Banach Journal of Mathematical Analysis
- Banach J. Math. Anal.
- Volume 13, Number 1 (2019), 113-132.
Analytic aspects of evolution algebras
We prove that every evolution algebra is a normed algebra, for an -norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra is a Banach algebra if and only if , where is finite-dimensional and is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator of with respect to a natural basis , and we show that need not be continuous. Moreover, if is finite-dimensional and , then is given by , where and is the multiplication operator , for . We establish necessary and sufficient conditions for convergence of , for all , in terms of the multiplicative spectrum of . Namely, converges, for all , if and only if and , where denotes the index of in the spectrum of .
Banach J. Math. Anal., Volume 13, Number 1 (2019), 113-132.
Received: 10 February 2018
Accepted: 26 May 2018
First available in Project Euclid: 28 November 2018
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Mellon, P.; Velasco, M. Victoria. Analytic aspects of evolution algebras. Banach J. Math. Anal. 13 (2019), no. 1, 113--132. doi:10.1215/17358787-2018-0018. https://projecteuclid.org/euclid.bjma/1543395629