Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero
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- by Chi-Wai Leung, Chi-Keung Ng and Ngai-Ching Wong PDF
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Abstract:
Let $A$ be a $C^*$-algebra with real rank zero. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $\theta : E\to F$ is a linear map preserving orthogonality, i.e., \[ \langle \theta (x),\theta (y)\rangle \ =\ 0\quad \text {whenever}\quad \langle x,y\rangle \ =\ 0. \] We show in this article that if $\theta$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $u\in M(A)$ such that \[ \langle \theta (x), \theta (y)\rangle \ =\ u \langle x, y\rangle \qquad (x,y\in E). \] In the case when $A$ is a standard $C^*$-algebra, when $A$ is a real rank zero properly infinite unital $C^*$-algebra, or when $A$ is a $W^*$-algebra, we also get the same conclusion with the assumption of $\theta$ being an $A$-module map weakened to being a local map.References
- J. Alaminos, M. Brešar, M. Černe, J. Extremera, and A. R. Villena, Zero product preserving maps on $C^1[0,1]$, J. Math. Anal. Appl. 347 (2008), no. 2, 472–481. MR 2440343, DOI 10.1016/j.jmaa.2008.06.037
- Jesús Araujo, Linear biseparating maps between spaces of vector-valued differentiable functions and automatic continuity, Adv. Math. 187 (2004), no. 2, 488–520. MR 2078345, DOI 10.1016/j.aim.2003.09.007
- A. Blanco and A. Turnšek, On maps that preserve orthogonality in normed spaces, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 4, 709–716. MR 2250441, DOI 10.1017/S0308210500004674
- Lawrence G. Brown, Semicontinuity and multipliers of $C^*$-algebras, Canad. J. Math. 40 (1988), no. 4, 865–988. MR 969204, DOI 10.4153/CJM-1988-038-5
- Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
- Jacek Chmieliński, Linear mappings approximately preserving orthogonality, J. Math. Anal. Appl. 304 (2005), no. 1, 158–169. MR 2124655, DOI 10.1016/j.jmaa.2004.09.011
- Michael Frank, Alexander S. Mishchenko, and Alexander A. Pavlov, Orthogonality-preserving, $C^*$-conformal and conformal module mappings on Hilbert $C^*$-modules, J. Funct. Anal. 260 (2011), no. 2, 327–339. MR 2737403, DOI 10.1016/j.jfa.2010.10.009
- Stanisław Goldstein and Adam Paszkiewicz, Linear combinations of projections in von Neumann algebras, Proc. Amer. Math. Soc. 116 (1992), no. 1, 175–183. MR 1094501, DOI 10.1090/S0002-9939-1992-1094501-9
- Dijana Ilišević and Aleksej Turnšek, Approximately orthogonality preserving mappings on $C^*$-modules, J. Math. Anal. Appl. 341 (2008), no. 1, 298–308. MR 2394085, DOI 10.1016/j.jmaa.2007.10.028
- Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. I, Graduate Studies in Mathematics, vol. 15, American Mathematical Society, Providence, RI, 1997. Elementary theory; Reprint of the 1983 original. MR 1468229, DOI 10.1090/gsm/015
- Robert Kantrowitz and Michael M. Neumann, Disjointness preserving and local operators on algebras of differentiable functions, Glasg. Math. J. 43 (2001), no. 2, 295–309. MR 1838633, DOI 10.1017/S0017089501020134
- C.-W. Leung, C.-K. Ng and N.-C. Wong, Automatic continuity and $C_0(\Omega )$-linearity of linear maps between $C_0(\Omega )$-modules, J. Aust. Math. Soc. 89 (2010), 245–254.
- Chi-Wai Leung, Chi-Keung Ng, and Ngai-Ching Wong, Linear orthogonality preservers of Hilbert bundles, J. Aust. Math. Soc. 89 (2010), no. 2, 245–254. MR 2769139, DOI 10.1017/S1446788710001515
- Bing Ren Li, Introduction to operator algebras, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1194183
- Ying-Fen Lin and Martin Mathieu, Jordan isomorphism of purely infinite $C^*$-algebras, Q. J. Math. 58 (2007), no. 2, 249–253. MR 2334865, DOI 10.1093/qmath/hal024
- Raghavan Narasimhan, Analysis on real and complex manifolds, Advanced Studies in Pure Mathematics, Vol. 1, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1968. MR 0251745
- Carl Pearcy and David Topping, Sums of small numbers of idempotents, Michigan Math. J. 14 (1967), 453–465. MR 218922
- Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
- Jaak Peetre, Réctification à l’article “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand. 8 (1960), 116–120 (French). MR 124611, DOI 10.7146/math.scand.a-10598
- Jürgen Schweizer, Interplay between noncommutative topology and operators on C*-algebras, PhD thesis, Univ. Tuebingen (1996).
- Jürgen Schweizer, An analogue of Peetre’s theorem in non-commutative topology, Q. J. Math. 52 (2001), no. 4, 499–506. MR 1874495, DOI 10.1093/qjmath/52.4.499
Additional Information
- Chi-Wai Leung
- Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR, People’s Republic of China
- Email: cwleung@math.cuhk.edu.hk
- Chi-Keung Ng
- Affiliation: Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
- Email: ckng@nankai.edu.cn
- Ngai-Ching Wong
- Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, Republic of China
- Email: wong@math.nsysu.edu.tw
- Received by editor(s): May 25, 2010
- Received by editor(s) in revised form: March 22, 2011
- Published electronically: January 6, 2012
- Additional Notes: The authors are supported by The Chinese University of Hong Kong Direct Grant (2060389), National Natural Science Foundation of China (10771106), and Taiwan NSC grant (NSC96-2115-M-110-004-MY3).
- Communicated by: Marius Junge
- © Copyright 2012 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 3151-3160
- MSC (2010): Primary 46L08, 46H40
- DOI: https://doi.org/10.1090/S0002-9939-2012-11260-2
- MathSciNet review: 2917088