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Representations of algebras by continuous sections. (English) Zbl 0237.16018


MSC:

16Gxx Representation theory of associative rings and algebras
16W80 Topological and ordered rings and modules
Full Text: DOI

References:

[1] Donald H. Adams, Semigroups with no non-zero nilpotent elements, Math. Z. 123 (1971), 168 – 176. · Zbl 0212.35803 · doi:10.1007/BF01110115
[2] Richard F. Arens and Irving Kaplansky, Topological representation of algebras, Trans. Amer. Math. Soc. 63 (1948), 457 – 481. · Zbl 0032.00702
[3] Bernhard Banaschewski, Maximal rings of quotients of semi-simple commutative rings, Arch. Math. (Basel) 16 (1965), 414 – 420. · Zbl 0135.07901 · doi:10.1007/BF01220051
[4] George M. Bergman, Hereditary commutative rings and centres of hereditary rings, Proc. London Math. Soc. (3) 23 (1971), 214 – 236. · Zbl 0219.13018 · doi:10.1112/plms/s3-23.2.214
[5] George M. Bergman, Boolean rings of projection maps, J. London Math. Soc. (2) 4 (1972), 593 – 598. · Zbl 0205.03805 · doi:10.1112/jlms/s2-4.4.593
[6] George M. Bergman, Hereditarily and cohereditarily projective modules, Ring theory (Proc. Conf., Park City, Utah, 1971) Academic Press, New York, 1972, pp. 29 – 62. · Zbl 0206.32501
[7] G. M. Bergman, Notes on epimorphisms of rings, Seminar Notes.
[8] Rudolphe Bkouche, Idéaux mous d’un anneau commutatif. Applications aux anneaux de fonctions, C. R. Acad. Sci. Paris 260 (1965), 6496 – 6498 (French). · Zbl 0142.28901
[9] Rudolphe Bkouche, Pureté, mollesse et paracompacité, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1653 – A1655 (French). · Zbl 0194.34802
[10] Rudolphe Bkouche, Couples spectraux et faisceaux associés. Applications aux anneaux de fonctions, Bull. Soc. Math. France 98 (1970), 253 – 295 (French). · Zbl 0201.37204
[11] N. Bourbaki, Éléments de mathématique. Fasc. XXVII. Algèbre commutative, Actualités Sci. Indust., no. 1290, Hermann, Paris, 1961. MR 36 # 146. · Zbl 0108.04002
[12] Alexandru Brezuleanu and Radu Diaconescu, Sur la duale de la catégorie des treillis, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A291 – A293 (French).
[13] Alexandru Brezuleanu, Sur les schémas de treillis, Rev. Roumaine Math. Pures Appl. 14 (1969), 949 – 954 (French). · Zbl 0208.29501
[14] P. M. Cohn, Skew fields of fractions, and the prime spectrum of a general ring, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I), Springer, Berlin, 1972, pp. 1 – 71. Lecture Notes in Math., Vol. 246.
[15] Stephen D. Comer, A sheaf-theoretic duality theory for cylindric algebras, Trans. Amer. Math. Soc. 169 (1972), 75 – 87. · Zbl 0264.02052
[16] Stephen D. Comer, Representations by algebras of sections over Boolean spaces, Pacific J. Math. 38 (1971), 29 – 38. Stephen D. Comer, Elementary properties of structures of sections, Bol. Soc. Mat. Mexicana (2) 19 (1974), no. 2, 78 – 85. · Zbl 0219.08002
[17] John Dauns, Representation of \?-rings, Bull. Amer. Math. Soc. 74 (1968), 249 – 252. · Zbl 0155.07801
[18] John Dauns, Representation of \?-groups and \?-rings, Pacific J. Math. 31 (1969), 629 – 654. · Zbl 0192.36402
[19] John Dauns and Karl Heinrich Hofmann, The representation of biregular rings by sheaves, Math. Z. 91 (1966), 103 – 123. · Zbl 0178.37003 · doi:10.1007/BF01110158
[20] John Dauns and Karl Heinrich Hofmann, Representation of rings by sections, Memoirs of the American Mathematical Society, No. 83, American Mathematical Society, Providence, R.I., 1968. · Zbl 0174.05703
[21] John Dauns and Karl Heinrich Hofmann, Spectral theory of algebras and adjunction of identity, Math. Ann. 179 (1969), 175 – 202. · Zbl 0169.36003 · doi:10.1007/BF01358486
[22] G. Davis, Rings with orthogonality relations, Bull. Austral. Math. Soc. 4 (1971), 163 – 178. , https://doi.org/10.1017/S0004972700046426 G. Davis, Representation and extension of semi-prime rings, J. Austral. Math. Soc. 15 (1973), 353 – 365. · Zbl 0203.33901
[23] Jacques Dixmier, Les \?*-algèbres et leurs représentations, Deuxième édition. Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). · Zbl 0288.46055
[24] J. Dixmier, Champs continus d’espaces hilbertiens et de \?*-algèbres. II, J. Math. Pures Appl. (9) 42 (1963), 1 – 20 (French). · Zbl 0127.33301
[25] J. Dixmier, Ideal center of a \?*-algebra, Duke Math. J. 35 (1968), 375 – 382. · Zbl 0179.18004
[26] Jacques Dixmier and Adrien Douady, Champs continus d’espaces hilbertiens et de \?*-algèbres, Bull. Soc. Math. France 91 (1963), 227 – 284 (French). · Zbl 0127.33102
[27] H. Evans, Various topics concerning Baer rings, Dissertation, Tulane University, New Orleans, La., 1972.
[28] J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233 – 280. · Zbl 0101.09301 · doi:10.1007/BF02545788
[29] J. M. G. Fell, Algebras and fiber bundles, Pacific J. Math. 16 (1966), 497 – 503. · Zbl 0151.18604
[30] J. M. G. Fell, An extension of Mackey’s method to algebraic bundles over finite groups, Amer. J. Math. 91 (1969), 203 – 238. · Zbl 0191.02502 · doi:10.2307/2373278
[31] J. M. G. Fell, An extension of Mackey’s method to Banach * algebraic bundles, Memoirs of the American Mathematical Society, No. 90, American Mathematical Society, Providence, R.I., 1969. · Zbl 0194.44301
[32] Bernard R. Gelbaum, Banach algebra bundles, Pacific J. Math. 28 (1969), 337 – 349. · Zbl 0172.41002
[33] Robin Giles, Foundations for quantum mechanics, J. Mathematical Phys. 11 (1970), 2139 – 2160. · Zbl 0195.28103 · doi:10.1063/1.1665373
[34] L. Gillman, Rings with Hausdorff structure space, Fund. Math. 45 (1957), 1 – 16. · Zbl 0079.26301
[35] James Glimm, A Stone-Weierstrass theorem for \?*-algebras, Ann. of Math. (2) 72 (1960), 216 – 244. · Zbl 0097.10705 · doi:10.2307/1970133
[36] R. Godement, Sur la théorie des représentations unitaires, Ann. of Math. (2) 53 (1951), 68 – 124 (French). · Zbl 0042.34606 · doi:10.2307/1969343
[37] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French).
[38] M. Henriksen and M. Jerison, The space of minimal prime ideals of a commutative ring, Trans. Amer. Math. Soc. 115 (1965), 110 – 130. · Zbl 0147.29105
[39] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43 – 60. · Zbl 0184.29401
[40] M. Hochster, Totally integrally closed rings and extremal spaces, Pacific J. Math. 32 (1970), 767 – 779. · Zbl 0192.38502
[41] M. Hochster, The minimal prime spectrum of a commutative ring, Canad. J. Math. 23 (1971), 749 – 758. · Zbl 0216.19304 · doi:10.4153/CJM-1971-083-8
[42] K. H. Hofmann, Gelfand-Naĭmark theorem for non-commutative topological rings, Proc. Second Sympos. General Topology Appl. (Prague, 1966), Prague, 1967, pp. 184-189.
[43] K. H. Hofmann, Extending C*-algebras by adjoining an identity, Contributions to Extension Theory of Topological Structures, Proc. Sympos. (Berlin, 1967), Deutscher Verlag Wissenschaften, Berlin, 1969, pp. 119-125.
[44] Karl Heinrich Hofmann, Representations of algebras by continuous sections, Bull. Amer. Math. Soc. 78 (1972), 291 – 373. · Zbl 0237.16018
[45] Karl Heinrich Hofmann and Klaus Keimel, A general character theory for partially ordered sets and lattices, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 122. · Zbl 0243.18005
[46] Irving Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951), 219 – 255. · Zbl 0042.34901
[47] Klaus Keimel, Représentation d’anneaux réticulés dans des faisceaux, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A124 – A127 (French). · Zbl 0155.07604
[48] Klaus Keimel, Anneaux réticulés quasi-réguliers et hyperarchimédiens, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A524 – A525 (French). · Zbl 0155.07701
[49] Klaus Keimel, Darstellung von Halbgruppen und universellen Albebren durch Schnitte in Garben; beireguläre Halbgruppen, Math. Nachr. 45 (1970), 81 – 96 (German). · Zbl 0181.02401 · doi:10.1002/mana.19700450105
[50] K. Keimel, Représentation d’anneaux et de groupes réticulés par des section dans des faisceaux, Thèse, Paris, 1970.
[51] Klaus Keimel, Algèbres commutatives engendrées par leurs éléments idempotents, Canad. J. Math. 22 (1970), 1071 – 1078 (French). · Zbl 0203.34702 · doi:10.4153/CJM-1970-123-5
[52] Klaus Keimel, Baer extensions of rings and Stone extensions of semi-groups, Semigroup Forum 2 (1971), no. 1, 55 – 63. · Zbl 0225.20041 · doi:10.1007/BF02572272
[53] K. Keimel, A unified theory of minimal prime ideals, Acta Math. Acad. Sci. Hungar. 23 (1972), 51 – 69. · Zbl 0265.06016 · doi:10.1007/BF01889903
[54] Klaus Keimel, The representation of lattice-ordered groups and rings by sections in sheaves, Lectures on the applications of sheaves to ring theory (Tulane Univ. Ring and Operator Theory Year, 1970 – 1971, Vol. III), Springer, Berlin, 1971, pp. 1 – 98. Lecture Notes in Math., Vol. 248. · Zbl 0231.06023
[55] Joseph Kist, Compact spaces of minimal prime ideals, Math. Z. 111 (1969), 151 – 158. · Zbl 0177.06404 · doi:10.1007/BF01111196
[56] J. Kist, Representing rings by sections; complexes, J. Austral. Math. Soc. (to appear). · Zbl 0144.02602
[57] Kwangil Koh, On functional representations of a ring without nilpotent elements, Canad. Math. Bull. 14 (1971), 349 – 352. · Zbl 0217.34004 · doi:10.4153/CMB-1971-063-7
[58] Kwangil Koh, On a representation of a strongly harmonic ring by sheaves, Pacific J. Math. 41 (1972), 459 – 468. · Zbl 0207.04804
[59] Kwangil Koh and Jiang Luh, Maximal regular right ideal space of a primitive ring, Trans. Amer. Math. Soc. 170 (1972), 269 – 277. · Zbl 0251.16005
[60] Joachim Lambek, Lectures on rings and modules, With an appendix by Ian G. Connell, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1966. · Zbl 0365.16001
[61] J. Lambek, On the representation of modules by sheaves of factor modules, Canad. Math. Bull. 14 (1971), 359 – 368. · Zbl 0217.34005 · doi:10.4153/CMB-1971-065-1
[62] I. G. Macdonald, Algebraic geometry. Introduction to schemes, W. A. Benjamin, Inc., New York-Amsterdam, 1968. John Mack, Fields of topological spaces, Pacific J. Math. 46 (1973), 457 – 466.
[63] Anastasios Mallios, On topological algebra sheaves of a nuclear type, Studia Math. 38 (1970), 215 – 220. · Zbl 0233.46060
[64] A. C. Mewborn, Some conditions on commutative semiprime rings, J. Algebra 13 (1969), 422 – 431. · Zbl 0184.06603 · doi:10.1016/0021-8693(69)90084-2
[65] Ancel C. Mewborn, Regular rings and Baer rings, Math. Z. 121 (1971), 211 – 219. · Zbl 0215.38102 · doi:10.1007/BF01111594
[66] G. O. Michler and O. E. Villamayor, On rings whose simple modules are injective, J. Algebra 25 (1973), 185 – 201. · Zbl 0258.16023 · doi:10.1016/0021-8693(73)90088-4
[67] C. J. Mulvey, On ringed spaces, Dissertation, University of Sussex, 1970. · Zbl 0211.32503
[68] C. J. Mulvey, Représentation des produits sous-directs d’anneaux par espaces annelés, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A564-A567. MR 41 # 1808. · Zbl 0191.31602
[69] C. J. Mulvey, A condition for a ringed space to be a generator in its category of modules, J. Algebra 15 (1970), 312 – 313. · Zbl 0211.32503 · doi:10.1016/0021-8693(70)90060-8
[70] M. A. Naĭmark, On a continuous analogue of Schur’s lemma, Dokl. Akad. Nauk SSSR (N.S.) 98 (1954), 185 – 188 (Russian). · Zbl 0070.02601
[71] David E. Peercy, Complexes and the complete Baer extension of a commutative ring (to appear).
[72] R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. · Zbl 0152.02601
[73] Yann Quentel, Sur la compacité du spectre minimal d’un anneau, Bull. Soc. Math. France 99 (1971), 265 – 272 (French). · Zbl 0215.36803
[74] Jan-Erik Roos, Locally distributive spectral categories and strongly regular rings, Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 156 – 181. · Zbl 1375.18005
[75] S. A. Selesnick, Lattice schemes and maps of pretopological spaces (to appear).
[76] T. P. Speed, A note on commutative Baer rings, J. Austral. Math. Soc. 14 (1972), 257 – 263. · Zbl 0242.13003
[77] G. Spirason and E. Strelecki, A note on P (to appear). · Zbl 0253.06020
[78] Richard G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264 – 277. · Zbl 0109.41601
[79] A. Takahashi, Fields of Hilbert modules, Dissertation, Tulane University, New Orleans, La., 1971.
[80] Silviu Teleman, Analyse harmonique dans les algèbres régulières, Rev. Roumaine Math. Pures Appl. 13 (1968), 691 – 750 (French). · Zbl 0174.07003
[81] Silviu Teleman, La représentation des anneaux tauberiens discrets par des faisceaux, Rev. Roumaine Math. Pures Appl. 14 (1969), 249 – 264 (French). Silviu Teleman, La représentation des anneaux réguliers par les faisceaux, Rev. Roumaine Math. Pures Appl. 14 (1969), 703 – 717 (French). · Zbl 0185.09601
[82] Silviu Teleman, La représentation des anneaux tauberiens discrets par des faisceaux, Rev. Roumaine Math. Pures Appl. 14 (1969), 249 – 264 (French). Silviu Teleman, La représentation des anneaux réguliers par les faisceaux, Rev. Roumaine Math. Pures Appl. 14 (1969), 703 – 717 (French). · Zbl 0185.09601
[83] Silviu Teleman, Représentation par faisceaux des modules sur les anneaux harmoniques, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A753 – A756 (French). · Zbl 0185.09701
[84] Silviu Teleman, La représentation par faisceaux des modules sur les algèbres harmoniques, Rev. Roumaine Math. Pures Appl. 16 (1971), 1247 – 1284 (French). · Zbl 0289.16021
[85] Silviu Teleman, La représentation des algèbres de von Neumann finies par faisceaux, Rev. Roumaine Math. Pures Appl. 15 (1970), 143 – 151 (French). · Zbl 0195.42005
[86] Silviu Teleman, Sur les anneaux réguliers, Rev. Roumaine Math. Pures Appl. 15 (1970), 407 – 434 (French). · Zbl 0198.36401
[87] Silviu Teleman, On the regular rings of John von Neumann, Rev. Roumaine Math. Pures Appl. 15 (1970), 735 – 742. · Zbl 0216.33401
[88] Silviu Teleman, Théorème de de Rham pour les algèbres harmoniques, C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A1119 – A1121 (French). · Zbl 0188.05102
[89] Silviu Teleman, The theorem of de Rham for harmonic algebras, J. Algebra 23 (1972), 271 – 290. · Zbl 0256.18013 · doi:10.1016/0021-8693(72)90132-9
[90] S. Teleman, Representations of von Neumann algebras by sheaves, Hilbert space operators and operator algebras (Proc. Internat. Conf., Tihany, 1970) North-Holland, Amsterdam, 1972, pp. 519 – 538. Colloq. Math. Soc. János Bolyai, No. 5. · Zbl 0252.46077
[91] S. Teleman, Algebraic reduction of von Neumann algebras (to appear). · Zbl 0252.46077
[92] Silviu Teleman, Theory of harmonic algebras with applications to von Neumann algebras and cohomology of locally compact spaces (de Rham’s theorem), Lectures on the applications of sheaves to ring theory (Tulane Univ. Ring and Operator Theory Year, 1970 – 1971, Vol. III), Springer, Berlin, 1971, pp. 99 – 315. Lecture Notes in Math., Vol. 248. · Zbl 0339.16008
[93] Jun Tomiyama and Masamichi Takesaki, Applications of fibre bundles to the certain class of \?*-algebras, Tôhoku Math. J. (2) 13 (1961), 498 – 522. · Zbl 0113.09701 · doi:10.2748/tmj/1178244253
[94] Jun Tomiyama, Topological representation of \?*-algebras, Tôhoku Math. J. (2) 14 (1962), 187 – 204. · Zbl 0216.16201 · doi:10.2748/tmj/1178244174
[95] Jože Vrabec, Adjoining a unit to a biregular ring, Math. Ann. 188 (1970), 219 – 226. · Zbl 0191.03402 · doi:10.1007/BF01350238
[96] Roger Wiegand, Modules over universal regular rings, Pacific J. Math. 39 (1971), 807 – 819. · Zbl 0224.13009
[97] J. Vrabec, Regular preschemes (to appear).
[98] Roger Wiegand, Globalization theorems for locally finitely generated modules, Pacific J. Math. 39 (1971), 269 – 274. · Zbl 0224.13005
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