# Every synaptic algebra has the monotone square root property

Every synaptic algebra has the monotone square root property A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW $$^{*}$$ ∗ -algebra. In this paper we prove that a synaptic algebra A has the monotone square root property, i.e., if $$0\le a,b\in A$$ 0 ≤ a , b ∈ A , then $$a\le b \Rightarrow a^{1/2}\le b^{1/2}$$ a ≤ b ⇒ a 1 / 2 ≤ b 1 / 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Every synaptic algebra has the monotone square root property

, Volume 21 (3) – Sep 1, 2016
12 pages

/lp/springer_journal/every-synaptic-algebra-has-the-monotone-square-root-property-y0EcTY52JE
Publisher
Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0443-z
Publisher site
See Article on Publisher Site

### Abstract

A synaptic algebra is a common generalization of several ordered algebraic structures based on algebras of self-adjoint operators, including the self-adjoint part of an AW $$^{*}$$ ∗ -algebra. In this paper we prove that a synaptic algebra A has the monotone square root property, i.e., if $$0\le a,b\in A$$ 0 ≤ a , b ∈ A , then $$a\le b \Rightarrow a^{1/2}\le b^{1/2}$$ a ≤ b ⇒ a 1 / 2 ≤ b 1 / 2 .

### Journal

PositivitySpringer Journals

Published: Sep 1, 2016

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