𝐶*-algebras generated by a subnormal operator

Chan, Kit; C̆uc̆ković, Z̆eljko

doi:10.1090/S0002-9947-99-02389-2

$C^\ast$-algebras generated by a subnormal operator
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by Kit C. Chan and Z̆eljko C̆uc̆ković

Trans. Amer. Math. Soc. 351 (1999), 1445-1460

DOI: https://doi.org/10.1090/S0002-9947-99-02389-2

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Abstract:

Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if $T$ is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under $T$ cannot be orthogonal to each other. Then we show that the $C^*$-algebra generated by $T$ and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that $C^*$-algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn’s theorems for the Hardy space and the Bergman space of the unit ball.

References

Sheldon Axler and Allen Shields, Algebras generated by analytic and harmonic functions, Indiana Univ. Math. J. 36 (1987), no. 3, 631–638. MR 905614, DOI 10.1512/iumj.1987.36.36034
Hari Bercovici, Ciprian Foias, and Carl Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Mathematics, vol. 56, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1985. MR 787041, DOI 10.1090/cbms/056
John Bunce, Characters on singly generated $C^{\ast }$-algebras, Proc. Amer. Math. Soc. 25 (1970), 297–303. MR 259622, DOI 10.1090/S0002-9939-1970-0259622-4
John Bunce, The joint spectrum of commuting nonnormal operators, Proc. Amer. Math. Soc. 29 (1971), 499–505. MR 283602, DOI 10.1090/S0002-9939-1971-0283602-7
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
John B. Conway, Subnormal operators, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR 634507
A. T. Dash, Joint spectra, Studia Math. 45 (1973), 225–237. MR 336381, DOI 10.4064/sm-45-3-225-237
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 0458185
Richard Frankfurt, Subnormal weighted shifts and related function spaces, J. Math. Anal. Appl. 52 (1975), no. 3, 471–489. MR 482334, DOI 10.1016/0022-247X(75)90074-8
Richard Frankfurt, Subnormal weighted shifts and related function spaces. II, J. Math. Anal. Appl. 55 (1976), no. 1, 2–17. MR 482335, DOI 10.1016/0022-247X(76)90273-0
Richard Frankfurt, Function spaces associated with radially symmetric measures, J. Math. Anal. Appl. 60 (1977), no. 2, 502–541. MR 507896, DOI 10.1016/0022-247X(77)90039-7
T. W. Gamelin, Uniform Algebras, Chelsea, New York, 1984.
P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity, Second Edition, Chelsea, New York, 1957.
Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952, DOI 10.1007/978-1-4684-9330-6
Nicholas P. Jewell, Multiplication by the coordinate functions on the Hardy space of the unit sphere in $\textbf {C}^{n}$, Duke Math. J. 44 (1977), no. 4, 839–851. MR 463966
Gerard E. Keough, Subnormal operators, Toeplitz operators and spectral inclusion, Trans. Amer. Math. Soc. 263 (1981), no. 1, 125–135. MR 590415, DOI 10.1090/S0002-9947-1981-0590415-6
Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
Arthur Lubin, Weighted shifts and products of subnormal operators, Indiana Univ. Math. J. 26 (1977), no. 5, 839–845. MR 448139, DOI 10.1512/iumj.1977.26.26067
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6
Donald Sarason, The $H^{p}$ spaces of an annulus, Mem. Amer. Math. Soc. 56 (1965), 78. MR 188824
Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899