Order-Unit Spaces which are Banach Dual Spaces
Spaces of selfadjoint elements of a C*-algebra or a von Neumann algebra, and also JB- and JBW-algebras are examples of order-unit spaces. A von Neumann algebra and a JBW-algebra possess predual spaces, but, generally speaking, a JB-algebra and a C*-algebra don't have this property. In this work...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
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| Cite this: | Order-Unit Spaces which are Banach Dual Spaces / M.A. Berdikulov // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 130-137. — Бібліогр.: 11 назв. — англ. |
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| citation_txt | Order-Unit Spaces which are Banach Dual Spaces / M.A. Berdikulov // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 130-137. — Бібліогр.: 11 назв. — англ. |
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| description | Spaces of selfadjoint elements of a C*-algebra or a von Neumann algebra, and also JB- and JBW-algebras are examples of order-unit spaces. A von Neumann algebra and a JBW-algebra possess predual spaces, but, generally speaking, a JB-algebra and a C*-algebra don't have this property. In this work, conditions are found for an order-unit space to possess a predual space. Moreover, a condition is obtained characterizing JBW-algebras among order-unit spaces having a predual space.
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Journal of Mathematical Physics, Analysis, Geometry
2006, vol. 2, No. 2, pp. 130�137
Order-Unit Spaces which are Banach Dual Spaces
M.A. Berdikulov
Institute of Mathematics, Uzbek Academy of Sciences
29 F. Khodjaev Str., Tashkent, 700125, Uzbekistan
E-mail:mathinst@uzsci.net
Received August 4, 2005
Spaces of selfadjoint elements of a C�-algebra or a von Neumann alge-
bra, and also JB- and JBW -algebras are examples of order-unit spaces.
A von Neumann algebra and a JBW -algebra possess predual spaces, but,
generally speaking, a JB-algebra and a C�-algebra don't have this property.
In this work, conditions are found for an order-unit space to possess a pre-
dual space. Moreover, a condition is obtained characterizing JBW -algebras
among order-unit spaces having a predual space.
Key words: order-unit space, predual space, state, trace, generalized
spin-factor, L1-norm.
Mathematics Subject Classi�cation 2000: 46B40, 46L50, 46G12, 46B10.
1. Preliminaries
Let A be a real ordered linear space. We denote by A+ the set of positive
elements of A: An element e 2 A+ is called order unit if for every a 2 A there
exists a number � 2 R
+ such, that ��e � a � �e: If the order is Archimedean
then the mapping a ! kak = inff� > 0 : ��e � a � �eg is a norm. If A is
a Banach space with respect to this norm, we say that (A; e) is an order-unit
space with the order unit e:
Let (A; e) be an order-unit space. An element � 2 A� is called positive if
�(a) � 0 for all a 2 A+; in this case one writes � � 0: A positive linear functional
is called a state if k�k = 1: This is equivalent to �(e) = 1: We denote by S(A) the
set of all states on A and call S(A) the states space of A: It is known that S(A)
is a �-weakly closed subset in A�:
As we know, the pair (A;A�
) is a dual pair. Following the work by E. Alfsen,
F. Shultz [1], we suppose that A and A� are in spectral duality. In this case
every element a 2 A has a spectral resolution with respect to projective units. We
denote P and U a set of P -projections and projective units of A, respectively.
c
M.A. Berdikulov, 2006
Order-Unit Spaces which are Banach Dual Spaces
Generally speaking, spectral duality in [1] is de�ned between A and a subspace
V � A�: Further, if the opposite is not supposed, spectral duality (A; V ) means
the case V � A�:
A P -projection R is called central if R + R0
= I: Here R0 is the quasi-
complement of R: A projective unit u = Re is called central if R is a central
P -projection.
An order-unit space (A; e) is said to be factor if it contains no central projective
units except 0 and e:
A projective unit u = Re is called Abelian if imR = R(A) is a vector lattice.
One says that an order-unit space A has type I if for any central P -projection
R in A, the subspace imR contains an Abelian projective unit.
An element u 2 U is called an atom if u is the minimal element of the latticeU:
If A is a factor of type I and u is an atom, then there is a unique continuous
linear functional bu on A corresponding to u: This functional is the extremal point
in S(A) with properties: hu; bui = 1; kuk = 1: The P -projection R corresponding
to u is of the form: Ra = ha; buiu:
Spaces of selfadjoint elements of a C�-algebra, a von Neumann algebra, and
JB- and JBW -algebras are the examples of order-unit spaces.
Let K be a compact convex subset of a local convex Hausdor� space V: We
denote by A(K) the space of all continuous a�ne functions, and by Ab
(K) the
space of all bounded a�ne functions on K: Then A(K) and Ab
(K) are order-unit
spaces. The role of unit plays the a�ne function identically equal to 1 on K:
It is known that a von Neumann algebra, a JBW -algebra and the space Ab
(K)
possess predual spaces, but this is not true for JB-algebras, C�-algebras and A(K)
[2].
A state � on A is called normal if �(a�)! 0 for any net fa�g � A monotoni-
cally decreasing to zero (a� # 0).
Theorem (F. Shultz [2, 3]). JB-algebra A has a predual space, i.e. it is
a JBW-algebra if and only if it has a separating space of normal states.
As it turns out, a similar result is valid for order-unit spaces, too.This work
is devoted to this result. Moreover, in the end of the paper, we prove a theorem
characterizing JBW -algebras among order-unit spaces possessing a predual space.
2. Main Results
2.1. Existence of a Predual Space
We start by studying one example of an order-unit space from [4] and prove
an analog of the Shultz theorem in this case. Spaces considered in [4] and called
there generalized spin-factors are constructed by the following way.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 131
M.A. Berdikulov
Let X and Y be real Banach spaces in separating duality [5]. Then A = R�X
and V = R � Y form a dual pair with respect to duality:
ha; �i = �� + hx; yi;
for a = � + x 2 A and � = � + y 2 V; where hx; yi is the duality between X
and Y:
The order and norm on A (on V ) are de�ned as:
a = �+ x � 0
def
, � � kxk
�
� = � + y � 0
def
, � � kyk
�
;
kak = j�j+ kxk
�
k�k = max (j�j; kyk)
�
:
Let A have a predual space. Then X = Y � and any functional � 2 V is
normal.
Indeed, let a� # 0; then �� # 0 and kx�k ! 0 since a� = �� + x� : Let
� = � + y 2 V; then j�(a�)j = j��� + hx� ; yij � �� j�j+ kx�k � kyk ! 0: Therefore
�(a�) ! 0: Since (A; V ) is a dual pair, then V separates points of A: Hence, V is
a separating space of normal functionals for A:
Conversely, let A have a separating space of normal functionals of V; i.e. a� # 0
follows �(a�) ! 0 for any � 2 V and there exists � 2 V for any a 6= 0 such, that
�(a) 6= 0: Since V � A�; then an arbitrary element � 2 V is of the form � = �+y;
where � 2 R; y 2 Y � X�: Since A and V are a dual pair, then X and Y are
a dual pair. As it is proved in [5, Th. 1, �3, III] Y �
= X: Hence, generalized
spin-factors possess a predual space when they have a separating space of normal
states.
Let us consider the general case. Let A be an order-unit space in spectral
duality, and S(A) the space of normal states on A. We denote V = lin(S(A)) the
linear hull of the normal states space. It is obvious, that V � A�: Let J = V 0 be
the polar of V in A��:
Theorem 1. There exists a central P -projection R in A��
such, that J =
R0
(A��
); where R0
is a quasicomlement of R and the mapping a 7! Ra is an
isomorphism of A onto R(A��
):
P r o o f. Let H be an arbitrary P -projection in A: Then H�
(V ) � V:
Indeed, let a� " a in A and � 2 S(A): Then H��(x) = �(Hx) for all x 2 A:
Since the P -projection H is positive and normal, �(Ha�) ! �(Ha): Therefore
H�� 2 V: Now it follows that if H is a P -projection in A and x 2 J; then
H��
(x)(�) = x(H��) = 0: Hence, H��
(J) � J for any P -projection H in A � A��:
This means that the set J is "invariant" with respect to P: By virtue of continuity
132 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Order-Unit Spaces which are Banach Dual Spaces
of P -projections, we conclude that J is invariant with respect to P -projections
in A��: Note that A�� �
= Ab
(S(A)) [2] and therefore A�� is an order-unit space in
spectral duality.
Before continuing the proof of Th. 1, we prove the following result.
Lemma. Let J be a weakly closed subspace invariant with respect to P -
projections in A: Then there is a central P -projection H in A such, that J = H(A):
P r o o f. We denote by h the order unit J: By the condition of Lemma, J is
invariant with respect to P -projections in A; then Rh 2 J for any R 2 P: Since
h is the unit in J then Rh � h: By Proposition 5.1 in [1], we conclude that R
is compatible with h: Since R is arbitrary it follows that h is a central element.
Thus there is a central P -projection H such, that h = He: Therefore J = H(A):
Lemma is proved.
Return to the proof of Th. 1. By lemma, there exists a central P -projection
H such, that J = H(A��
):
Let u = e� h: Then u is a central projective unit in A��: Hence, R is homo-
morphism of A�� into itself where Re = u: Since id = R + H; then the kernel
of R is J: Further, since the space of normal states of A is separating we have
that A\ J = A\ V 0
= f0g: Hence, R is a one-to-one mapping of A into R(A��
):
Theorem 1 is proved.
Theorem 2. Weakly �-continuous extensions of states from A onto A��
are
normal.
P r o o f. By Proposition 1.2.11 [2], A�� is monotone complete and order
isomorphic to Ab
(S(A)): It is known from Cor. 1.1.22 in [2] that an arbitrary state
� on A can be uniquely extended to a state � on A��: Let fa�g be a bounded
increasing net in A�� with the least upper bound a: Since a� " a implies a�jS(A) !
ajS(A) pointwise by virtue of A
�� �
= Ab
(S(A)); so we have �(a�) = a�(�) ! a(�) =
�(a): Hence, � is a normal state on A��: Theorem 2 is proved.
Theorem 3. If A has a predual space V (V � �
= A), then elements of V are
normal functionals on A:
P r o o f. If � 2 V; then it is obvious that � 2 A�; and its extension is
a normal functional on A�� by Th. 2. Hence, � is also a normal functional on
R(A��
); where R is a P -projection from Th. 1. Since a 7! Ra is an isomorphism
of A onto R(A��
) and �(a) = �(Ra) for all a 2 A; then � is normal on A = R(A��
):
Theorem 3 is proved.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 133
M.A. Berdikulov
Theorem 4. Let A be a monotone complete order-unit space in spectral dual-
ity. Then A has a predual space if and only if it has separating space of normal
states. In this case the predual space is unique and coincides with a space of
normal linear functionals on A:
P r o o f. Let A have a separating space of normal states S(A). Recall that
V = lin(S(A); J = V 0. By Theorem 1, there is a central P -projection R such,
that A �
= R(A��
) and J = R0
(A��
). In [6] G. Godefroy has proved the following
fact: a Banach space E has a predual space if and only if there exists a closed
linear subspace F in E�� such, that i(E) � F = E�� (Prop. 1 in [6]).
In our case, the role of the subspace F plays the subspace J: From this we
conclude that A has a predual space.
Conversely, let A have a predual space, i.e. there exists a subspace V � A�
such, that A = V �. Then V separates the points of A, i.e. A and V are a dual
pair. Further, by Th. 3 elements of V are normal functionals. Hence, A has a
separating space of normal functionals.
Later, if � 2 V; then � 2 A� and it is normal on A�� by Th. 2. Since a 7! Ra
is an isomorphism of A onto R(A��
) and �(a) = �(Ra) for all a 2 A; then � is
normal on A:
Conversely, if � 2 A� is normal, then the extension � on A�� has the form
� = �R: Since R is an isomorphism between A and R(A��
); then � is normal on
A�� and is equal to zero on R0
(A��
): Thus, � is equal to zero on J = V 0; and thus
it belongs to V:
This proves that a predual space to A is unique and coincides with the space
of normal functionals. Theorem 4 is proved.
2.2. Characterization of JBW -Algebra among Order-Unit Spaces
Having a Predual Space
Note that JB-algebras are examples of order-unit spaces. Various authors have
investigated conditions under which an order-unit space becomes a JB-algebra.
For example, in [7] it is shown that if a state space S(A) of a spectral order-unit
space A has the Hilbert ball property then A is a JB-algebra. In [8] geometric
conditions on S(A) are found: a spectral order-unit space A to be a JB-algebra
if and only if S(A) is symmetric.
Here, it was found another condition in this circle of problems: let a spectral
order-unit space A has a predual space V (V �
= A). If the spaces L1(�) and V
are order and isometrically isomorphic then A is a JBW -algebra.
A positive linear functional � is called a trace on an order-unit space (A; e) if
it satis�es the following condition:
�(a) = �(Ra) + �(R0a) 8a 2 A; R 2 P:
134 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Order-Unit Spaces which are Banach Dual Spaces
Let A be an order-unit space, � be a faithful trace on A. For a 2 A, we put
kak1 = �(jaj), where jaj = a+ + a� is the module of the element a: The following
result is proved in [9].
Theorem 5. The mapping k � k1 : A! R is a norm on A:
A mapping k � k1 : A ! R is said to be L1-norm on A: We denote L1(�) the
completion of A by L1-norm.
Let A be an order-unit space of type I having a predual space, i.e. there is a
space V such, that V �
= A:
Consider relation between L1(�) and V:
Theorem 6. The spaces L1(�) and V are order and isometrically isomorphic
if and only if A is a JBW -algebra.
P r o o f. It is known [10], if A is a JBW -algebra with a trace � , then spaces
L1(�) and V are order and isometrically isomorphic.
Conversely, suppose that L1(�) and V are isometrically isomorphic.
By Lemma 7.1 in [7], any order-unit space of type I can be reduced to factors
of type I: Therefore we shall prove the theorem for factors of type I:
For an atom u 2 U; u = Re; where R 2 P; we assume
'u(x) = �(Rx) = R��(x):
It is obvious, that 'u is a positive functional on A, i.e it is an element of V:
Functionals of the form R�� were called in [11] projective traces. If v = Qe is
another atom orthogonal to u; then the element h = u + v corresponds to a P -
projection H = R _Q = R+Q and the functional 'h = H�� = R�� +Q�� . It is
natural, that to their linear combination a = �u+ �v corresponds the functional
'a = �R�� + �Q�� . This process can be done for an arbitrary �nite number
of orthogonal atoms. Since A is a spectral order-unit space, then by assumption
of theorem, an arbitrary element of L1(�) can be approximated by �nite linear
combinations of functionals of type R��:
From the above, one can determine the following order and isometrical iso-
morphism between spaces L1(�) and V :
If fuig is a family of orthogonal atoms then for a =
P
�iui 2 L1(�), we de�ne
'a(x) =
X
�i�(Rix); (1)
where ui = Rie:
Let b =
P
�jvj be an element of L1(�): We de�ne for b by formula (1) the
functional 'b(x) =
P
�j�(Qjx); where vj = Qje are atoms.
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 135
M.A. Berdikulov
Then
'a(b) =
X
�i�(Rib) =
XX
�i�j�(Rivj) =
XX
�i�j�(RiQje);
'b(a) =
X
�j�(Qja) =
XX
�j�i�(Qjui) =
XX
�i�j�(QjRie):
In order to functionals be well de�ned by formula (1), the values of 'a(b) and
'b(a) have to be equal.That's why we have
�(RQe) = �(QRe)
for all atoms u = Re and v = Qe:
The last equality means that �(Rv) = �(Qu); i.e.
�(hv; buiu) = �(hu; bviv):
Since the trace on factors of type I takes equal values on atoms, we have
hv; bui = hu; bvi
for all atoms u and v: But this is the Hilbert ball property. By Proposition 6.14
from [7], we conclude that A is a JBW -factor. Theorem 5 is proved.
References
[1] E.M. Alfsen and F.W. Shultz, Noncommutative spectral theory for a�ne function
spaces on convex sets. Mem. AMS. V. 172, Providence, RI, 1976.
[2] H. Hanche-Olsen and E.O. Stormer, Jordan Operator Algebras. Pitman APP,
London, 1984.
[3] F.W. Shultz, On Normed Jordan Algebras which are Banach Dual Spaces. � J.
Funct. Anal. 31 (1979), 360�376.
[4] M.A. Berdikulov and S.T. Odilov, Generalized Spin-Factors. � Uzb. Math. J. 1
(1995), 10�15. (Russian)
[5] A.V. Kantorovich and G.P. Akilov, Functional Analysis. Nauka, Moscow, 1977.
(Russian)
[6] G. Godefroy, Espaces de Banach: Existence et Unicite de Certains Preduaux. �
Ann. Inst. Fourier, Grenoble 28 (1978), No. 3, 87�105.
[7] E.M. Alfsen and F.W. Shultz, State Spases of Jordan Algebras. � Acta Math. 140
(1978), No. 3�4, 155�190.
[8] Sh.A. Ayupov, B. Iochum B., and N.J. Yadgorov, The Geometry of Spaces of States
of Finite-Dimensional Jordan Algebras. � Izv. AN UzSSR, Ser. Phys.-Mat. (1990),
No. 3, 19�22. (Russian)
136 Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2
Order-Unit Spaces which are Banach Dual Spaces
[9] M.A. Berdikulov, Concept of a Trace on the Order-Unit Spaces and the Space of
Integrable Elements. � Mat. Trudy 8 (2005), No. 2, 39�48. (Russian)
[10] Sh.A. Ayupov, Integration on Jordan Algebras. � Izv. AN USSR, Ser. Mat. 47
(1983), No. 1, 3�25. (Russian)
[11] O.E. Tikhonov, Spectral theory for base-norm spaces. � Konstr. Teor. Funkts. i
Funkts. Anal., Kazan 8 (1992), 76�91. (Russian)
Journal of Mathematical Physics, Analysis, Geometry, 2006, vol. 2, No. 2 137
|
| id | nasplib_isofts_kiev_ua-123456789-106587 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T15:53:25Z |
| publishDate | 2006 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Berdikulov, M.A. 2016-09-30T20:35:22Z 2016-09-30T20:35:22Z 2006 Order-Unit Spaces which are Banach Dual Spaces / M.A. Berdikulov // Журнал математической физики, анализа, геометрии. — 2006. — Т. 2, № 2. — С. 130-137. — Бібліогр.: 11 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106587 Spaces of selfadjoint elements of a C*-algebra or a von Neumann algebra, and also JB- and JBW-algebras are examples of order-unit spaces. A von Neumann algebra and a JBW-algebra possess predual spaces, but, generally speaking, a JB-algebra and a C*-algebra don't have this property. In this work, conditions are found for an order-unit space to possess a predual space. Moreover, a condition is obtained characterizing JBW-algebras among order-unit spaces having a predual space. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Order-Unit Spaces which are Banach Dual Spaces Article published earlier |
| spellingShingle | Order-Unit Spaces which are Banach Dual Spaces Berdikulov, M.A. |
| title | Order-Unit Spaces which are Banach Dual Spaces |
| title_full | Order-Unit Spaces which are Banach Dual Spaces |
| title_fullStr | Order-Unit Spaces which are Banach Dual Spaces |
| title_full_unstemmed | Order-Unit Spaces which are Banach Dual Spaces |
| title_short | Order-Unit Spaces which are Banach Dual Spaces |
| title_sort | order-unit spaces which are banach dual spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106587 |
| work_keys_str_mv | AT berdikulovma orderunitspaceswhicharebanachdualspaces |