varvec{\varepsilon }$$-Shading Operator on Riesz Spaces and Order Continuity of Orthogonally Additive Operators
Given a Riesz space E and $$0 < e \in E$$ 0 < e ∈ E , we introduce and study an order continuous orthogonally additive operator which is an $$\varepsilon $$ ε -approximation of the principal lateral band projection $$Q_e$$ Q e (the order discontinuous lattice homomorphism $$Q_e :E \rightarrow...
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Veröffentlicht in: | Resultate der Mathematik 2022-10, Vol.77 (5), Article 209 |
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Hauptverfasser: | , |
Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | Given a Riesz space
E
and
$$0 < e \in E$$
0
<
e
∈
E
, we introduce and study an order continuous orthogonally additive operator which is an
$$\varepsilon $$
ε
-approximation of the principal lateral band projection
$$Q_e$$
Q
e
(the order discontinuous lattice homomorphism
$$Q_e :E \rightarrow E$$
Q
e
:
E
→
E
which assigns to any element
$$x \in E$$
x
∈
E
the maximal common fragment
$$Q_e(x)$$
Q
e
(
x
)
of
e
and
x
). This gives a tool for constructing an order continuous orthogonally additive operator with given properties. Using it, we provide the first example of an order discontinuous orthogonally additive operator which is both uniformly-to-order continuous and horizontally-to-order continuous. Another result gives sufficient conditions on Riesz spaces
E
and
F
under which such an example does not exist. Our next main result asserts that, if
E
has the principal projection property and
F
is a Dedekind complete Riesz space then every order continuous regular orthogonally additive operator
$$T :E \rightarrow F$$
T
:
E
→
F
has order continuous modulus |
T
|. Finally, we provide an example showing that the latter theorem is not true for
$$E = C[0,1]$$
E
=
C
[
0
,
1
]
and some Dedekind complete
F
. The above results answer two problems posed in a recent paper by O. Fotiy, I. Krasikova, M. Pliev and the second named author. |
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ISSN: | 1422-6383 1420-9012 |
DOI: | 10.1007/s00025-022-01742-0 |