# zbMATH — the first resource for mathematics

Orthogonality preservers in C$$^*$$-algebras, JB$$^*$$-algebras and JB$$^*$$-triples. (English) Zbl 1156.46045
Let $$A$$ be a $$C^*$$-algebra or a $$JB^*$$-algebra, and let $$E$$ be a $$JB^*$$-triple. The authors give a description of those bounded linear operators $$T: A\rightarrow E$$ with the property of preserving orthogonality, which means that $$\{T(a),T(b),y\}=0$$ for each $$y\in E$$ whenever $$a,b\in A$$ are such that $$\{a,b,x\}=0$$ for each $$x\in A$$ (where $$\{\cdot,\cdot,\cdot\}$$ stands for the triple product in both $$A$$ and $$E$$). It is shown that if $$T''(\mathbf{1})$$ is a von Neumann regular element in the bidual $$E''$$ of $$E$$, then $$T(A)$$ is contained in the Peirce subspace $$E_2''(r(T''(\mathbf{1})))$$ of the range tripotent $$r(T''(\mathbf{1}))$$ of $$T''(\mathbf{1})$$ and there exists a triple homomorphism $$S\colon A\rightarrow E_2''(r(T''(\mathbf{1})))$$ such that $$T(x)=\{T''(\mathbf{1}),r(T''(\mathbf{1})),S(x)\}$$ for each $$x\in A$$. As a consequence, it is shown that a bounded linear operator $$T: A\rightarrow B$$ between two $$C^*$$-algebras $$A$$ and $$B$$ preserves orthogonality in the usual sense ($$a,b\in A, \;ab^*=b^*a=0 \;\Rightarrow \;T(a)T(b)^*=T(b)^*T(a)=0$$) if and only if there exists a triple homomorphism $$S\colon A\rightarrow B''$$ such that $$T''(\mathbf{1})^*S(x)=S(x^*)^*T''(\mathbf{1})$$, $$T''(\mathbf{1})S(x^*)^*=S(x)T''(\mathbf{1})^*$$, and $$T(x)=T''(\mathbf{1})r(T''(\mathbf{1}))^*S(x)$$ for each $$x\in A$$.

##### MSC:
 46L70 Nonassociative selfadjoint operator algebras 46L05 General theory of $$C^*$$-algebras 46H70 Nonassociative topological algebras 46K70 Nonassociative topological algebras with an involution
Full Text:
##### References:
 [1] Abramovich, Y.A., Multiplicative representation of disjointness preserving operators, Nederl. akad. wetensch. indag. math., 45, 3, 265-279, (1983), MR0718068 (85f:47040) · Zbl 0527.47025 [2] J. Alaminos, M. Brešar, J. Extremera, A.R. Villena, Maps preserving zero products, preprint, 2007 [3] Araujo, J.; Jarosz, K., Biseparating maps between operator algebras, J. math. anal. appl., 282, 1, 48-55, (2003), MR2000328 (2004f:47049) · Zbl 1044.47029 [4] Arendt, W., Spectral properties of Lamperti operators, Indiana univ. math. J., 32, 2, 199-215, (1983), MR0690185 (85d:47040) · Zbl 0488.47016 [5] Barton, T.J.; Friedman, Y., Bounded derivations of $$\operatorname{JB}^\ast$$-triples, Quart. J. math. Oxford ser. (2), 41, 163, 255-268, (1990), MR1067482 (91j:46086) · Zbl 0728.46046 [6] Barton, T.; Timoney, R.M., Weak^∗-continuity of Jordan triple products and its applications, Math. scand., 59, 177-191, (1986), MR0884654 (88d:46129) · Zbl 0621.46044 [7] Bunce, L.J.; Chu, C.H.; Zalar, B., Structure spaces and decomposition in $$\operatorname{JB}^\ast$$-triples, Math. scand., 86, 1, 17-35, (2000), MR1738513 (2001g:46148) · Zbl 1073.46513 [8] Bunce, L.J.; Fernández-Polo, F.; Martínez-Moreno, J.; Peralta, A.M., Saitô-tomita – lusin theorem for $$\operatorname{JB}^\ast$$-triple and applications, Quart. J. math. Oxford, 57, 37-48, (2006), MR2204259 (2006k:46117) · Zbl 1123.46053 [9] Burgos, M.; Kaidi, A.; Morales, A.; Peralta, A.M.; Ramírez, M., Von Neumann regularity and quadratic conorms in $$\operatorname{JB}^\ast$$-triples and $$\operatorname{C}^\ast$$-algebras, Acta math. sinica, 24, 2, 185-200, (2008), MR2383347 · Zbl 1157.46042 [10] Chan, J.T., Operators with the disjoint support property, J. operator theory, 24, 383-391, (1990), MR1150627 (93c:47035) · Zbl 0815.47037 [11] Chebotar, M.A.; Ke, W.-F.; Lee, P.-H.; Wong, N.-C., Mappings preserving zero products, Studia math., 155, 1, 77-94, (2003), MR1961162 (2003m:47066) · Zbl 1032.46063 [12] Chebotar, M.A.; Ke, W.-F.; Lee, P.-H.; Zhang, R., On maps preserving zero Jordan products, Monatsh. math., 149, 2, 91-101, (2006), MR2264576 (2007g:47054) · Zbl 1109.16030 [13] Dineen, S., The second dual of a $$\operatorname{JB}^\ast$$-triple system, (), 67-69, MR0893410 (88f:46097) [14] Edwards, C.M.; Rüttimann, G.T., Compact tripotents in bi-dual $$\operatorname{JB}^\ast$$-triples, Math. proc. Cambridge philos. soc., 120, 1, 155-173, (1996), MR1373355 (96k:46131) · Zbl 0853.46070 [15] Fernández López, A.; García Rus, E.; Sánchez Campos, E.; Siles Molina, M., Strong regularity and generalized inverses in Jordan systems, Comm. algebra, 20, 7, 1917-1936, (1992), MR1167082 (93h:17072) · Zbl 0759.17019 [16] Fernández-Polo, F.J.; Peralta, A.M., Closed tripotents and weak compactness in the dual space of a $$\operatorname{JB}^\ast$$-triple, J. London math. soc., 74, 75-92, (2006), MR2254553 (2007g:46104) · Zbl 1108.46049 [17] Fernández-Polo, F.J.; Peralta, A.M., Compact tripotents and the stone – weierstrass theorem for $$\operatorname{C}^\ast$$-algebras and $$\operatorname{JB}^\ast$$-triples, J. operator theory, 58, 1, 157-173, (2007), MR2336049 · Zbl 1150.46027 [18] Friedman, Y.; Russo, B., Structure of the predual of a $$\operatorname{JBW}^\ast$$-triple, J. reine angew. math., 356, 67-89, (1985), MR0779376 (86f:46073) · Zbl 0547.46049 [19] Font, J.J.; Hernández, S., On separating maps between locally compact spaces, Arch. math. (basel), 63, 158-165, (1994), MR1289298 (95k:46083) · Zbl 0805.46049 [20] Goldstein, S., Stationarity of operator algebras, J. funct. anal., 118, 2, 275-308, (1993), MR1250265 (94k:46135) · Zbl 0796.46041 [21] Haagerup, U.; Laustsen, N.J., Weak amenability of $$C^\ast$$-algebras and a theorem of goldstein, (), 223-243, MR1656608 (99k:46096) · Zbl 1034.46506 [22] Hanche-Olsen, H.; Størmer, E., Jordan operator algebras, Monogr. stud. in math., vol. 21, (1984), Pitman (Advanced Publishing Program) Boston, MA, MR0755003 (86a:46092) · Zbl 0561.46031 [23] Jarosz, K., Automatic continuity of separating linear isomorphisms, Canad. math. bull., 33, 2, 139-144, (1990), MR1060366 (92j:46049) · Zbl 0714.46040 [24] Jeang, J.-S.; Wong, N.-C., Weighted composition operators of $$C_0(X)$$’s, J. math. anal. appl., 201, 981-993, (1996), MR1400575 (97f:47029) · Zbl 0936.47011 [25] Kaup, W., Algebraic characterization of symmetric complex Banach manifolds, Math. ann., 228, 39-64, (1977), MR0454091 (56:12342) · Zbl 0335.58005 [26] Kaup, W., A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z., 183, 503-529, (1983), MR0710768 (85c:46040) · Zbl 0519.32024 [27] Kaup, W., On spectral and singular values in $$\operatorname{JB}^\ast$$-triples, Proc. roy. irish acad. sect. A, 96, 1, 95-103, (1996), MR1644656 (99f:46099) · Zbl 0904.46039 [28] Kaup, W., On real Cartan factors, Manuscripta math., 92, 191-222, (1997), MR1428648 (97m:46109) · Zbl 0881.17033 [29] Kaup, W., On Grassmannians associated with $$\operatorname{JB}^\ast$$-triples, Math. Z., 236, 567-584, (2001), MR1821305 (2002b:46110) · Zbl 0988.46048 [30] Loos, O., Jordan pairs, Lecture notes in math., vol. 460, (1975), Springer-Verlag Berlin · Zbl 0301.17003 [31] Martínez, J.; Peralta, A.M., Separate weak^∗-continuity of the triple product in dual real JB∗-triples, Math. Z., 234, 635-646, (2000), MR1778403 (2001h:46123) · Zbl 0977.17032 [32] Molnár, L., Selected preserver problems on algebraic structures of linear operators and on function spaces, Lecture notes in math., vol. 1895, (2007), Springer-Verlag Berlin, MR2267033 (2007g:47056) · Zbl 1119.47001 [33] C. Palazuelos, A.M. Peralta, I. Villanueva, Orthogonally additive polynomials on $$\operatorname{C}^\ast$$-algebras, Q. J. Math., in press · Zbl 1159.46035 [34] Peralta, A.M.; Rodríguez Palacios, A., Grothendieck’s inequalities for real and complex $$\operatorname{JBW}^\ast$$-triples, Proc. London math. soc. (3), 83, 3, 605-625, (2001), MR1851084 (2002g:46121) · Zbl 1037.46058 [35] Rodríguez, A., On the strong^∗ topology of a JBW∗-triple, Quart. J. math. Oxford ser. (2), 42, 165, 99-103, (1991), MR1094345 (92j:46123) · Zbl 0723.17026 [36] Topping, D., Jordan algebras of self-adjoint operators, Mem. amer. math. soc., 53, (1965), MR0190788 (32:8198) · Zbl 0149.09801 [37] Wolff, M., Disjointness preserving operators in $$\operatorname{C}^\ast$$-algebras, Arch. math., 62, 248-253, (1994), MR1259840 (94k:46122) · Zbl 0803.46069 [38] Wong, N.C., Triple homomorphisms of $$\operatorname{C}^\ast$$-algebras, Southeast Asian bull. math., 29, 401-407, (2005), MR2217546 (2007c:46054) · Zbl 1108.46041 [39] Wong, N.C., Zero products preservers of $$\operatorname{C}^\ast$$-algebras, (), 377-380 [40] Wright, J.D.M., Jordan $$\operatorname{C}^\ast$$-algebras, Michigan math. J., 24, 291-302, (1977), MR0487478 (58:7108)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.