Geodesic
On a class of inner maps
Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 4, pp. 215-226.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Let $f$ be a continuous map of the closure $\overline{\Delta}$ of the open unit disc $\Delta$ of $\mathbb{C}$ into a unital associative Banach algebra $\mathcal{A}$, whose restriction to $\Delta$ is holomorphic, and which satisfies the condition whereby $0 \notin \sigma(f(z)) \subset \overline{\Delta}$ for all $z \in \Delta$ and $\sigma(f(z)) \subset \partial \Delta$ whenever $z \in \partial \Delta$ (where $\sigma(x)$ is the spectrum of any $x \in \mathcal{A}$). One of the basic results of the present paper is that $f$ is , that is to say, $\sigma(f(z))$ is then a compact subset of $\partial \Delta$ that does not depend on $z$ for all $z \in \overline{\Delta}$. This fact will be applied to holomorphic self-maps of the open unit ball of some $J^{*}$-algebra and in particular of any unital $C^{*}$-algebra, investigating some cases in which not only the spectra but the maps themselves are necessarily constant. 
@article{RLIN_2005_9_16_4_a0,
     author = {Vesentini, Edoardo},
     title = {On a class of inner maps},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {215--226},
     publisher = {mathdoc},
     volume = {Ser. 9, 16},
     number = {4},
     year = {2005},
     zbl = {1215.46030},
     mrnumber = {549769},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_4_a0/}
}
TY  - JOUR
AU  - Vesentini, Edoardo
TI  - On a class of inner maps
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
PY  - 2005
SP  - 215
EP  - 226
VL  - 16
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_4_a0/
LA  - en
ID  - RLIN_2005_9_16_4_a0
ER  - 
%0 Journal Article
%A Vesentini, Edoardo
%T On a class of inner maps
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2005
%P 215-226
%V 16
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_4_a0/
%G en
%F RLIN_2005_9_16_4_a0
Vesentini, Edoardo. On a class of inner maps. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 16 (2005) no. 4, pp. 215-226. http://geodesic.mathdoc.fr/item/RLIN_2005_9_16_4_a0/

[1] B. Aupetit, Propriétés spectrales des algèbres de Banach. Lecture Notes in Mathematics, 735, Springer-Verlag, Berlin-Heidelberg-New York, 1979. | MR | Zbl

[2] T. Franzoni - E. Vesentini, Holomorphic maps and invariant distances. North Holland, Amsterdam-New York-Oxford, 1980. | MR | Zbl

[3] L.A. Harris, Bounded symmetric homogeneous domains in infinite dimensional spaces. In: T.L. HAYDEN - T.J. SUFFRIDGE (eds.), Proceedings on infinite dimensional holomorphy, University of Kentucky 1973. Lecture Notes in Mathematics, 364, Springer, Berlin 1974, 13-40. | MR | Zbl

[4] K. Hoffman, Banach spaces of analytic functions. Prentice-Hall, Englewood Cliffs, N.J., 1962. | MR | Zbl

[5] M. Jarnicki - P. Pflug, Invariant distances and metrics in complex analysis. Walter de Gruyter, Berlin-New York 1993. | DOI | MR | Zbl

[6] K. Oka, Note sur les familles de fonctions analytiques multiformes etc. J. Sci. Hiroshima Univ. Ser. A, 4, 1934, 93-98. | Zbl

[7] W. Rudin, Function theory in the unit ball of $\mathbb{C}^{n}$. Springer-Verlag, New York-Heidelberg-Berlin 1980. | MR | Zbl

[8] W. Rudin, New constructions of functions holomorphic in the unit ball $\mathbb{C}^{n}$. Conference Board Math. Sci., 63, 1985. | MR | Zbl

[9] S. Sakai, $C^{*}$-algebras and $W^{*}$-algebras. Springer-Verlag, New York-Heidelberg-Berlin 1971. | MR | Zbl

[10] Z. Slodkowski, Analytic set-valued functions and spectra. Math. Ann., 256, 1981, 363-386. | fulltext EuDML | DOI | MR | Zbl

[11] E. Vesentini, Maximum theorems for spectra. Essays on topology and related topics, Mémoires dédiés à Georges de Rham, Springer-Verlag, Berlin-Heidelberg-New York 1970, 111-117. | MR | Zbl

[12] E. Vesentini, Maximum theorems for vector valued holomorphic functions. University of Maryland Technical Report, 69-132, 1969; Rend. Sem. Mat. Fis. Milano, 40, 1970, 1-34. | MR | Zbl

[13] E. Vesentini, Complex geodesies. Compositio Math., 44, 1981, 375-394. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[14] E. Vesentini, Holomorphic isometries of spin-factors. Rend. Sem. Mat. Univ. Politec. Torino, 50, 4, 1992, 427-455. | MR | Zbl