## Banach Journal of Mathematical Analysis

### On the unit sphere of positive operators

Antonio M. Peralta

#### Abstract

Given a $C^{*}$-algebra $A$, let $S(A^{+})$ denote the set of positive elements in the unit sphere of $A$. Let $H_{1}$, $H_{2}$, $H_{3}$, and $H_{4}$ be complex Hilbert spaces, where $H_{3}$ and $H_{4}$ are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry $\Delta:S(B(H_{1})^{+})\to S(B(H_{2})^{+})$ (resp., $\Delta:S(K(H_{3})^{+})\to S(K(H_{4})^{+})$) admits a unique extension to a surjective complex linear isometry from $B(H_{1})$ onto $B(H_{2})$ (resp., from $K(H_{3})$ onto $K(H_{4})$). 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
Accepted: 21 May 2018
First available in Project Euclid: 30 October 2018

https://projecteuclid.org/euclid.bjma/1540865071

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

Mathematical Reviews number (MathSciNet)
MR3892338

Zentralblatt MATH identifier
07002033

#### 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

#### 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.