62, 441-454, (2008) · Zbl 1191.47050  Šemrl, P., Symmetries on bounded observables: a unified approach based on adjacency preserving maps, Integral Equ. Oper. Theory, 72, 7-66, (2012) · Zbl 1246.81053  Tonev, T., Spectral conditions for almost composition operators between algebras of functions, Proc. Am. Math. Soc., 142, 2721-2732, (2014) · Zbl 1325.46052  Tonev, T.; Yates, R., Norm-linear and norm-additive operators between uniform algebras, J. Math. Anal. Appl., 357, 45-53, (2009) · Zbl 1171.47032
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