Some exact sequences for Toeplitz algebras of spherical isometries

Prunaru, Bebe

doi:10.1090/S0002-9939-07-08893-4

Some exact sequences for Toeplitz algebras of spherical isometries
HTML articles powered by AMS MathViewer

by Bebe Prunaru PDF

Proc. Amer. Math. Soc. 135 (2007), 3621-3630 Request permission

Abstract:

A family $\{T_{j}\}_{j\in J}$ of commuting bounded operators on a Hilbert space $H$ is said to be a spherical isometry if $\sum _{j\in J}T^{*}_{j}T_{j}=1$ in the weak operator topology. We show that every commuting family $\mathcal {F}$ of spherical isometries is jointly subnormal, which means that it has a commuting normal extension $\widehat {\mathcal {F}}$ on some Hilbert space $\widehat {H}\supset H.$ Suppose now that the normal extension $\widehat {\mathcal {F}}$ is minimal. Then we show that every bounded operator $X$ in the commutant of $\mathcal {F}$ has a unique norm preserving extension to an operator $\widehat {X}$ in the commutant of $\widehat {\mathcal {F}}.$ Moreover, if $\mathcal {C}$ is the commutator ideal in $C^{*}(\mathcal {F}),$ then $C^{*}(\mathcal {F})/{\mathcal {C}}$ is *-isomorphic to $C^{*}(\widehat {\mathcal {F}}).$ We also show that the commutant of the minimal normal extension is completely isometric, via the compression mapping, to the space of Toeplitz-type operators associated to $\mathcal {F}.$ We apply these results to construct exact sequences for Toeplitz algebras on generalized Hardy spaces associated to strictly pseudoconvex domains.

References

M. B. Abrahamse, Toeplitz operators in multiply connected regions, Amer. J. Math. 96 (1974), 261–297. MR 361891, DOI 10.2307/2373633
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
William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
Ameer Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (1990), no. 2, 339–350. MR 1066811
Ameer Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (1990), no. 2, 339–350. MR 1066811
K. R. M. Attele and A. R. Lubin, Dilations and commutant lifting for jointly isometric operators—a geometric approach, J. Funct. Anal. 140 (1996), no. 2, 300–311. MR 1409039, DOI 10.1006/jfan.1996.0109
L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators, Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR 620794, DOI 10.1515/9781400881444
D. Beltiţă, B. Prunaru, Amenability, completely bounded projections, dynamical systems and smooth orbits, Integral Equations Operator Theory 57 (2007), no. 1, 1–17.
C. A. Berger and L. A. Coburn, One-parameter semigroups of isometries, Bull. Amer. Math. Soc. 76 (1970), 1125–1129. MR 265984, DOI 10.1090/S0002-9904-1970-12590-3
C. A. Berger, L. A. Coburn, and A. Lebow, Representation and index theory for $C^*$-algebras generated by commuting isometries, J. Functional Analysis 27 (1978), no. 1, 51–99. MR 0467392, DOI 10.1016/0022-1236(78)90019-8
Arlen Brown and P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/64), 89–102. MR 160136, DOI 10.1007/978-1-4613-8208-9_{1}9
Man Duen Choi and Edward G. Effros, Injectivity and operator spaces, J. Functional Analysis 24 (1977), no. 2, 156–209. MR 0430809, DOI 10.1016/0022-1236(77)90052-0
L. A. Coburn, Singular integral operators and Toeplitz operators on odd spheres, Indiana Univ. Math. J. 23 (1973/74), 433–439. MR 322595, DOI 10.1512/iumj.1973.23.23036
A. M. Davie and N. P. Jewell, Toeplitz operators in several complex variables, J. Functional Analysis 26 (1977), no. 4, 356–368. MR 0461195, DOI 10.1016/0022-1236(77)90020-9
Michael Didas, Spherical isometries are reflexive, Integral Equations Operator Theory 52 (2005), no. 4, 599–604. MR 2184607, DOI 10.1007/s00020-005-1354-8
Jacques Dixmier, Les moyennes invariantes dans les semi-groupes et leurs applications, Acta Sci. Math. (Szeged) 12 (1950), 213–227 (French). MR 37470
R. G. Douglas, On the operator equation $S^{\ast } XT=X$ and related topics, Acta Sci. Math. (Szeged) 30 (1969), 19–32. MR 250106
R. G. Douglas, On the operator equation $S^{\ast } XT=X$ and related topics, Acta Sci. Math. (Szeged) 30 (1969), 19–32. MR 250106
R. G. Douglas, On the operator equation $S^{\ast } XT=X$ and related topics, Acta Sci. Math. (Szeged) 30 (1969), 19–32. MR 250106
Jörg Eschmeier, Algebras of subnormal operators on the unit ball, J. Operator Theory 42 (1999), no. 1, 37–76. MR 1694801
Jörg Eschmeier, Algebras of subnormal operators on the unit ball, J. Operator Theory 42 (1999), no. 1, 37–76. MR 1694801
Jörg Eschmeier, Algebras of subnormal operators on the unit ball, J. Operator Theory 42 (1999), no. 1, 37–76. MR 1694801
Jörg Eschmeier and Mihai Putinar, Some remarks on spherical isometries, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 271–291. MR 1882699
Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
Takasi Itô, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958), 1–15. MR 0107177
J. Janas, Toeplitz spectral inclusion and generalized in modulus property, Proc. Amer. Math. Soc. 104 (1988), no. 1, 231–234. MR 958073, DOI 10.1090/S0002-9939-1988-0958073-8
Richard V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325–338. MR 43392, DOI 10.2307/1969534
Erik Løw, Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls, Math. Z. 190 (1985), no. 3, 401–410. MR 806898, DOI 10.1007/BF01215140
G. J. Murphy, Ordered groups and Toeplitz algebras, J. Operator Theory 18 (1987), no. 2, 303–326. MR 915512
G. J. Murphy, Ordered groups and Toeplitz algebras, J. Operator Theory 18 (1987), no. 2, 303–326. MR 915512
W. Forrest Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, DOI 10.1090/S0002-9939-1955-0069403-4
Carl Sundberg, Exact sequences for generalized Toeplitz operators, Proc. Amer. Math. Soc. 101 (1987), no. 4, 634–636. MR 911023, DOI 10.1090/S0002-9939-1987-0911023-1
Jun Tomiyama and Kôzô Yabuta, Toeplitz operators for uniform algebras, Tohoku Math. J. (2) 30 (1978), no. 1, 117–129. MR 482220, DOI 10.2748/tmj/1178230102
Harald Upmeier, Toeplitz operators and index theory in several complex variables, Operator Theory: Advances and Applications, vol. 81, Birkhäuser Verlag, Basel, 1996. MR 1384981, DOI 10.1016/s0168-9274(96)90019-7