Banach Journal of Mathematical Analysis

On domains of unbounded derivations of generalized B-algebras

Martin Weigt and Ioannis Zarakas

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Abstract

We study properties under which the domain of a closed derivation δ:D(δ)A of a generalized B-algebra A remains invariant under analytic functional calculus. For a complete, generalized B-algebra with jointly continuous multiplication, two sufficient conditions are assumed: that the unit of A belongs to the domain of the derivation, along with a condition related to the coincidence σA(x)=σD(δ)(x) of the (Allan) spectra for every element xD(δ). Certain results are derived concerning the spectra for a general element of the domain, in the realm of a domain which is advertibly complete or enjoys the Q-property. For a closed -derivation δ of a complete GB-algebra with jointly continuous multiplication such that 1D(δ) and x a normal element of the domain, f(x)D(δ) for every analytic function on a neighborhood of the spectrum of x. We also give an example of a closed derivation of a GB-algebra which does not contain the identity element. A condition for a closed derivation of a GB-algebra A to be the generator of a one-parameter group of automorphisms of A is provided along with a generalization of the Lumer–Phillips theorem for complete locally convex spaces.

Article information

Source
Banach J. Math. Anal., Volume 12, Number 4 (2018), 873-908.

Dates
Received: 17 May 2017
Accepted: 1 November 2017
First available in Project Euclid: 20 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1524211222

Digital Object Identifier
doi:10.1215/17358787-2017-0060

Mathematical Reviews number (MathSciNet)
MR3858753

Zentralblatt MATH identifier
06946295

Subjects
Primary: 46H05: General theory of topological algebras
Secondary: 46H35: Topological algebras of operators [See mainly 47Lxx] 46K05: General theory of topological algebras with involution 46L05: General theory of $C^*$-algebras

Keywords
$\mathrm{GB}^{*}$-algebra topological algebra derivation

Citation

Weigt, Martin; Zarakas, Ioannis. On domains of unbounded derivations of generalized B $^{*}$ -algebras. Banach J. Math. Anal. 12 (2018), no. 4, 873--908. doi:10.1215/17358787-2017-0060. https://projecteuclid.org/euclid.bjma/1524211222


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