## Rocky Mountain Journal of Mathematics

### On the algebra of WCE operators

Yousef Estaremi

#### 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

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

#### 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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