Difference between revisions of "Yosida representation theorem"
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==The Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905022.png" />.== | ==The Riesz space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905022.png" />.== | ||
− | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905023.png" /> be a compact Hausdorff space. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905024.png" /> is an Archimedean Riesz space and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905027.png" />, is a strong unit. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905028.png" /> be a second compact Hausdorff space. The Banach–Stone theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905030.png" /> are isomorphic as Riesz spaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905032.png" /> are homeomorphic. As immediate corollaries one obtains that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905034.png" /> are isomorphic as algebras (pointwise multiplication), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905036.png" /> are homeomorphic; also, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905038.png" /> are isomorphic as Banach spaces (sup-norm), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905040.png" /> are homeomorphic. | + | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905023.png" /> be a compact Hausdorff space. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905024.png" /> is an Archimedean Riesz space and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905027.png" />, is a strong unit. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905028.png" /> be a second compact Hausdorff space. The Banach–Stone theorem says that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905030.png" /> are isomorphic as Riesz spaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905032.png" /> are homeomorphic. As immediate corollaries one obtains that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905034.png" /> are isomorphic as algebras ([[pointwise multiplication]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905036.png" /> are homeomorphic; also, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905038.png" /> are isomorphic as Banach spaces (sup-norm), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905040.png" /> are homeomorphic. |
A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905041.png" /> is extremely disconnected if every open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905042.png" /> has an open closure (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905043.png" /> is both open and closed). Nakano's theorem says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905044.png" /> is Dedekind complete if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905045.png" /> is extremely disconnected. It was also obtained independently by T. Ogasawara and M.H. Stone, cf. [[#References|[a2]]]. | A topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905041.png" /> is extremely disconnected if every open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905042.png" /> has an open closure (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905043.png" /> is both open and closed). Nakano's theorem says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905044.png" /> is Dedekind complete if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099050/y09905045.png" /> is extremely disconnected. It was also obtained independently by T. Ogasawara and M.H. Stone, cf. [[#References|[a2]]]. |
Revision as of 18:25, 1 December 2014
Let be a topological space,
the set of continuous real-valued functions on
(cf. Continuous functions, space of). Using the pointwise defined partial order:
if and only if
for all
,
becomes a Riesz space. The question arises whether it is possible to represent an arbitrary Riesz space by continuous functions with this order relation where, possibly, more general (extended) functions that can also take the values
and
may be used. Answers are given by various representation theorems. Below the Yosida representation theorem for the case of Archimedean Riesz spaces with a strong unit is described. For the Yosida representation theorem for Riesz spaces
, where
is a weak unit, see [a1], and for the more general Johnson–Kist representation theorem, see [a2].
A strong unit in a Riesz space is an element
such that for all
there is an
such that
, i.e. the principal ideal generated by
should equal the whole space. A weak unit in
is an element of
such that the principal band generated by
is all of
.
The Riesz space
.
Let be a compact Hausdorff space. Then
is an Archimedean Riesz space and the function
,
for all
, is a strong unit. Let
be a second compact Hausdorff space. The Banach–Stone theorem says that if
and
are isomorphic as Riesz spaces, then
and
are homeomorphic. As immediate corollaries one obtains that if
and
are isomorphic as algebras (pointwise multiplication), then
and
are homeomorphic; also, if
and
are isomorphic as Banach spaces (sup-norm), then
and
are homeomorphic.
A topological space is extremely disconnected if every open subset
has an open closure (i.e.
is both open and closed). Nakano's theorem says that
is Dedekind complete if and only if
is extremely disconnected. It was also obtained independently by T. Ogasawara and M.H. Stone, cf. [a2].
Representation of Archimedean Riesz spaces with strong unit.
Let be an Archimedean Riesz space. Let
be the set of maximal ideals of
. For each
, let
. Define a topology on
by taking the
as a subbase (cf. Pre-base). The closed sets of
are the sets
, where
runs through all subsets of
. This topology is called the hull-kernel topology on
. The construction is a fairly familiar one and occurs in several parts of mathematics. It is originally due to M.H. Stone. Depending on which sets of ideals are used, the mathematical specialism involved, and the ideosyncracies of authors it is also called the Zariski topology, Gel'fand topology, Gel'fand–Kolmogorov topology, Jacobson topology, Grothendieck topology, etc.
From now on, let have a strong unit
. For each
,
and there is a unique homomorphism of Riesz spaces
such that
. Using this, one defines for every
a function
by
. The number
can also be described as the unique real number such that
. One now has the following representation theorem (K. Yosida, S. Kakutani, M.G. Krein, S.G. Krein, H. Nakano). Let
,
,
be as just described. Then
defines a continuous function on
and the mapping
is a Riesz isomorphism of
onto a Riesz subspace
. There are a number of complementary facts. Using the Stone–Weierstrass theorem one obtains that
is norm dense in
; it is then also order dense.
Given , where
is a weak unit,
defines a metric on
, called the uniform metric. If
is complete with respect to this metric,
is called uniformly closed. A further addition to the representation theorem is then that
is isomorphic to
if and only if
is a strong unit and
is uniformly closed. This last statement, together with that fact that
is isomorphic to a sub-Riesz space of
(if
is a strong unit), is also referred to as the Krein–Kakutani theorem.
A final complement to the Yosida representation theorem is that if has the principal projection property, i.e.
for every principal band
, then
is zero dimensional and
contains all locally constant functions on
. This can also be called the Freudenthal spectral theorem, [a3], in the sense that that theorem in its traditional formulation is an immediate consequence of this result, cf. (the editorial comments to) Riesz space.
References
[a1] | A.W. Hager, L.C. Robertson, "Representing and ringifying a Riesz space" , Symp. Math. INDAM , 21 , Acad. Press (1977) pp. 411–432 |
[a2] | W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971) pp. Chapt. 7 |
[a3] | E. de Jonge, A.C.M. van Rooy, "Introduction to Riesz spaces" , Tracts , 8 , Math. Centre (1977) |
Yosida representation theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Yosida_representation_theorem&oldid=35263