Banach Journal of Mathematical Analysis

On the unit sphere of positive operators

Antonio M. Peralta

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Abstract

Given a C-algebra A, let S(A+) denote the set of positive elements in the unit sphere of A. Let H1, H2, H3, and H4 be complex Hilbert spaces, where H3 and H4 are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry Δ:S(B(H1)+)S(B(H2)+) (resp., Δ:S(K(H3)+)S(K(H4)+)) admits a unique extension to a surjective complex linear isometry from B(H1) onto B(H2) (resp., from K(H3) onto K(H4)). This provides a positive answer to a conjecture recently posed by Nagy.

Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 91-112.

Dates
Received: 13 April 2018
Accepted: 21 May 2018
First available in Project Euclid: 30 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1540865071

Digital Object Identifier
doi:10.1215/17358787-2018-0017

Mathematical Reviews number (MathSciNet)
MR3892338

Zentralblatt MATH identifier
07002033

Subjects
Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 46A22: Theorems of Hahn-Banach type; extension and lifting of functionals and operators [See also 46M10] 46B20: Geometry and structure of normed linear spaces 46B04: Isometric theory of Banach spaces 46A16: Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.) 46E40: Spaces of vector- and operator-valued functions

Keywords
Tingley’s problem extension of isometries isometries positive operators operator norm

Citation

Peralta, Antonio M. On the unit sphere of positive operators. Banach J. Math. Anal. 13 (2019), no. 1, 91--112. doi:10.1215/17358787-2018-0017. https://projecteuclid.org/euclid.bjma/1540865071


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References

  • [1] J. F. Aarnes, Quasi-states on $C^{*}$-algebras, Trans. Amer. Math. Soc. 149 (1970), no. 2, 601–625.
  • [2] L. J. Bunce and J. D. M. Wright, The Mackey-Gleason problem, Bull. Amer. Math. Soc. (N.S.) 26 (1992), no. 2, 288–293.
  • [3] L. Cheng and Y. Dong, On a generalized Mazur-Ulam question: Extension of isometries between unit spheres of Banach spaces, J. Math. Anal. Appl. 377 (2011), no. 2, 464–470.
  • [4] G. G. Ding, The isometric extension problem in the spheres of $l^{p}(\Gamma)$ ($p>1$) type spaces, Sci. China Ser. A 46 (2003), no. 3, 333–338.
  • [5] G. G. Ding, The representation theorem of onto isometric mappings between two unit spheres of $l^{\infty}$-type spaces and the application on isometric extension problem, Sci. China Ser. A 47 (2004), no. 5, 722–729.
  • [6] G. G. Ding, The representation theorem of onto isometric mappings between two unit spheres of $l^{1}(\Gamma)$ type spaces and the application to the isometric extension problem, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 6, 1089–1094.
  • [7] F. J. Fernández-Polo, J. J. Garcés, A. M. Peralta, and I. Villanueva, Tingley’s problem for spaces of trace class operators, Linear Algebra Appl. 529 (2017), 294–323.
  • [8] F. J. Fernández-Polo and A. M. Peralta, Tingley’s problem through the facial structure of an atomic $\mathrm{JBW}^{*}$-triple, J. Math. Anal. Appl. 455 (2017), no. 1, 750–760.
  • [9] F.J. Fernández-Polo and A. M. Peralta, On the extension of isometries between the unit spheres of a $C^{*}$-algebra and $B(H)$, Trans. Amer. Math. Soc. Ser. B 5 (2018), 63–80.
  • [10] F. J. Fernández-Polo and A. M. Peralta, On the extension of isometries between the unit spheres of von Neumann algebras, J. Math. Anal. Appl. 466 (2018), no. 1, 127–143.
  • [11] F. J. Fernández-Polo and A. M. Peralta, Partial isometries: A survey, Adv. Oper. Theory 3 (2018), no. 1, 75–116.
  • [12] F. J. Fernández-Polo and A. M. Peralta, Low rank compact operators and Tingley’s problem, Adv. Math. 338 (2018), 1–40.
  • [13] J. M. Isidro and Á. Rodríguez-Palacios, Isometries of $\mathrm{JB}$-algebras, Manuscripta Math. 86 (1995), no. 3, 337–348.
  • [14] R. V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325–338.
  • [15] P. Mankiewicz, On extension of isometries in normed linear spaces, Bull. Pol. Acad. Sci. Math. 20 (1972), 367–371.
  • [16] L. Molnár and G. Nagy, Isometries and relative entropy preserving maps on density operators, Linear Multilinear Algebra 60 (2012), no. 1, 93–108.
  • [17] L. Molnár and W. Timmermann, Isometries of quantum states, J. Phys. A 36 (2003), no. 1, 267–273.
  • [18] M. Mori, Tingley’s problem through the facial structure of operator algebras, J. Math. Anal. Appl. 466 (2018), no. 2, 1281–1298.
  • [19] G. Nagy, Isometries on positive operators of unit norm, Publ. Math. Debrecen 82 (2013), no. 1, 183–192.
  • [20] G. Nagy, Isometries of spaces of normalized positive operators under the operator norm, Publ. Math. Debrecen 92 (2018), no. 1–2, 243–254.
  • [21] G. K. Pedersen, $C^{*}$-Algebras and Their Automorphism Groups, London Math. Soc. Monogr 14, Academic Press, London, 1979.
  • [22] A. M. Peralta, Characterizing projections among positive operators in the unit sphere, Adv. Oper. Theory 3 (2018), no. 3, 731–744.
  • [23] A. M. Peralta, A survey on Tingley’s problem for operator algebras, Acta Sci. Math. (Szeged) 84 (2018), no. 1–2, 81–123.
  • [24] A. M. Peralta and R. Tanaka, A solution to Tingley’s problem for isometries between the unit spheres of compact $C^{*}$-algebras and $\mathrm{JB}^{*}$-triples, to appear in Sci. China Math., preprint, arXiv:1608.06327v2 [mathFA].
  • [25] S. Sakai, $C^{*}$-Algebras and $W^{*}$-Algebras, Ergeb. Math. Grenzgeb. (3) 60, Springer, New York, 1971.
  • [26] D. N. Tan, Extension of isometries on unit sphere of $L^{\infty}$, Taiwanese J. Math. 15 (2011), no. 2, 819–827.
  • [27] D. N. Tan, On extension of isometries on the unit spheres of $L^{p}$-spaces for $0<p\leq1$, Nonlinear Anal. 74 (2011), no. 18, 6981–6987.
  • [28] D. N. Tan, Extension of isometries on the unit sphere of $L^{p}$ spaces, Acta Math. Sin. (Engl. Ser.) 28 (2012), no. 6, 1197–1208.
  • [29] R. Tanaka, A further property of spherical isometries, Bull. Aust. Math. Soc. 90 (2014), no. 2, 304–310.
  • [30] R. Tanaka, Spherical isometries of finite dimensional $C^{*}$-algebras, J. Math. Anal. Appl. 445 (2017), no. 1, 337–341.
  • [31] R. Tanaka, Tingley’s problem on finite von Neumann algebras, J. Math. Anal. Appl. 451 (2017), no. 1, 319–326.
  • [32] R. S. Wang, Isometries between the unit spheres of $C_{0}(\Omega)$ type spaces, Acta Math. Sci. (Engl. Ed.) 14 (1994), no. 1, 82–89.
  • [33] J. D. M. Wright and M. A. Youngson, On isometries of Jordan algebras, J. Lond. Math. Soc. (2) 17 (1978), no. 2, 339–344.