Rocky Mountain Journal of Mathematics

On the algebra of WCE operators

Yousef Estaremi

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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

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

First available in Project Euclid: 4 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Conditional expectation operator commutant commutator Von Neumann algebras triangularizable algebra


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.

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