Iwasawa decompositions of some infinite-dimensional Lie groups

Beltiţă, Daniel

doi:10.1090/S0002-9947-09-04824-7

Iwasawa decompositions of some infinite-dimensional Lie groups
HTML articles powered by AMS MathViewer

by Daniel Beltiţă PDF

Trans. Amer. Math. Soc. 361 (2009), 6613-6644 Request permission

Abstract:

We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. This provides a method to construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals $\mathfrak {S}_p(\mathcal {H})$ on a complex separable Hilbert space $\mathcal {H}$ if $1<p<\infty$.

References

Jonathan Arazy, Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces, Integral Equations Operator Theory 1 (1978), no. 4, 453–495. MR 516764, DOI 10.1007/BF01682937
William B. Arveson, Analyticity in operator algebras, Amer. J. Math. 89 (1967), 578–642. MR 223899, DOI 10.2307/2373237
William Arveson, Interpolation problems in nest algebras, J. Functional Analysis 20 (1975), no. 3, 208–233. MR 0383098, DOI 10.1016/0022-1236(75)90041-5
V. K. Balachandran, Simple $L^{\ast }$-algebras of classical type, Math. Ann. 180 (1969), 205–219. MR 243362, DOI 10.1007/BF01350738
Vladimir Balan and Josef Dorfmeister, Birkhoff decompositions and Iwasawa decompositions for loop groups, Tohoku Math. J. (2) 53 (2001), no. 4, 593–615. MR 1862221, DOI 10.2748/tmj/1113247803
Daniel Beltiţă, On Banach-Lie algebras, spectral decompositions and complex polarizations, Recent advances in operator theory, operator algebras, and their applications, Oper. Theory Adv. Appl., vol. 153, Birkhäuser, Basel, 2005, pp. 13–38. MR 2105467, DOI 10.1007/3-7643-7314-8_{2}
Daniel Beltiţă, Smooth homogeneous structures in operator theory, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 137, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2188389
D. Beltiţă, Functional analytic background for a theory of infinite-dimensional reductive Lie groups, in: K.-H. Neeb, A. Pianzola (eds.), Developments and Trends in Infinite Dimensional Lie Theory, Progress in Mathematics, Birkhäuser Verlag, Basel 2009.
Daniel Beltiţă and Bebe Prunaru, Amenability, completely bounded projections, dynamical systems and smooth orbits, Integral Equations Operator Theory 57 (2007), no. 1, 1–17. MR 2294192, DOI 10.1007/s00020-006-1446-0
D. Beltiţă and T. S. Ratiu, Symplectic leaves in real Banach Lie-Poisson spaces, Geom. Funct. Anal. 15 (2005), no. 4, 753–779. MR 2221149, DOI 10.1007/s00039-005-0524-9
Daniel Beltiţă and Tudor S. Ratiu, Geometric representation theory for unitary groups of operator algebras, Adv. Math. 208 (2007), no. 1, 299–317. MR 2304319, DOI 10.1016/j.aim.2006.02.009
Daniel Beltiţă and Mihai Şabac, Lie algebras of bounded operators, Operator Theory: Advances and Applications, vol. 120, Birkhäuser Verlag, Basel, 2001. MR 1825892, DOI 10.1007/978-3-0348-8332-0
A. M. Bloch, H. Flaschka, and T. Ratiu, A Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus, Invent. Math. 113 (1993), no. 3, 511–529. MR 1231835, DOI 10.1007/BF01244316
Robert P. Boyer, Representation theory of the Hilbert-Lie group $\textrm {U}({\mathfrak {H}})_{2}$, Duke Math. J. 47 (1980), no. 2, 325–344. MR 575900
Robert P. Boyer, Representation theory of infinite-dimensional unitary groups, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 381–391. MR 1216198, DOI 10.1090/conm/145/1216198
A. L. Carey, Some homogeneous spaces and representations of the Hilbert Lie group ${\scr U}(H)_2$, Rev. Roumaine Math. Pures Appl. 30 (1985), no. 7, 505–520. MR 826232
Gustavo Corach and José E. Galé, On amenability and geometry of spaces of bounded representations, J. London Math. Soc. (2) 59 (1999), no. 1, 311–329. MR 1688504, DOI 10.1112/S0024610798006759
Kenneth R. Davidson, Nest algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. Triangular forms for operator algebras on Hilbert space. MR 972978
Ivan Dimitrov, Ivan Penkov, and Joseph A. Wolf, A Bott-Borel-Weil theory for direct limits of algebraic groups, Amer. J. Math. 124 (2002), no. 5, 955–998. MR 1925340
Ken Dykema, Tadeusz Figiel, Gary Weiss, and Mariusz Wodzicki, Commutator structure of operator ideals, Adv. Math. 185 (2004), no. 1, 1–79. MR 2058779, DOI 10.1016/S0001-8708(03)00141-5
J. A. Erdos, The triangular factorization of operators on Hilbert space, Indiana Univ. Math. J. 22 (1972/73), 939–950. MR 336404, DOI 10.1512/iumj.1973.22.22078
J. A. Erdos, Triangular integration on symmetrically normed ideals, Indiana Univ. Math. J. 27 (1978), no. 3, 401–408. MR 473892, DOI 10.1512/iumj.1978.27.27029
J. A. Erdos and W. E. Longstaff, The convergence of triangular integrals of operators on Hilbert space, Indiana Univ. Math. J. 22 (1972/73), 929–938. MR 336403, DOI 10.1512/iumj.1973.22.22077
José E. Galé, Geometry of orbits of representations of amenable groups and algebras, Rev. R. Acad. Cienc. Exactas Fís. Quím. Nat. Zaragoza (2) 61 (2006), 7–46 (Spanish, with Spanish summary). MR 2311998
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
I. C. Gohberg and M. G. Kreĭn, Theory and applications of Volterra operators in Hilbert space, Translations of Mathematical Monographs, Vol. 24, American Mathematical Society, Providence, R.I., 1970. Translated from the Russian by A. Feinstein. MR 0264447
Hendrik Grundling, Generalising group algebras, J. London Math. Soc. (2) 72 (2005), no. 3, 742–762. MR 2190335, DOI 10.1112/S0024610705007003
Pierre de la Harpe, Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space, Lecture Notes in Mathematics, Vol. 285, Springer-Verlag, Berlin-New York, 1972. MR 0476820
L. A. Harris and W. Kaup, Linear algebraic groups in infinite dimensions, Illinois J. Math. 21 (1977), no. 3, 666–674. MR 460551
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001. Corrected reprint of the 1978 original. MR 1834454, DOI 10.1090/gsm/034
Kenkichi Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558. MR 29911, DOI 10.2307/1969548
Victor Kaftal and Gary Weiss, Traces, ideals, and arithmetic means, Proc. Natl. Acad. Sci. USA 99 (2002), no. 11, 7356–7360. MR 1907839, DOI 10.1073/pnas.112074699
V. Kaftal, G. Weiss, $B(H)$ lattices, density and arithmetic mean ideals, preprint, 2006.
Peter Kellersch, Eine Verallgemeinerung der Iwasawa Zerlegung in Loop Gruppen, DGDS. Differential Geometry—Dynamical Systems. Monographs, vol. 4, Geometry Balkan Press, Bucharest, 2004 (German, with German summary). Dissertation, Technische Universität München, Munich, 1999; Available electronically at http://vectron.mathem.pub.ro/dgds/mono/dgdsmono.htm. MR 2158164
A. A. Kirillov, Representations of the infinite-dimensional unitary group, Dokl. Akad. Nauk. SSSR 212 (1973), 288–290 (Russian). MR 0340487
Anthony W. Knapp, Lie groups beyond an introduction, Progress in Mathematics, vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1399083, DOI 10.1007/978-1-4757-2453-0
Bertram Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413–455 (1974). MR 364552
Serge Lang, Fundamentals of differential geometry, Graduate Texts in Mathematics, vol. 191, Springer-Verlag, New York, 1999. MR 1666820, DOI 10.1007/978-1-4612-0541-8
David R. Larson, Nest algebras and similarity transformations, Ann. of Math. (2) 121 (1985), no. 3, 409–427. MR 794368, DOI 10.2307/1971180
Jiang-Hua Lu and Tudor Ratiu, On the nonlinear convexity theorem of Kostant, J. Amer. Math. Soc. 4 (1991), no. 2, 349–363. MR 1086967, DOI 10.1090/S0894-0347-1991-1086967-2
Paul S. Muhly, Kichi-Suke Saito, and Baruch Solel, Coordinates for triangular operator algebras, Ann. of Math. (2) 127 (1988), no. 2, 245–278. MR 932297, DOI 10.2307/2007053
Loki Natarajan, Enriqueta Rodríguez-Carrington, and Joseph A. Wolf, The Bott-Borel-Weil theorem for direct limit groups, Trans. Amer. Math. Soc. 353 (2001), no. 11, 4583–4622. MR 1650034, DOI 10.1090/S0002-9947-01-02452-7
Karl-Hermann Neeb, Holomorphic highest weight representations of infinite-dimensional complex classical groups, J. Reine Angew. Math. 497 (1998), 171–222. MR 1617431, DOI 10.1515/crll.1998.038
Karl-Hermann Neeb, Classical Hilbert-Lie groups, their extensions and their homotopy groups, Geometry and analysis on finite- and infinite-dimensional Lie groups (Będlewo, 2000) Banach Center Publ., vol. 55, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 87–151. MR 1911982, DOI 10.4064/bc55-0-6
Karl-Hermann Neeb, A Cartan-Hadamard theorem for Banach-Finsler manifolds, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), 2002, pp. 115–156. MR 1950888, DOI 10.1023/A:1021221029301
Karl-Hermann Neeb, Infinite-dimensional groups and their representations, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 213–328. MR 2042690
Karl-Hermann Neeb and Bent Ørsted, Unitary highest weight representations in Hilbert spaces of holomorphic functions on infinite-dimensional domains, J. Funct. Anal. 156 (1998), no. 1, 263–300. MR 1632917, DOI 10.1006/jfan.1997.3233
Andreas Neumann, An infinite-dimensional version of the Schur-Horn convexity theorem, J. Funct. Anal. 161 (1999), no. 2, 418–451. MR 1674643, DOI 10.1006/jfan.1998.3348
Andreas Neumann, An infinite dimensional version of the Kostant convexity theorem, J. Funct. Anal. 189 (2002), no. 1, 80–131. MR 1887630, DOI 10.1006/jfan.2000.3739
G. I. Ol′šanskiĭ, Unitary representations of the infinite-dimensional classical groups $\textrm {U}(p,\,\infty )$, $\textrm {SO}_{0}(p,\,\infty )$, $\textrm {Sp}(p,\,\infty )$, and of the corresponding motion groups, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 32–44, 96 (Russian). MR 509382
G. I. Ol′shanskiĭ, The method of holomorphic extensions in the theory of unitary representations of infinite-dimensional classical groups, Funktsional. Anal. i Prilozhen. 22 (1988), no. 4, 23–37, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 4, 273–285 (1989). MR 976993, DOI 10.1007/BF01077419
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029
Doug Pickrell, Separable representations for automorphism groups of infinite symmetric spaces, J. Funct. Anal. 90 (1990), no. 1, 1–26. MR 1047575, DOI 10.1016/0022-1236(90)90078-Y
David R. Pitts, Factorization problems for nests: factorization methods and characterizations of the universal factorization property, J. Funct. Anal. 79 (1988), no. 1, 57–90. MR 950084, DOI 10.1016/0022-1236(88)90030-4
S. C. Power, Factorization in analytic operator algebras, J. Funct. Anal. 67 (1986), no. 3, 413–432. MR 845465, DOI 10.1016/0022-1236(86)90033-9
I. E. Segal, The structure of a class of representations of the unitary group on a Hilbert space, Proc. Amer. Math. Soc. 8 (1957), 197–203. MR 84122, DOI 10.1090/S0002-9939-1957-0084122-8
Şerban Strătilă and Dan Voiculescu, Representations of AF-algebras and of the group $U(\infty )$, Lecture Notes in Mathematics, Vol. 486, Springer-Verlag, Berlin-New York, 1975. MR 0458188
A.B. Tumpach, Variétés Kählériennes et Hyperkählériennes de Dimension Infinie, Ph.D. Thesis, École Polytechnique, Paris, 2005.
A.B. Tumpach, Mostow decomposition theorems for $L^*$-groups and applications to affine coadjoint orbits and stable manifolds, preprint math-ph/0605039.
Harald Upmeier, Symmetric Banach manifolds and Jordan $C^\ast$-algebras, North-Holland Mathematics Studies, vol. 104, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 96. MR 776786
Gary Weiss, $B(H)$-commutators: a historical survey, Recent advances in operator theory, operator algebras, and their applications, Oper. Theory Adv. Appl., vol. 153, Birkhäuser, Basel, 2005, pp. 307–320. MR 2105485, DOI 10.1007/3-7643-7314-8_{2}0
Joseph A. Wolf, Principal series representations of direct limit groups, Compos. Math. 141 (2005), no. 6, 1504–1530. MR 2188447, DOI 10.1112/S0010437X05001430