Available in electronic format
Available in print format
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
     

Isomorphisms of function modules, and generalized approximation in modulus

Author(s): David Blecher; Krzysztof Jarosz
Journal: Trans. Amer. Math. Soc. 354 (2002), 3663-3701.
MSC (2000): Primary 46H25, 47L30, 46J10; Secondary 46L07
Posted: May 8, 2002
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: For a function algebra $A$ we investigate relations between the following three topics: isomorphisms of singly generated $A$-modules, Morita equivalence bimodules, and ``real harmonic functions'' with respect to $A$. We also consider certain groups which are naturally associated with a uniform algebra $A$. We illustrate the notions considered with several examples.


References:

1.
W. B. Arveson, Subalgebras of $C^{*}-$algebras, I, Acta Math. 123 (1969), 141-224; II 128 (1972), 271-308. MR 40:6274; MR 52:15038

2.
E. Behrends and M. Cambern, An isomorphic Banach-Stone theorem, Studia Math. 90 (1988), 15-26. MR 89h:46020

3.
B. Blackadar, K-theory for Operator Algebras, 2nd Ed., Math. Sci. Research Inst. Publications, Cambridge University Press (1998). MR 99c:46104

4.
D. P. Blecher, A generalization of Hilbert modules, J. Funct. Analysis, 136 (1996), 365-421. MR 97g:46071

5.
D. P. Blecher, A new approach to Hilbert $C^*$-modules, Math Ann. 307 (1997), 253-290. MR 98d:46063

6.
D. P. Blecher, Some general theory of operator algebras and their modules, in Operator algebras and applications, A. Katavalos (editor), NATO ASIC, Vol. 495, Kluwer, Dordrecht, 1997, pp. 113-143. MR 98g:46079

7.
D. P. Blecher, Modules over operator algebras, and the maximal $C^*$-dilation, J. Funct. Anal., 169 (1999), 251-288. MR 2001j:47122

8.
D. P. Blecher, The Shilov boundary of an operator space--and the characterization theorems, J. Funct. Anal. 182 (2001), 280-343.

9.
D. P. Blecher and C. Le Merdy, On function and operator modules, Proc. Amer. Math. Soc. 129 (2001), 833-844. MR 2001k:47100

10.
D. P. Blecher, P. S. Muhly and Q. Na, Morita equivalence of operator algebras and their $C^*$-envelopes, Bull. London Math. Soc. 31 (1999), 581-591. MR 2001i:46046

11.
D. P. Blecher, P. S. Muhly and V. I. Paulsen, Categories of operator modules - Morita equivalence and projective modules, (1998 Revision), Memoirs of the American Mathematical Society, Vol. 143, number 681 (Jan, 2000). MR 2001j:46132

12.
L. Brown, P. Green, and M. Rieffel, Stable isomorphism and strong Morita equivalence of $C^{\ast }-$algebras, Pacific J. Math. 71 (1977), 349-363. MR 57:3866

13.
R. G. Douglas and V. I. Paulsen, Hilbert modules over function algebras, Pitman Res. Notes in Math., vol. 217, Longman Sci. Tech., Harlow, and Wiley, New York, 1989. MR 91g:46084
14.
M. J. Dupre and R. M. Gillette, Banach bundles, Banach modules and automorphisms of $C^*$-algebras, Res. Notes in Math., vol. 93, Pitman (1983). MR 85j:46127

15.
E. G. Effros and Z.-J. Ruan, On nonselfadjoint operator algebras, Proc. Amer. Math. Soc., 110(1990), 915-922. MR 91c:47086

16.
C. Faith, Algebra I: Rings, Modules, and Categories, Springer-Verlag, Berlin Heidelberg New York (1981). MR 82g:16001

17.
T. W. Gamelin, Uniform algebras, Prentice Hall (1969). MR 53:14137
18.
T. W. Gamelin, Uniform Algebras and Jensen Measures, Cambridge Univ. Press, 1978. MR 81a:46058

19.
T. W. Gamelin and N. Sibony, Subharmonicity for uniform algebras, J. Funct. Anal. 35 (1980), 64-108. MR 81f:46062
20.
A. M. Gleason, Function algebras, Seminar on Analytic Functions, vol. II, Inst. for Adv. Study, Princeton, N.J. 1957.

21.
P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Springer-Verlag, Berlin - New York (1993). MR 94k:46022

22.
K. Jarosz, Perturbations of uniform algebras, Bull. London Math. Soc. 15 (1983), 133-138. MR 84e:46056

23.
K. Jarosz, Perturbations of Banach Algebras, Springer-Verlag Lecture Notes in Math. 1120 (1985). MR 86k:46074
24.
K. Jarosz, Small isomorphisms of $C(X,E)$ spaces, Pacific J. Math. 138 (1989), 295-315. MR 90f:46056
25.
K. Hoffman, Analytic functions and logmodular Banach algebras, Acta Math. 108 (1962), 271-317. MR 26:6820
26.
K. Hoffman, Banach spaces of analytic functions, Dover (1988). MR 92d:46066
27.
E. C. Lance, Hilbert $C^*$-modules - A toolkit for operator algebraists, London Math. Soc. Lecture Notes, Cambridge University Press (1995). MR 96k:46100

28.
M. Nagasawa, Isomorphisms between commutative Banach algebras with application to rings of analytic functions, Kodai Math. Sem. Rep. Math. 11 (1959), 182-188. MR 22:12379
29.
I. Raeburn, On the Picard group of continuous trace $C^*$-algebras, Trans. Amer. Math. Soc. 263 (1981), 183-205. MR 82b:46090
30.
C. E. Rickart, Plurisubharmonic functions and convexity properties for general function algebras, Trans. Amer. Math. Soc., 169 (1972), 1-24. MR 47:5603
31.
M. Rieffel, Morita equivalence for operator algebras, Proceedings of Symposia in Pure Mathematics 38 Part 1 (1982), 285-298. MR 84k:46045

32.
R. Rochberg, Deformation of uniform algebras on Riemann surfaces, Pacific J. Math. 121 (1986), 135-181. MR 87d:46060

33.
R. R. Smith, An addendum to M-ideal structure in Banach algebras, J. Funct. Anal. 32 (1979), 269-271. MR 80j:46087

34.
E. L. Stout, The theory of uniform algebras, Bogden and Quigley, Tarrytown-on-Hudson, NY (1971). MR 54:11066
35.
I. Suciu, Function algebras, Noordhoff (1975). MR 51:6428


Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46H25, 47L30, 46J10, 46L07

Retrieve articles in all Journals with MSC (2000): 46H25, 47L30, 46J10, 46L07


Additional Information:

David Blecher
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: dblecher@math.uh.edu

Krzysztof Jarosz
Affiliation: Department of Mathematics and Statistics, Southern Illinois University, Edwards- ville, Illinois 62026-1653
Email: kjarosz@siue.edu

DOI: 10.1090/S0002-9947-02-03016-7
PII: S 0002-9947(02)03016-7
Received by editor(s): September 14, 1999
Received by editor(s) in revised form: November 26, 2001
Posted: May 8, 2002
Copyright of article: Copyright 2002, American Mathematical Society


  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement
Search the AMSPowered by Google