Geodesic
Diagonalization of compact operators on Hilbert modules over $C^*$-algebras of real rank zero
Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 865-870.

Voir la notice de l'article provenant de la source Math-Net.Ru

The classical Hilbert–Schmidt theorem can be extended to compact operators on Hilbert $\mathscr A$-modules over $W^*$-algebras of finite type; i.e., with minor restrictions, compact operators on $\mathscr H_\mathscr A^*$ can be diagonalized over $\mathscr A$. We show that if $B$ is a weakly dense $C^*$-subalgebra of $\mathscr A$ with real rank zero and if some additional condition holds, then the natural extension from $\mathscr H_\mathscr B$ to $\mathscr H_\mathscr A^*\supset\mathscr H_\mathscr B$ of a compact operator can be diagonalized so that the diagonal elements belong to the original $C^*$-algebra $\mathscr B$. 
@article{MZM_1997_62_6_a6,
     author = {V. M. Manuilov},
     title = {Diagonalization of compact operators on {Hilbert} modules over $C^*$-algebras of real rank zero},
     journal = {Matemati\v{c}eskie zametki},
     pages = {865--870},
     publisher = {mathdoc},
     volume = {62},
     number = {6},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a6/}
}
TY  - JOUR
AU  - V. M. Manuilov
TI  - Diagonalization of compact operators on Hilbert modules over $C^*$-algebras of real rank zero
JO  - Matematičeskie zametki
PY  - 1997
SP  - 865
EP  - 870
VL  - 62
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a6/
LA  - ru
ID  - MZM_1997_62_6_a6
ER  - 
%0 Journal Article
%A V. M. Manuilov
%T Diagonalization of compact operators on Hilbert modules over $C^*$-algebras of real rank zero
%J Matematičeskie zametki
%D 1997
%P 865-870
%V 62
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a6/
%G ru
%F MZM_1997_62_6_a6
V. M. Manuilov. Diagonalization of compact operators on Hilbert modules over $C^*$-algebras of real rank zero. Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 865-870. http://geodesic.mathdoc.fr/item/MZM_1997_62_6_a6/

[1] Paschke W. L., “Inner product modules over $B^*$-algebras”, Trans. Amer. Math. Soc., 182 (1973), 443–468 | DOI | MR | Zbl

[2] Kasparov G. G., “Topologicheskie invarianty ellipticheskikh operatorov. I: $K$-gomologii”, Izv. AN SSSR. Ser. matem., 39 (1975), 796–838 | MR | Zbl

[3] Mischenko A. S., Fomenko A. T., “Indeks ellipticheskikh operatorov nad $C^*$-algebrami”, Izv. AN SSSR. Ser. matem., 43 (1979), 831–859 | MR | Zbl

[4] Paschke W. L., “The double $B$-dual of an inner product module over a $C^*$-algebra”, Canad. J. Math., 26 (1974), 1272–1280 | MR | Zbl

[5] Kadison R. V., “Diagonalizing matrices”, Amer. J. Math., 106 (1984), 1451–1468 | DOI | MR | Zbl

[6] Grove K., Pedersen G. K., “Diagonalizing matrices over $C(X)$”, J. Funct. Anal., 59 (1984), 64–89 | DOI | MR

[7] Murphy Q. J., “Diagonality in $C^*$-algebras”, Math. Z., 199 (1990), 279–284 | DOI | MR

[8] Frank M., Manuilov V. M., “Diagonalizing “compact” operators on Hilbert $W^*$-modules”, Z. Anal. Anwendungen, 14 (1995), 33–41 | MR | Zbl

[9] Sunder V. S., Thomsen K., “Unitary orbits of selfadjoints in some $C^*$-algebras”, Houston J. Math., 18 (1992), 127–137 | MR | Zbl

[10] Manuilov V. M., “Diagonalizatsiya kompaktnykh operatorov v gilbertovykh modulyakh nad $W^*$-algebrami konechnogo tipa”, UMN, 49:2 (1994), 159–160 | MR | Zbl

[11] Manuilov V. M., “Diagonalization of compact operators in Hilbert modules over finite $W^*$-algebras”, Ann. Global Anal. Geom., 13:3 (1995), 207–226 | DOI | MR | Zbl

[12] Brown L. G., Pedersen G. K., “$C^*$-algebras of real rank zero”, J. Funct. Anal., 99 (1991), 131–149 | DOI | MR | Zbl

[13] Murray F. J., von Neumann J., “On rings of operators”, Ann. of Math., 37 (1936), 116–229 | DOI | MR | Zbl

[14] Takesaki M., Theory of Operator Algebras, V. I, Springer, New York–Heidelberg–Berlin, 1979

[15] Zhang S., “Diagonalizing projections”, Pacific J. Math., 145 (1990), 181–200 | MR | Zbl

[16] Choi M.-D., Elliot G. A., “Density of self-adjoint elements with finite spectrum in an irrational rotation $C^*$-algebra”, Math. Scand., 67 (1990), 73–86 | MR | Zbl

[17] Brenken B., “Representations and automorphisms of the irrational rotation algebra”, Pacific J. Math., 111 (1984), 257–282 | MR | Zbl