Algebraic theory of measure algebras
A. Horn and A. Tarski initiated the abstract theory of measure algebras. Independently V. Sushchansky, B. Oliynyk and P. Cameron studied the direct limits of Hamming spaces. In the current paper, we introduce new examples of locally standard measure algebras and complete the classification of coun...
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| Дата: | 2023 |
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| Мова: | Англійська |
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Видавничий дім "Академперіодика" НАН України
2023
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Algebraic theory of measure algebras / O.O. Bezushchak, B.V. Oliynyk // Доповіді Національної академії наук України. — 2023. — № 2. — С. 3-9. — Бібліогр.: 13 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859957164390481920 |
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| author | Bezushchak, O.O. Oliynyk, B.V. |
| author_facet | Bezushchak, O.O. Oliynyk, B.V. |
| citation_txt | Algebraic theory of measure algebras / O.O. Bezushchak, B.V. Oliynyk // Доповіді Національної академії наук України. — 2023. — № 2. — С. 3-9. — Бібліогр.: 13 назв. — англ. |
| collection | DSpace DC |
| container_title | Доповіді НАН України |
| description | A. Horn and A. Tarski initiated the abstract theory of measure algebras. Independently V. Sushchansky, B. Oliynyk
and P. Cameron studied the direct limits of Hamming spaces. In the current paper, we introduce new examples of
locally standard measure algebras and complete the classification of countable locally standard measure algebras.
Countable unital locally standard measure algebras are in one-to-one correspondence with Steinitz numbers. Given
a Steinitz number s such measure algebra is isomorphic to the Boolean algebra of s-periodic sequences of 0 and 1.
Nonunital locally standard measure algebras are parametrized by pairs (s, r), where s is a Steinitz number and r
is a real number greater or equal to 1. We also show that an arbitrary (not necessarily locally standard) measure
algebra is embeddable in a metric ultraproduct of standard Hamming spaces. In other words, an arbitrary measure
algebra is sofic.
Абстрактна теорія алгебр з мірою була започаткована А. Хорном і А. Тарським. Незалежно від них В. Сущанський, Б. Олійник і П. Камерон досліджували прямі границі просторів Хемінга. У цій статті наведено нові приклади локально стандартних алгебр з мірою та завершено класифікацію зліченних локально стандартних алгебр з мірою. Зліченні унітальні локально стандартні алгебри з мірою знаходяться у взаємно
однозначній відповідності з числами Стейніца. Для даного числа Стейніца s така алгебра з мірою ізоморфна булевій алгебрі s-періодичних послідовностей iз 0 та 1. Неунітальні локально стандартні алгебри з мірою параметризуються парами (s, r), де s — число Стейніца, а r — дійсне число, яке більше або дорівнює 1. Також показано, що довільна (не обов’язково локально стандартна) алгебра з мірою занурюється в метричний ультрадобуток стандартних алгебр з мірою. Іншими словами, довільна алгебра з мірою є софічною.
|
| first_indexed | 2025-12-07T16:20:27Z |
| format | Article |
| fulltext |
3
ОПОВІДІ
НАЦІОНАЛЬНОЇ
АКАДЕМІЇ НАУК
УКРАЇНИ
ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 2: 3—9
C i t a t i o n: Bezushchak O.O., Oliynyk B.V. Algebraic theory of measure algebras. Dopov. Nac. akad. nauk Ukr.
2023. No 2. P. 3—9. https://doi.org/10.15407/dopovidi2023.02.003
© Видавець ВД «Академперіодика» НАН України, 2023. Стаття опублікована за умовами відкритого до-
ступу за ліцензією CC BY-NC-ND (https://creativecommons.org/licenses/by-nc-nd/4.0/)
https://doi.org/10.15407/dopovidi2023.02.003
UDC 512.552, 512.552.13, 512.563.2, 512.717
O.O. Bezushchak1, https://orcid.org/0000-0003-3654-6753
B.V. Oliynyk2,3, https://orcid.org/0000-0001-8721-4850
1 Taras Shevchenko National University of Kyiv, Kyiv
2 Silesian University of Technology, Gliwice, Poland
3 National University of Kyiv-Mohyla Academy, Kyiv
E-mail: bezushchak@knu.ua, boliynyk@polsl.pl
Algebraic theory of measure algebras
Presented by Academician of the NAS of Ukraine M.O. Perestyuk
A. Horn and A. Tarski initiated the abstract theory of measure algebras. Independently V. Sushchansky, B. Oliynyk
and P. Cameron studied the direct limits of Hamming spaces. In the current paper, we introduce new examples of
locally standard measure algebras and complete the classification of countable locally standard measure algebras.
Countable unital locally standard measure algebras are in one-to-one correspondence with Steinitz numbers. Given
a Steinitz number s such measure algebra is isomorphic to the Boolean algebra of s-periodic sequences of 0 and 1.
Nonunital locally standard measure algebras are parametrized by pairs (s, r), where s is a Steinitz number and r
is a real number greater or equal to 1. We also show that an arbitrary (not necessarily locally standard) measure
algebra is embeddable in a metric ultraproduct of standard Hamming spaces. In other words, an arbitrary measure
algebra is sofic.
Keywords: measure algebra, locally matrix algebra, Boolean algebra, Hamming spaces, Steinitz number.
Let F2 be the field of order 2. By a Boolean algebra we mean an associative commutative algebra
over the field F2 satisfying the identity x2 = x.
Let [0, ∞) denote the set of nonnegative real numbers. Let H be a Boolean algebra. We call a
function μ : H → [0, ∞) a measure if
(1) μ (a) = 0 if and only if a = 0, a H∈ ;
(2) if ,a b H∈ and 0a b⋅ = , then ( ) ( ) ( )a b a bμ + = μ + μ .
Following A. Horn and A. Tarski [1], we call a Boolean algebra H with a measure μ : H→ [0, ∞) a
measure algebra. For more information on measure algebras (see [1—3]). If (H, μ) is a measure
algebra, then the distance dH (a, b) = μ (a – b) makes it a metric space.
МАТЕМАТИКА
MATHEMATICS
4 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 2
O.O. Bezushchak, B.V. Oliynyk
Example 1. The Boolean algebra 2 2, {0,1},n
n = =St F F with the function
1 1
1
( , ..., ) ( )n n nx x x x
n
μ = + + for all 1, ..., {0,1}nx x ∈
is a measure algebra. We call the measure algebra ( , )n nμSt standard. For all elements , na b ∈St
the distance ( , )
n
d a bSt equals the number of coordinates, where a and b differ, divided by n.
Example 2. Let N be the set of positive integers. For a sequence 1 2( , , ...) {0,1}Na a= ∈a define
the pseudomeasure function
→∞
μ = 1
1
( ) sup ( , , ).limn n
n
a a
n
a
Then { {0,1} | ( ) 0}NI = ∈ μ =a a is an ideal of the Boolean algebra 2
NF .
Consider the Boolean algebra 2 /NB I= F and the measure
μ + = μ( ) ( ),Ia a ∈ 2 .Na F
The measure algebra (B, μ) is called Besicovich measure algebra (see [4]).
Example 3. Let X be an infinite set and let H be the Boolean algebra of finite subsets of X,
including the empty one. The measure ( ) , ,#a a a Hμ = ∈ makes ( , )H μ a measure algebra. If the
set X is countable, then we denote the measure algebra ( , )H μ as H (∞).
In order to introduce the next series of examples we need to start with the concept of a Stein-
itz number.
Definition 1. A Steinitz number [5] is an infinite formal product of the form
,pr
p
p
∈Ρ
∏
where Ρ is the set of all primes, {0, }pr N∈ ∪ ∞ for all p ∈Ρ . We can define the product of two
Steinitz numbers by the rule:
,p p p pr k r k
p p p
p p p
+
∈Ρ ∈Ρ ∈Ρ
⋅ =∏ ∏ ∏ , {0, }p pr k N∈ ∪ ∞ ,
where we assume, that
if and ,
in other cases.
p p p p
p p
r k r k
r k
+ < ∞ < ∞⎧⎪+ = ⎨
∞⎪⎩
By symbol NS we denote the set of all Steinitz numbers. The set of all positive integers N is
the subset of NS . The elements of the set \N NS are called infinite Steinitz numbers.
Example 4. An infinite sequence 1 2( , , ...) {0,1}Na a= ∈a is said to be periodic if there exists a
positive integer k N∈ such that the equality ai = ai+k holds for all i N∈ . In this case the number
k is called a period of the sequence a.
Let s be a Steinitz number. A periodic sequence a is called s-periodic if its minimum period is
a divisor of s.
5ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 2
Algebraic theory of measure algebras
Let ( )s be the set of all s-periodic sequences. Clearly, ( )s is a Boolean subalgebra of
{0,1}N. The function
) 2( 1 1
1
( , , ...) ( ),ka a a a
k
μ = + +s
where k is a period of the sequence (a1, a2, ...), makes ( )( , )( ) μ ss a measure algebra.
A measure algebra (H, μ) is called unital if the Boolean algebra H contains 1 and μ (1) = 1. In
this case, it is easy to see that ( ) [0,1]Hμ ⊆ and 1 is the only element of measure 1.
The measure algebras of examples 1, 2, and 4 are unital. The measure algebras of example 3
are not unital.
If (H, μ) is a measure algebra and h H∈ is a nonzero element, then hH is a unital Boolean
algebra. The function
: [0,1],h hHμ →
( )
( ) ,
( )h
a
a
h
μμ =
μ
,a hH∈
makes Hh = (hH, μh) a unital measure algebra.
Definition 2. We say that two measure algebras (H1, μ1) and (H2, μ2) are scalar equivalent if
there exists a positive number α > 0 and an isomorphism ϕ : H1 → H2 of Boolean algebras such that
μ2 (ϕ(a)) = αμ1(a) for an arbitrary element 1a H∈ .
If measure algebras are scalar equivalent and unital, then they are isomorphic.
Definition 3. We call a measure algebra (H, μ) locally standard if every finite subset of H is
contained in a measure subalgebra of (H, μ) that is scalar equivalent to Stn for some 1n .
If the measure algebra (H, μ) is unital and locally standard, then every infinite subset of H is
contained in a measure subalgebra that is isomorphic to Stn for some 1.n ≥
The measure algebras of Examples 1, 3, and 4 are locally standard. The Besicovich measure
algebra is not locally standard because it contains elements of irrational measure.
In Sec. 1, we review the classification of unital countable locally standard measure algebras
and their connections to locally matrix algebras. In Sec. 2, we introduce new examples of non-
unital locally standard measure algebras and proceed with the classification of countable, not
necessarily unital, locally standard measure algebras. In Sec. 3, we discuss the property of unital
measure algebras, not necessarily locally standard, to be sofic.
1. Unital locally standard algebras.
Definition 4. Let H be a unital locally standard measure algebras, and let ( )D H =
{ 1 |1 , }nn H H H′ ′= ∈ ⊂ ≅ St . The least common multiple of the set D(H) is called the Steinitz
number of the measure algebra H and is denoted as st(H).
Theorem 1. If H is a countable unital locally measure algebra and st(H) = s, then ( )H ≅ s
(see Example 4 above).
In particular, every countable unital locally standard measure algebra is uniquely determined
by its Steinitz number.
The theory of locally standard measure algebras is parallel to the theory of locally matrix al-
gebras. From this point of view, Theorem 1 is an analogue of the theorem of J.G. Glimm [6].
Definition 5. Let F be a field. An associative F-algebra A is called a locally matrix algebra if
an arbitrary finite collection of elements 1, ..., na a A∈ is contained in a subalgebra A A′ ⊂ that is
6 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 2
O.O. Bezushchak, B.V. Oliynyk
isomorphic to a matrix algebra Mn(F) for some 1n . If 1 ∈ A, then we say that A is a unital locally
matrix algebra.
Definition 6. For a unital locally matrix algebra A, let D(A) be the set of all positive integers
n N∈ such that there exists a subalgebra , 1 , ( )A A A A M F′ ′ ′∈ ⊂ ≅ . The least common mul-
tiple of the set D(A) is called the Steinitz number st(A) of the algebra A (see [7, 8]).
J.G. Glimm [6] showed that if 0dimF A ℵ and
( ) ,
i
i
p
A p
∈Ρ
= ∏st then ( )
i
i
p
p
A M F
∈Ρ
≅ ⊗ .
In particular, every countable-dimensional unital locally matrix algebra is uniquely determined
by its Steinitz number.
For an element a of a unital locally matrix algebra A choose a subalgebra A A′ ⊂ such that
1 , A A′ ′∈ ≅ Mn (F). Let ( )Ar a′ be the range of the matrix a in ( )nM F . As shown by V.M. Kuroch-
kin [9], the ratio
1
( ) ( )Ar a r a
n ′=
does not depend on the choice of the subalgebra A′ . We call r(a) the relative range of the element
a. If ,a b A∈ are orthogonal idempotents, then r (a + b) = r (a) + r (b).
Let C be a commutative subalgebra of a locally matrix algebra A and 1 C∈ . Let E(C) be
the set of all idempotents from C. For , ( )e f E C∈ let ef and e + f – 2ef be their Boolean prod-
uct and Boolean sum, respectively. The Boolean algebra E (C) with the relative range function
r : E (C) → [0, 1] is a measure algebra.
A subalgebra H of the matrix algebra Mn(F) is called a Cartan subalgebra if H F F≅ ⊕ ⊕
(n summands), in other words, H is spanned by n pairwise orthogonal idempotents. It is well
known that every Cartan subalgebra is a conjugate of the diagonal subalgebra of Mn (F).
A Cartan subalgebra of A is a subalgebra H A⊂ with decompositions
1
i
i
A A
∞
=
= ⊗ and
1
i
i
H H
∞
=
= ⊗
in which all Ai are finite-dimensional matrix algebras and Hi are Cartan subalgebras of Ai. Any two
Cartan subalgebras of A are conjugate via an automorphism.
In [10], it is shown that an arbitrary countable unital locally standard measure algebra M is
isomorphic to E (C), where C is a Cartan subalgebra of a countable-dimensional unital locally
matrix algebra A and st(M) = st(A).
2. Classification of non-unital locally standard measure algebras. In [10], we showed that
given two measure algebras (H1, μ1) and (H2, μ2) there exists a unique measure μ on the Boolean
algebra ⊗
21 2H HF such that 1 2( ) ( ) ( )a b a bμ ⊗ = μ μ for arbitrary elements 1,a H∈ 2.b H∈
Remark 1. In [10], we assumed the unitality of the Boolean algebra H1 and H2. However, this
unitality has never been used in the definition of tensor product.
Example 5. Let s be an infinite Steinitz number and let 1 r < ∞ be a real number. Choose a
sequence b1, b2, ... of positive integers such that bi divides bi+1, 1i , and all these numbers divide s.
7ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 2
Algebraic theory of measure algebras
Let mi = [r bi], 1i , and let
[ ] if ,
1 if .
i i
i
i i
rb rb N
m
rb rb N
+ ∉⎧
= ⎨ − ∈⎩
For each 1i consider the unital countable measure algebras H (s/bi). Let
( / ),
ii m iM H s b= ⊗St ( / ),
i
i im
M H s b+
+ = ⊗St 1i .
The locally standard unital measure algebras H (s/bi) and
1/ 1( / )
i ib b iH s b
+ +⊗St have equal Stei-
nitz numbers. Hence,
1/ 1( / ) ( / )
i ii b b iH s b H s b
+ +≅ ⊗St .
This implies
1 1/ 1 1( / ) ( / ) ( / )
i i i i i
i
i
i m i m b b i b i
m
b
M H s b H s b H s b
+ ++ += ⊗ ≅ ⊗ ⊗ ≅ ⊗St St St St ,
and, similarly,
1 1( / )
i
i
i
i b i
m
b
M H s b
+ +
+
+≅ ⊗St .
We have
1
1,i
i i
i
b
m m
b
+
+⋅ 1
1.i
i i
i
b
m m
b
+ ++
+⋅
Let
1
1
(1,1, ...,1, 0, 0, ..., 0) ,
i
i
i
i
i m
b
m
b
e
+
+⋅
= ∈St
1
1
(1,1, ...,1, 0, 0, ..., 0) .
i
i
i
i
i m
b
m
b
e +
+
+ +
+
⋅
= ∈St Then
1 1
,
i i
i
i
b i m i
m
b
e e
+ +
≅St St
1 1
,
i ii
i
b i imm
b
e e++ + +
+ +≅St St
and, therefore, the measure algebra Mi (resp. iM + ) is isomorphic to the corner
1( 1) ( 1)i i ie M e+⊗ ⊗ (resp. 1( 1) ( 1)i i ie M e+ + +
+⊗ ⊗ ).
Let
1
( , ) ,i
i
H r s M=
1
( , ) .i
i
H r s M+ +=
Theorem 2. Any countable locally standard measure algebra is scalar equivalent to one of the
following measure algebras: Stn; H (∞) ⊗ ( )s , s N∈S ; H (r, s), H+(r, s), [0, )r ∈ ∞ , \s N N∈S .
8 ISSN 1025-6415. Dopov. Nac. akad. nauk Ukr. 2023. No 2
O.O. Bezushchak, B.V. Oliynyk
Remark 2. Unital measure algebra ( )s appear as H (r, s), where \s N N∈S , r = u/v, ,u v N∈ ,
and v divides s.
3. Sofic measure algebras.
Definition 7. A group is called sofic if it is embeddable in a metric ultraproduct of symmetric
groups with normalized Hamming distances.
Equivalently, it is sofic if it is locally ε-embeddable in symmetric groups (see [11, 12]). We
formulate a similar concept for measure algebras.
Let (H, μ) be a unital measure algebra; and let X H⊂ be a finite subset containing 1. Let
ε > 0. As above, (Stn, μn) is the standard measure algebra with normalized Hamming distance
dn (a, b) = μn (a – b).
Definition 8. A mapping ϕ : X → Stn is called an ε-embedding if
(1) ( ( ) ( ), ( ))nd a b a bϕ + ϕ ϕ + ε as long as , ,a b a b X+ ∈ ;
(2) ( ( ) ( ), ( ))nd a b abϕ ϕ ϕ ε as long as , ,a b ab X∈ ;
(3) ϕ (1) = 1;
(4) ( ( ), ( )) 1 4nd a bϕ ϕ for all distinct elements ,a b X∈ .
Let I be an infinite set, and let be a non-principle ultrafilter on I (see [13]). Let ( , )
i in n i I∈μSt
be a family of standard measure algebras. Consider the ultraproduct
/ .
in
i I
U
∈
= ∏St
For an element ( ) / ,i i Ia ∈=a
ii na ∈St , define
(a) lim ( )
in iaμ = μ
.
Then
∈
= ∈ μ =∏ { / | ( ) 0}
in
i I
R a St a
is an ideal in the Boolean algebra U and μ + = μ( ) ( )Ra a is a measure. We call the Boolean algebra
U with the measure μ a metric ultraproduct of standard measure algebras
inSt .
Proposition 1. The following two conditions on a unital measure algebra (H, μ) are equivalent:
(a) for an arbitrary finite subset 1 X H∈ ⊂ and an arbitrary ε > 0 there exists an ε-embedding
of X in a standard measure algebra,
(b) (H, μ) is embeddable in a metric ultraproduct of standard measure algebras.
The problem is if all groups are sofic is still open. The general expectation is that there is a
counterexample. The answer for measure algebras, however, is positive.
Theorem 3. All unital measure algebras are sofic.
The first author was supported by the program PAUSE (France), and was partly supported by
UMR 5208 du CNRS and by MES of Ukraine: Grant for the perspective development of the scientific
direction “Mathematical sciences and natural sciences” at TSNUK.
9ISSN 1025-6415. Допов. Нац. акад. наук Укр. 2023. № 2
Algebraic theory of measure algebras
REFERENCES
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https://doi.org/10.2307/1990396
2. Jech, T. (2008). Algebraic characterizations of measure algebras. Proc. Amer. Math. Soc., 136, pp. 1285-1294.
3. Maharam, D. (1947). An algebraic characterization of measure algebras. Ann. Math. Ser. 2., 48, pp. 154-167.
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4. Vershik, A. M. (1995). Theory of decreasing sequences of measurable partitions. St. Petersburg Math. J., 6,
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5. Steinitz, E. (1910). Algebraische Theorie der Körper. J. Reine Angew. Math., 137, pp. 167-309.
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6. Glimm, J. G. (1960). On a certain class of operator algebras. Trans. Amer. Math. Soc., 95, No. 2, pp. 318-340.
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7. Bezushchak, O. & Oliynyk, B. (2020). Primary decompositions of unital locally matrix algebras. Bull. Math.
Sci., 10, No. 1. https://doi.org/10.1142/S166436072050006X
8. Bezushchak, O. & Oliynyk, B. (2020). Unital locally matrix algebras and Steinitz numbers. J. Algebra Appl.,
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9. Kurochkin, V. M. (1948). On the theory of locally simple and locally normal algebras. Mat. Sb., Nov. Ser.,
22(64), No. 3, pp. 443-454 (in Russian).
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No. 8. https://doi.org/10.1142/S0219498821501474
11. Elek, G. & Szabó, E. (2006). On sofic groups. J. Group Theory, 9, No. 2, pp. 161-171.
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12. Gromov, M. (1999). Endomorphism of symbolic algebraic varieties. J. Eur. Math. Soc., 1, No. 2, pp. 109-197.
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13. Mal’cev, A. I. (1973). Algebraic system. Berlin, Heidelberg: Springer.
Received 18.11.2022
О.О. Безущак1, https://orcid.org/0000-0003-3654-6753
Б.В. Олійник2,3, https://orcid.org/0000-0001-8721-4850
1Київський національний університет ім. Тараса Шевченка, Київ
2Сілезький технологічний університет, Глівіце, Польща
3Національний університет “Києво-Могилянська академія”, Київ
E-mail: bezushchak@knu.ua, boliynyk@polsl.pl
АЛГЕБРАЇЧНА ТЕОРІЯ АЛГЕБР З МІРОЮ
Абстрактна теорія алгебр з мірою була започаткована А. Хорном і А. Тарським. Незалежно від них В. Су-
щанський, Б. Олійник і П. Камерон досліджували прямі границі просторів Хемінга. У цій статті наведено
нові приклади локально стандартних алгебр з мірою та завершено класифікацію зліченних локально стан-
дартних алгебр з мірою. Зліченні унітальні локально стандартні алгебри з мірою знаходяться у взаємно
однозначній відповідності з числами Стейніца. Для даного числа Стейніца s така алгебра з мірою ізоморф-
на булевій алгебрі s-періодичних послідовностей iз 0 та 1. Неунітальні локально стандартні алгебри з мі-
рою параметризуються парами (s, r), де s — число Стейніца, а r — дійсне число, яке більше або дорівнює 1.
Також показано, що довільна (не обов’язково локально стандартна) алгебра з мірою занурюється в метрич-
ний ультрадобуток стандартних алгебр з мірою. Іншими словами, довільна алгебра з мірою є софічною.
Ключові слова: алгебра з мірою, локально матрична алгебра, булева алгебра, простір Хемінга, число
Стейніца.
|
| id | nasplib_isofts_kiev_ua-123456789-192996 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1025-6415 |
| language | English |
| last_indexed | 2025-12-07T16:20:27Z |
| publishDate | 2023 |
| publisher | Видавничий дім "Академперіодика" НАН України |
| record_format | dspace |
| spelling | Bezushchak, O.O. Oliynyk, B.V. 2023-07-30T13:19:44Z 2023-07-30T13:19:44Z 2023 Algebraic theory of measure algebras / O.O. Bezushchak, B.V. Oliynyk // Доповіді Національної академії наук України. — 2023. — № 2. — С. 3-9. — Бібліогр.: 13 назв. — англ. 1025-6415 DOI: doi.org/10.15407/dopovidi2023.02.003 https://nasplib.isofts.kiev.ua/handle/123456789/192996 512.552, 512.552.13, 512.563.2, 512.71 A. Horn and A. Tarski initiated the abstract theory of measure algebras. Independently V. Sushchansky, B. Oliynyk and P. Cameron studied the direct limits of Hamming spaces. In the current paper, we introduce new examples of locally standard measure algebras and complete the classification of countable locally standard measure algebras. Countable unital locally standard measure algebras are in one-to-one correspondence with Steinitz numbers. Given a Steinitz number s such measure algebra is isomorphic to the Boolean algebra of s-periodic sequences of 0 and 1. Nonunital locally standard measure algebras are parametrized by pairs (s, r), where s is a Steinitz number and r is a real number greater or equal to 1. We also show that an arbitrary (not necessarily locally standard) measure algebra is embeddable in a metric ultraproduct of standard Hamming spaces. In other words, an arbitrary measure algebra is sofic. Абстрактна теорія алгебр з мірою була започаткована А. Хорном і А. Тарським. Незалежно від них В. Сущанський, Б. Олійник і П. Камерон досліджували прямі границі просторів Хемінга. У цій статті наведено нові приклади локально стандартних алгебр з мірою та завершено класифікацію зліченних локально стандартних алгебр з мірою. Зліченні унітальні локально стандартні алгебри з мірою знаходяться у взаємно однозначній відповідності з числами Стейніца. Для даного числа Стейніца s така алгебра з мірою ізоморфна булевій алгебрі s-періодичних послідовностей iз 0 та 1. Неунітальні локально стандартні алгебри з мірою параметризуються парами (s, r), де s — число Стейніца, а r — дійсне число, яке більше або дорівнює 1. Також показано, що довільна (не обов’язково локально стандартна) алгебра з мірою занурюється в метричний ультрадобуток стандартних алгебр з мірою. Іншими словами, довільна алгебра з мірою є софічною. The first author was supported by the program PAUSE (France), and was partly supported by UMR 5208 du CNRS and by MES of Ukraine: Grant for the perspective development of the scientific direction “Mathematical sciences and natural sciences” at TSNUK. en Видавничий дім "Академперіодика" НАН України Доповіді НАН України Математика Algebraic theory of measure algebras Алгебраїчна теорія алгебр з мірою Article published earlier |
| spellingShingle | Algebraic theory of measure algebras Bezushchak, O.O. Oliynyk, B.V. Математика |
| title | Algebraic theory of measure algebras |
| title_alt | Алгебраїчна теорія алгебр з мірою |
| title_full | Algebraic theory of measure algebras |
| title_fullStr | Algebraic theory of measure algebras |
| title_full_unstemmed | Algebraic theory of measure algebras |
| title_short | Algebraic theory of measure algebras |
| title_sort | algebraic theory of measure algebras |
| topic | Математика |
| topic_facet | Математика |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/192996 |
| work_keys_str_mv | AT bezushchakoo algebraictheoryofmeasurealgebras AT oliynykbv algebraictheoryofmeasurealgebras AT bezushchakoo algebraíčnateoríâalgebrzmíroû AT oliynykbv algebraíčnateoríâalgebrzmíroû |