Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON FRAMES FOR COUNTABLY GENERATED HILBERT MODULES OVER LOCALLY C*-ALGEBRAS

  • Alizadeh, Leila (Department of Mathematics Mashhad Branch Islamic Azad University) ;
  • Hassani, Mahmoud (Department of Mathematics Mashhad Branch Islamic Azad University)
  • Received : 2017.05.09
  • Accepted : 2017.08.29
  • Published : 2018.04.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let $\mathcal{X}$ be a countably generated Hilbert module over a locally $C^*$-algebra $\mathcal{A}$ in multiplier module M($\mathcal{X}$) of $\mathcal{X}$. We propose the necessary and sufficient condition such that a sequence $\{h_n:n{{\in}}\mathbb{N}\}$ in M($\mathcal{X}$) is a standard frame of multipliers in $\mathcal{X}$. We also show that if T in $b(L_{\mathcal{A}}(\mathcal{X}))$, the space of bounded maps in set of all adjointable maps on $\mathcal{X}$, is surjective and $\{h_n:n{{\in}}\mathbb{N}\}$ is a standard frame of multipliers in $\mathcal{X}$, then $\{T{\circ}h_n:n{\in}\mathbb{N}}$ is a standard frame of multipliers in $\mathcal{X}$, too.

Keywords

References

  1. L. Arambasic, On frames for countably generated Hilbert $C^{\ast}$-modules, Proc. Amer. Math. Soc. 135 (2007), no. 2, 469-478. https://doi.org/10.1090/S0002-9939-06-08498-X
  2. I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), no. 5, 1271-1283. https://doi.org/10.1063/1.527388
  3. R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341-366. https://doi.org/10.1090/S0002-9947-1952-0047179-6
  4. M. Frank and D. R. Larson, Frames in Hilbert $C^{\ast}$-modules and $C^{\ast}$-algebras, J. Operator Theory 48 (2002), no. 2, 273-314.
  5. A. Inoue, Locally CSp∗-algebra, Mem. Fac. Sci. Kyushu Univ. Ser. A 25 (1971), 197-235.
  6. M. Joita, On bounded module maps between Hilbert modules over locally $C^{\ast}$-algebras, Acta Math. Univ. Comenian. (N.S.) 74 (2005), no. 1, 71-78.
  7. M. Joita, Hilbert modules over locally $C^{\ast}$-algebras, University of Bucharest Press, 2006.
  8. M. Joita, On frames in Hilbert modules over pro-$C^{\ast}$-algebras, Topology Appl. 156 (2008), no. 1, 83-92. https://doi.org/10.1016/j.topol.2007.12.015
  9. M. Joita, On multipliers of Hilbert modules over pro-$C^{\ast}$-algebras, Studia Math. 185 (2008), no. 3, 263-277. https://doi.org/10.4064/sm185-3-4
  10. N. C. Phillips, Inverse limits of $C^{\ast}$-algebras, J. Operator Theory 19 (1988), no. 1, 159-195.