On a class of inner maps

Vesentini, Edoardo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 16 (2005), p. 215-226 / Harvested from Biblioteca Digitale Italiana di Matematica

Résumé

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.

Détail

Publié le : 2005-12-01

MR 2255005 | Zbl 1215.46030

Bibliographie

[1] Aupetit, B., Propriétés spectrales des algèbres de Banach. Lecture Notes in Mathematics, 735, Springer-Verlag, Berlin-Heidelberg-New York, 1979. | MR 549769 | Zbl 0409.46054

[2] Franzoni, T. - Vesentini, E., Holomorphic maps and invariant distances. North Holland, Amsterdam-New York-Oxford, 1980. | MR 563329 | Zbl 0447.46040

[3] Harris, L.A., 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 407330 | Zbl 0293.46049

[4] Hoffman, K., Banach spaces of analytic functions. Prentice-Hall, Englewood Cliffs, N.J., 1962. | MR 133008 | Zbl 0117.34001

[5] Jarnicki, M. - Pflug, P., Invariant distances and metrics in complex analysis. Walter de Gruyter, Berlin-New York 1993. | MR 1242120 | Zbl 0789.32001

[6] Oka, K., Note sur les familles de fonctions analytiques multiformes etc. J. Sci. Hiroshima Univ. Ser. A, 4, 1934, 93-98. | Zbl 0011.31403

[7] Rudin, W., Function theory in the unit ball of $\mathbb{C}^{n}$. Springer-Verlag, New York-Heidelberg-Berlin1980. | MR 601594 | Zbl 1139.32001

[8] Rudin, W., New constructions of functions holomorphic in the unit ball $\mathbb{C}^{n}$. Conference Board Math. Sci., 63, 1985. | MR 840468 | Zbl 1187.32001

[9] Sakai, S., $C^{*}$-algebras and $W^{*}$-algebras. Springer-Verlag, New York-Heidelberg-Berlin1971. | MR 442701 | Zbl 1024.46001

[10] Slodkowski, Z., Analytic set-valued functions and spectra. Math. Ann., 256, 1981, 363-386. | MR 626955 | Zbl 0452.46028

[11] Vesentini, E., Maximum theorems for spectra. Essays on topology and related topics, Mémoires dédiés à Georges de Rham, Springer-Verlag, Berlin-Heidelberg-New York1970, 111-117. | MR 271731 | Zbl 0195.41903

[12] Vesentini, E., 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 287299 | Zbl 0221.58007

[13] Vesentini, E., Complex geodesies. Compositio Math., 44, 1981, 375-394. | MR 662466 | Zbl 0488.30015

[14] Vesentini, E., Holomorphic isometries of spin-factors. Rend. Sem. Mat. Univ. Politec. Torino, 50, 4, 1992, 427-455. | MR 1261453 | Zbl 0791.46031