# The order continuity in ordered algebras

The order continuity in ordered algebras Let A be an ordered algebra with a unit $$\mathbf{e}$$ e and a cone $$A^+$$ A + . The class of order continuous elements $$A_\mathrm{n}$$ A n of A is introduced and studied. If $$A=L(E)$$ A = L ( E ) , where E is a Dedekind complete Riesz space, this class coincides with the band $$L_\mathrm{n}(E)$$ L n ( E ) of all order continuous operators on E. Special subclasses of  $$A_\mathrm {n}$$ A n are considered. Firstly, the order ideal $$A_\mathbf{e}$$ A e generated by $$\mathbf{e}$$ e . It is shown that $$A_\mathbf{e}$$ A e can be embedded into the algebra of continuous functions and, in particular, is a commutative subalgebra of A. If A is an ordered Banach algebra with normal cone $$A^+$$ A + then $$A_\mathbf{e}$$ A e is an AM-space and is closed in A. Secondly, the notion of an orthomorphism in the ordered algebra A is introduced. Among others, the conditions under which orthomorphisms are order continuous, are considered. In the second part, the main emphasis will be on the case of an ordered $$C^*$$ C ∗ -algebra A and, in particular, on the case of the algebra B(H), where H is an ordered Hilbert space with self-adjoint cone $$H^+$$ H + . If the cone $$A^+$$ A + is normal then every element of $$A_\mathbf{e}$$ A e is hermitian. In H the operations are introduced which coincide with the lattice ones when H is a Riesz space. It is shown that every regular $$T\in B(H)$$ T ∈ B ( H ) is an order continuous element and operators $$T\in (B(H))_I$$ T ∈ ( B ( H ) ) I have properties which are analogous to the properties of orthomorphisms on Riesz spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# The order continuity in ordered algebras

, Volume 21 (2) – Mar 30, 2016
36 pages

/lp/springer_journal/the-order-continuity-in-ordered-algebras-it0ZeRlmAX
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/s11117-016-0406-4
Publisher site
See Article on Publisher Site

### Abstract

Let A be an ordered algebra with a unit $$\mathbf{e}$$ e and a cone $$A^+$$ A + . The class of order continuous elements $$A_\mathrm{n}$$ A n of A is introduced and studied. If $$A=L(E)$$ A = L ( E ) , where E is a Dedekind complete Riesz space, this class coincides with the band $$L_\mathrm{n}(E)$$ L n ( E ) of all order continuous operators on E. Special subclasses of  $$A_\mathrm {n}$$ A n are considered. Firstly, the order ideal $$A_\mathbf{e}$$ A e generated by $$\mathbf{e}$$ e . It is shown that $$A_\mathbf{e}$$ A e can be embedded into the algebra of continuous functions and, in particular, is a commutative subalgebra of A. If A is an ordered Banach algebra with normal cone $$A^+$$ A + then $$A_\mathbf{e}$$ A e is an AM-space and is closed in A. Secondly, the notion of an orthomorphism in the ordered algebra A is introduced. Among others, the conditions under which orthomorphisms are order continuous, are considered. In the second part, the main emphasis will be on the case of an ordered $$C^*$$ C ∗ -algebra A and, in particular, on the case of the algebra B(H), where H is an ordered Hilbert space with self-adjoint cone $$H^+$$ H + . If the cone $$A^+$$ A + is normal then every element of $$A_\mathbf{e}$$ A e is hermitian. In H the operations are introduced which coincide with the lattice ones when H is a Riesz space. It is shown that every regular $$T\in B(H)$$ T ∈ B ( H ) is an order continuous element and operators $$T\in (B(H))_I$$ T ∈ ( B ( H ) ) I have properties which are analogous to the properties of orthomorphisms on Riesz spaces.

### Journal

PositivitySpringer Journals

Published: Mar 30, 2016

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