## Abstract and Applied Analysis

### Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras

#### Abstract

We study holomorphic maps between C${\mathrm{}}^{\mathrm{\ast}}$-algebras $A$ and $B$, when $f:{B}_{A}(\mathrm{0},\varrho )\to B$ is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball $U={B}_{A}(\mathrm{0},\delta )$. If we assume that $f$ is orthogonality preserving and orthogonally additive on ${A}_{sa}\cap U$ and $f(U)$ contains an invertible element in $B$, then there exist a sequence $({h}_{n})$ in ${B}^{\mathrm{\ast}\mathrm{\ast}}$ and Jordan ${\mathrm{}}^{\mathrm{\ast}}$-homomorphisms $\mathrm{\Theta },\stackrel{~}{\mathrm{\Theta }}:M(A)\to {B}^{\mathrm{\ast}\mathrm{\ast}}$ such that $f(x)={\sum }_{n=\mathrm{1}}^{\infty }{h}_{n}\stackrel{~}{\mathrm{\Theta }}({a}^{n})={\sum }_{n=\mathrm{1}}^{\infty }\mathrm{\Theta }({a}^{n}){h}_{n}$ uniformly in $a\in U$. When $B$ is abelian, the hypothesis of $B$ being unital and $f(U)\cap \mathrm{}\text{i}\text{n}\text{v}\mathrm{}(B)\ne \mathrm{\varnothing }$ can be relaxed to get the same statement.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 415354, 9 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449674

Digital Object Identifier
doi:10.1155/2013/415354

Mathematical Reviews number (MathSciNet)
MR3147830

Zentralblatt MATH identifier
1292.32001

#### Citation

Garcés, Jorge J.; Peralta, Antonio M.; Puglisi, Daniele; Ramírez, María Isabel. Orthogonally Additive and Orthogonality Preserving Holomorphic Mappings between C*-Algebras. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 415354, 9 pages. doi:10.1155/2013/415354. https://projecteuclid.org/euclid.aaa/1393449674

#### References

• Y. Benyamini, S. Lassalle, and J. G. Llavona, “Homogeneous orthogonally additive polynomials on Banach Lattices,” Bulletin of the London Mathematical Society, vol. 38, no. 3, pp. 459–469, 2006.
• D. Pérez-García and I. Villanueva, “Orthogonally additive polynomials on spaces of continuous functions,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 97–105, 2005.
• C. Palazuelos, A. M. Peralta, and I. Villanueva, “Orthogonally additive polynomials on C$^{\ast\,\!}$-algebras,” Quarterly Journal of Mathematics, vol. 59, no. 3, pp. 363–374, 2008.
• K. Sundaresan, “Geometry of spaces of homogeneous polynomials on Banach lattices,” in Applied Geometry and Discrete Mathematics, Discrete Mathematics and Theoretical Computer Science, pp. 571–586, American Mathematical Society, Providence, RI, USA, 1991.
• D. Carando, S. Lassalle, and I. Zalduendo, “Orthogonally additive polynomials over C(K) are measures–-a short proof,” Integral Equations and Operator Theory, vol. 56, no. 4, pp. 597–602, 2006.
• M. Burgos, F. J. Fernández-Polo, J. J. Garcés, and A. M. Peralta, “Orthogonality preservers revisited,” Asian-European Journal of Mathematics, vol. 2, no. 3, pp. 387–405, 2009.
• Q. Bu and G. Buskes, “Polynomials on Banach lattices and positive tensor products,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 845–862, 2012.
• D. Carando, S. Lassalle, and I. Zalduendo, “Orthogonally additive holomorphic functions of bounded type over C(K),” Proceedings of the Edinburgh Mathematical Society, vol. 53, no. 3, pp. 609–618, 2010.
• J. Á. Jaramillo, Á. Prieto, and I. Zalduendo, “Orthogonally additive holomorphic functions on open subsets of C(K),” Revista Matematica Complutense, vol. 25, no. 1, pp. 31–41, 2012.
• A. M. Peralta and D. Puglisi, “Orthogonally additive holomorphic functions on C$^{\ast\,\!}$-algebras,” Operators and Matrices, vol. 6, no. 3, pp. 621–629, 2012.
• Q. Bu, M. H. Hsu, and N. Ch. Wong, “Zero products and norm preserving orthogonally additive homogeneous polynomials on C$^{\ast\,\!}$-algebras,” Preprint.
• M. Burgos, F. J. Fernández-Polo, J. J. Garcés, J. M. Moreno, and A. M. Peralta, “Orthogonality preservers in C$^{\ast\,\!}$-algebras, JB$^{\ast\,\!}$-algebras and JB$^{\ast\,\!}$-triples,” Journal of Mathematical Analysis and Applications, vol. 348, no. 1, pp. 220–233, 2008.
• A. C. Zaanen, “Examples of orthomorphisms,” Journal of Approximation Theory, vol. 13, no. 2, pp. 192–204, 1975.
• B. E. Johnson, “Local derivations on C$^{\ast\,\!}$-algebras are derivations,” Transactions of the American Mathematical Society, vol. 353, no. 1, pp. 313–325, 2001.
• S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer, 1999.
• T. W. Gamelin, “Analytic functions on Banach spaces,” in Complex Potential Theory, vol. 439 of NATO Advanced Science Institutes Series C, pp. 187–233, Kluwer Academic, Dodrecht, The Netherlands, 1994.
• M. Wolff, “Disjointness preserving operators on C$^{\ast\,\!}$-algebras,” Archiv der Mathematik, vol. 62, no. 3, pp. 248–253, 1994.
• S. Sakai, C$^{\ast\,\!}$-Algebras and W$^{\ast\,\!}$-Algebras, Springer, Berlin, Germany, 1971.
• S. Goldstein, “Stationarity of operator algebras,” Journal of Functional Analysis, vol. 118, no. 2, pp. 275–308, 1993.
• J. D. M. Wright, “Jordan C$^{\ast\,\!}$-algebras,” Michigan Mathematical Journal, vol. 24, pp. 291–302, 1977.
• D. Topping, “Jordan algebras of self-adjoint operators,” Memoirs of the American Mathematical Society, vol. 53, 1965. \endinput