# Holomorphic automorphisms and collective compactness in J*-algebras of operator

Open Mathematics (2007)

- Volume: 5, Issue: 3, page 512-522
- ISSN: 2391-5455

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topJosé Isidro. "Holomorphic automorphisms and collective compactness in J*-algebras of operator." Open Mathematics 5.3 (2007): 512-522. <http://eudml.org/doc/269061>.

@article{JoséIsidro2007,

abstract = {Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball \[B\_\mathfrak \{A\} \]
in a J*-algebra \[\mathfrak \{A\}\]
of operators. Let \[\mathfrak \{F\}\]
be the family of all collectively compact subsets W contained in \[B\_\mathfrak \{A\} \]
. We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family \[\mathfrak \{F\}\]
is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when \[\mathfrak \{A\}\]
is a Cartan factor.},

author = {José Isidro},

journal = {Open Mathematics},

keywords = {J*-algebras; Cartan factors; holomorphic automorphisms; Banach-Lie groups; collective compactness},

language = {eng},

number = {3},

pages = {512-522},

title = {Holomorphic automorphisms and collective compactness in J*-algebras of operator},

url = {http://eudml.org/doc/269061},

volume = {5},

year = {2007},

}

TY - JOUR

AU - José Isidro

TI - Holomorphic automorphisms and collective compactness in J*-algebras of operator

JO - Open Mathematics

PY - 2007

VL - 5

IS - 3

SP - 512

EP - 522

AB - Let G be the Banach-Lie group of all holomorphic automorphisms of the open unit ball \[B_\mathfrak {A} \]
in a J*-algebra \[\mathfrak {A}\]
of operators. Let \[\mathfrak {F}\]
be the family of all collectively compact subsets W contained in \[B_\mathfrak {A} \]
. We show that the subgroup F ⊂ G of all those g ∈ G that preserve the family \[\mathfrak {F}\]
is a closed Lie subgroup of G and characterize its Banach-Lie algebra. We make a detailed study of F when \[\mathfrak {A}\]
is a Cartan factor.

LA - eng

KW - J*-algebras; Cartan factors; holomorphic automorphisms; Banach-Lie groups; collective compactness

UR - http://eudml.org/doc/269061

ER -

## References

top- [1] P.M. Anselone and T.W. Palmer: “Collectively compact sets of linear operators“, Pac. J. Math., Vol. 25, (1968), pp. 417–422. Zbl0157.45202
- [2] L.A. Harris: “Bounded symmetric homogeneous domains in infinite-dimensional spaces“, In: Proceedings on Infinite Dimensional Holomorphy, Lecture Notes in Mathematics, Vol. 364, Springer-Verlag, 1974, pp. 13–40.
- [3] L.A. Harris: “A generalization of C*-algebras“, P. Lond. Math. Soc., Vol. 42, (1981), pp. 331–361. http://dx.doi.org/10.1112/plms/s3-42.2.331
- [4] L.A. Harris and W. Kaup: “Linear algebraic groups in infinite dimensions“, Illinois J.. Math., Vol. 21, (1977), pp. 666–674. Zbl0385.22011
- [5] T. Ho, J. Martinez Moreno, A. Peralta and B. Russo: “Derivations on real and complex JB*-triples“, J. Lond. Math. Soc., Vol. 65, (2002), pp. 85–102. http://dx.doi.org/10.1112/S002461070100271X Zbl1015.46041
- [6] J.M. Isidro and W. Kaup: “Weak continuity of holomorphic automorphisms in JB*-triples“, Math. Z., Vol. 210, (1992), pp. 277–288. http://dx.doi.org/10.1007/BF02571798 Zbl0812.46066
- [7] J.M. Isidro and L.L. Stachó: “Weakly and weakly** continuous elements in JBW*-triples“, Acta Sci. Math. (Szeged), Vol. 57, (1993), pp. 555–567. Zbl0834.17046
- [8] W. Kaup: “Uber die Automorphismen Grassmancher Mannigfaltigkeiten unendlicher Dimension“, Math. Z., Vol. 144, (1975), pp. 75–96. http://dx.doi.org/10.1007/BF01190938 Zbl0322.32014
- [9] W. Kaup: “A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces“, Math. Z., Vol. 183, (1983), pp. 503–529. http://dx.doi.org/10.1007/BF01173928 Zbl0519.32024
- [10] W. Kaup: “Hermitian Jordan Triple Systems and Automorphisms of Bounded Symmetric Domains“, In: Santoz González (Ed.): Non-Associative Algebras and Applications, Kluwer Academic Publishers, 1994, pp. 204–214.
- [11] T.W. Palmer: “Totally bounded sets of precompact linear operators“, P. Am. Math. Soc., Vol. 20, (1969), pp. 101–106. http://dx.doi.org/10.2307/2035969 Zbl0165.47603
- [12] L.L. Stachó and J.M. Isidro: “Algebraically compact elements in JB*-triples“, Acta Sci. Math. (Szeged), Vol. 54, (1990), pp. 171–190. Zbl0736.46053
- [13] H. Upmeier: “Symmetric Banach Manifolds and Jordan C*-Algebras“, In: North Holland Mathematics Studies, Vol. 104, North Holland, Amsterdam, 1985.
- [14] J.P. Viguée and J.M. Isidro: “Sur la topologie du groupe des automorphismes analytiques d’un domaine cerclé borné”, B. Sci. Math., Vol. 106, (1982), pp. 417–426. Zbl0546.32012

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