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

Positivity , Volume 21 (2) – Mar 30, 2016

Loading next page...
 
/lp/springer_journal/the-order-continuity-in-ordered-algebras-it0ZeRlmAX
Publisher
Springer Journals
Copyright
Copyright © 2016 by 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

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off