Rocky Mountain Journal of Mathematics

On the algebra of WCE operators

Yousef Estaremi

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Abstract

In this paper, we consider the algebra of WCE operators on $L^p$-spaces, and we investigate some al\-ge\-braic properties of it. For instance, we show that the set of normal WCE operators is a unital finite Von Neumann algebra, and we obtain the spectral measure of a normal WCE operator on $L^2(\mathcal {F})$. Then, we specify the form of projections in the Von Neumann algebra of normal WCE operators, and we obtain that, if the underlying measure space is purely atomic, then all projections are minimal. In the non-atomic case, there is no minimal projection. Also, we give a non-commutative operator algebra on which the spectral map is subadditive and submultiplicative. As a consequence, we obtain that the set of quasinilpotents is an ideal, and we get a relation between quasinilpotents and commutators. Moreover, we give some sufficient conditions for an algebra of WCE operators to be triangularizable, and consequently, that its quotient space over its quasinilpotents is commutative.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 2 (2018), 501-517.

Dates
First available in Project Euclid: 4 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1528077630

Digital Object Identifier
doi:10.1216/RMJ-2018-48-2-501

Mathematical Reviews number (MathSciNet)
MR3809155

Zentralblatt MATH identifier
06883478

Subjects
Primary: 47L80: Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)

Keywords
Conditional expectation operator commutant commutator Von Neumann algebras triangularizable algebra

Citation

Estaremi, Yousef. On the algebra of WCE operators. Rocky Mountain J. Math. 48 (2018), no. 2, 501--517. doi:10.1216/RMJ-2018-48-2-501. https://projecteuclid.org/euclid.rmjm/1528077630


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References

  • P.G. Dodds, C.B. Huijsmans and B. De Pagter, Characterizations of conditional expectation-type operators, Pacific J. Math. 141 (1990), 55–77.
  • R.G. Douglas, Contractive projections on an $L\sb{1}$ space, Pacific J. Math. 15 (1965), 443–462.
  • Y. Estaremi and M.R. Jabbarzadeh, Weighted Lambert type operators on $L^{p}$-spaces, Oper. Matrices 1 (2013), 101–116.
  • J.J. Grobler and B. de Pagter, Operators representable as multiplication-conditional expectation operators, J. Oper. Th. 48 (2002), 15–40.
  • A. Katavolos and H. Radjavi, Simultaneous triangularization of operators on a Banach space, J. Lond. Math. Soc. 41 (1990) 547–554.
  • A. Lambert, Conditional expectation related characterizations of the commutant of an abelian $W^*$-algebra, Far East J. Math. Sci. 2 (1994), 1–7.
  • Shu-Teh Chen Moy, Characterizations of conditional expectation as a transformation on function spaces, Pacific J. Math. 4 (1954), 47–63.
  • G.J. Murphy, $\C^{\ast}$-algebras and operator theory, Academic Press, Boston, 1990.
  • M.M. Rao, Conditional measure and applications, Marcel Dekker, New York, 1993.
  • A.C. Zaanen, Integration, North-Holland, Amsterdam, 1967.