A generalized dual for a 𝐶*-algebra

Halpern, Herbert

doi:10.1090/S0002-9947-1971-0270165-X

A generalized dual for a $C^*$-algebra
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Trans. Amer. Math. Soc. 153 (1971), 139-156 Request permission

Abstract:

Let $\mathcal {A}$ be a ${C^ \ast }$-algebra, let $\mathcal {B}$ be its enveloping von Neumann algebra, and let $\mathcal {F}$ be the center of $\mathcal {B}$. Let ${\mathcal {B}_ \sim }$ be the set of all $\sigma$-weakly continuous $\mathcal {F}$-module homomorphisms of the $\mathcal {F}$-module $\mathcal {B}$ into $\mathcal {F}$ and let ${\mathcal {A}^ \sim }$ be the set of all restrictions to $\mathcal {A}$ of elements of ${\mathcal {B}_ \sim }$. Then $\mathcal {A}$ is classified as CCR, GCR, and NGCR in terms of certain naturally occurring topologies on ${\mathcal {A}^ \sim }$.

References

Jacques Dixmier, Traces sur les $C^*$-algèbres, Ann. Inst. Fourier (Grenoble) 13 (1963), no. fasc. 1, 219–262 (French). MR 149317, DOI 10.5802/aif.139

Les algèbres d’opérateurs dans l’espace hilbertien, (Algèbres de von Neumann)

20

Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
J. M. G. Fell, The structure of algebras of operator fields, Acta Math. 106 (1961), 233–280. MR 164248, DOI 10.1007/BF02545788
James Glimm, A Stone-Weierstrass theorem for $C^{\ast }$-algebras, Ann. of Math. (2) 72 (1960), 216–244. MR 116210, DOI 10.2307/1970133
James Glimm, Type I $C^{\ast }$-algebras, Ann. of Math. (2) 73 (1961), 572–612. MR 124756, DOI 10.2307/1970319
R. Godement, Sur la théorie des représentations unitaires, Ann. of Math. (2) 53 (1951), 68–124 (French). MR 38571, DOI 10.2307/1969343
Herbert Halpern, An integral representation of a normal functional on a von Neumann algebra, Trans. Amer. Math. Soc. 125 (1966), 32–46. MR 198266, DOI 10.1090/S0002-9947-1966-0198266-3
Herbert Halpern, A spectral decomposition for self-adjoint elements in the maximum $\textrm {GCR}$ ideal of a von Neumann algebra with applications to noncommutative integration theory, Trans. Amer. Math. Soc. 133 (1968), 281–306. MR 230139, DOI 10.1090/S0002-9947-1968-0230139-1
Herbert Halpern, Module homomorphisms of a von Neumann algebra into its center, Trans. Amer. Math. Soc. 140 (1969), 183–193. MR 241986, DOI 10.1090/S0002-9947-1969-0241986-5
Herbert Halpern, Irreducible module homomorphisms of a von Neumann algebra into its center, Trans. Amer. Math. Soc. 140 (1969), 195–221. MR 241987, DOI 10.1090/S0002-9947-1969-0241987-7
Richard V. Kadison, Irreducible operator algebras, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 273–276. MR 85484, DOI 10.1073/pnas.43.3.273
Irving Kaplansky, A theorem on rings of operators, Pacific J. Math. 1 (1951), 227–232. MR 50181, DOI 10.2140/pjm.1951.1.227
Irving Kaplansky, The structure of certain operator algebras, Trans. Amer. Math. Soc. 70 (1951), 219–255. MR 42066, DOI 10.1090/S0002-9947-1951-0042066-0
Irving Kaplansky, Algebras of type $\textrm {I}$, Ann. of Math. (2) 56 (1952), 460–472. MR 50182, DOI 10.2307/1969654
Irving Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839–858. MR 58137, DOI 10.2307/2372552
J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578, DOI 10.1007/978-3-662-41914-4
Mitsuru Nakai, Some expectations in $AW^{\ast }$-algebras, Proc. Japan Acad. 34 (1958), 411–416. MR 95422
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Shôichirô Sakai, On type $I$ $C^{\ast }$-algebras, Proc. Amer. Math. Soc. 18 (1967), 861–863. MR 213887, DOI 10.1090/S0002-9939-1967-0213887-3
Kazuyuki Saitô, Non-commutative extension of Lusin’s theorem, Tohoku Math. J. (2) 19 (1967), 332–340. MR 226417, DOI 10.2748/tmj/1178243283
Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, No. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
I. E. Segal, Decompositions of operator algebras. I, Mem. Amer. Math. Soc. 9 (1951), 67. MR 44749
Erling Størmer, Positive linear maps of operator algebras, Acta Math. 110 (1963), 233–278. MR 156216, DOI 10.1007/BF02391860
Masamichi Takesaki, On the Hahn-Banach type theorem and the Jordan decomposition of module linear mapping over some operator algebras, K\B{o}dai Math. Sem. Rep. 12 (1960), 1–10. MR 115112
Jun Tomiyama and Masamichi Takesaki, Applications of fibre bundles to the certain class of $C^{\ast }$-algebras, Tohoku Math. J. (2) 13 (1961), 498–522. MR 139025, DOI 10.2748/tmj/1178244253
Minoru Tomita, Spectral theory of operator algebras. I, Math. J. Okayama Univ. 9 (1959/60), 63–98. MR 117581
G. Vincent-Smith, The Hahn-Banach theorem for modules, Proc. London Math. Soc. (3) 17 (1967), 72–90. MR 209870, DOI 10.1112/plms/s3-17.1.72
Harold Widom, Embedding in algebras of type I, Duke Math. J. 23 (1956), 309–324. MR 78668