## Banach Journal of Mathematical Analysis

### On domains of unbounded derivations of generalized B$^{*}$-algebras

#### Abstract

We study properties under which the domain of a closed derivation $\delta:D(\delta)\rightarrow A$ 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 $\sigma_{A}(x)=\sigma_{D(\delta)}(x)$ of the (Allan) spectra for every element $x\in D(\delta)$. 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 $\delta$ of a complete GB$^{*}$-algebra with jointly continuous multiplication such that $1\in D(\delta)$ and $x$ a normal element of the domain, $f(x)\in D(\delta)$ 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
Accepted: 1 November 2017
First available in Project Euclid: 20 April 2018

https://projecteuclid.org/euclid.bjma/1524211222

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

Mathematical Reviews number (MathSciNet)
MR3858753

Zentralblatt MATH identifier
06946295

#### 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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