×

On the unit sphere of positive operators. (English) Zbl 07002033

Summary: 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.

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
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
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] J. F. Aarnes, Quasi-states on \(C^*\)-algebras, Trans. Amer. Math. Soc. 149 (1970), no. 2, 601-625. · Zbl 0212.15403
[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. · Zbl 0759.46054
[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. · Zbl 1220.46006
[4] G. G. Ding, The isometric extension problem in the spheres of \(l^p(Γ) (p>1)\) type spaces, Sci. China Ser. A 46 (2003), no. 3, 333-338. · Zbl 1217.46010
[5] G. G. Ding, The representation theorem of onto isometric mappings between two unit spheres of \(l^∞\)-type spaces and the application on isometric extension problem, Sci. China Ser. A 47 (2004), no. 5, 722-729. · Zbl 1093.46007
[6] G. G. Ding, The representation theorem of onto isometric mappings between two unit spheres of \(l^1(Γ)\) type spaces and the application to the isometric extension problem, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 6, 1089-1094. · Zbl 1093.46008
[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. · Zbl 1388.46013
[8] F. J. Fernández-Polo and A. M. Peralta, Tingley’s problem through the facial structure of an atomic \(\text{JBW}^* \)-triple, J. Math. Anal. Appl. 455 (2017), no. 1, 750-760. · Zbl 1387.46015
[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. · Zbl 06843537
[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. · Zbl 1403.46009
[11] F. J. Fernández-Polo and A. M. Peralta, Partial isometries: A survey, Adv. Oper. Theory 3 (2018), no. 1, 75-116. · Zbl 1386.46042
[12] F. J. Fernández-Polo and A. M. Peralta, Low rank compact operators and Tingley’s problem, Adv. Math. 338 (2018), 1-40. · Zbl 1491.46004
[13] J. M. Isidro and Á. Rodríguez-Palacios, Isometries of \(\text{JB} \)-algebras, Manuscripta Math. 86 (1995), no. 3, 337-348. · Zbl 0834.17048
[14] R. V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325-338. · Zbl 0045.06201
[15] P. Mankiewicz, On extension of isometries in normed linear spaces, Bull. Pol. Acad. Sci. Math. 20 (1972), 367-371. · Zbl 0234.46019
[16] L. Molnár and G. Nagy, Isometries and relative entropy preserving maps on density operators, Linear Multilinear Algebra 60 (2012), no. 1, 93-108. · Zbl 1241.47034
[17] L. Molnár and W. Timmermann, Isometries of quantum states, J. Phys. A 36 (2003), no. 1, 267-273. · Zbl 1047.81017
[18] M. Mori, Tingley’s problem through the facial structure of operator algebras, J. Math. Anal. Appl. 466 (2018), no. 2, 1281-1298. · Zbl 1411.46007
[19] G. Nagy, Isometries on positive operators of unit norm, Publ. Math. Debrecen 82 (2013), no. 1, 183-192. · Zbl 1299.47072
[20] G. Nagy, Isometries of spaces of normalized positive operators under the operator norm, Publ. Math. Debrecen 92 (2018), no. 1-2, 243-254. · Zbl 1399.47110
[21] G. K. Pedersen, \(C^*\)-Algebras and Their Automorphism Groups, London Math. Soc. Monogr 14, Academic Press, London, 1979. · Zbl 0416.46043
[22] A. M. Peralta, Characterizing projections among positive operators in the unit sphere, Adv. Oper. Theory 3 (2018), no. 3, 731-744. · Zbl 06902464
[23] A. M. Peralta, A survey on Tingley’s problem for operator algebras, Acta Sci. Math. (Szeged) 84 (2018), no. 1-2, 81-123. · Zbl 1413.47065
[24] A. M. Peralta and R. Tanaka, A solution to Tingley’s problem for isometries between the unit spheres of compact \(C^*\)-algebras and \(\text{JB}^* \)-triples, to appear in Sci. China Math., preprint, arXiv:1608.06327v2 [mathFA]. · Zbl 07033414
[25] S. Sakai, \(C^*\)-Algebras and \(W^*\)-Algebras, Ergeb. Math. Grenzgeb. (3) 60, Springer, New York, 1971. · Zbl 0219.46042
[26] D. N. Tan, Extension of isometries on unit sphere of \(L^∞\), Taiwanese J. Math. 15 (2011), no. 2, 819-827. · Zbl 1244.46003
[27] D. N. Tan, On extension of isometries on the unit spheres of \(L^p\)-spaces for \(0<p≤1\), Nonlinear Anal. 74 (2011), no. 18, 6981-6987. · Zbl 1235.46005
[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. · Zbl 1271.46011
[29] R. Tanaka, A further property of spherical isometries, Bull. Aust. Math. Soc. 90 (2014), no. 2, 304-310. · Zbl 1312.46021
[30] R. Tanaka, Spherical isometries of finite dimensional \(C^*\)-algebras, J. Math. Anal. Appl. 445 (2017), no. 1, 337-341. · Zbl 1371.46008
[31] R. Tanaka, Tingley’s problem on finite von Neumann algebras, J. Math. Anal. Appl. 451 (2017), no. 1, 319-326. · Zbl 1371.46009
[32] R. S. Wang, Isometries between the unit spheres of \(C_0(Ω)\) type spaces, Acta Math. Sci. (Engl. Ed.) 14 (1994), no. 1, 82-89. · Zbl 0817.46027
[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. · Zbl 0384.46041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.