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7SUBADDITIVE MEASURE ON VON NEUMANN ALGEBRAS
KODIROV Komiljon NISHONBOYEV Azizbek RUZIKOV Maxamadjon TUXTASINOV Toxir
Ferghana State University, Uzbekistan Email: kodirov [email protected]
d https://doi.org/10.24412/2181-2993-2022-2-134-139
In the paper subadditive measure in the lattice of orthogonal projector ABSTRACT of von Neumann algebra is considered. The basic peoperties of the subadditive measure are determined and proved.
Keywords: The lattice of projectors, equivalent projectors, subadditive measure, finite measure, affiliated with the algebra, spectral decomposition.
В статье рассматривается субаддитивная мера в решетке АННОТАЦИЯ ортогонального проектора алгебры фон Неймана. Определены и
доказаны основные свойства субаддитивной меры.
Ключевые слова: решетка проекторов, эквивалентные проекторы, субаддитивная мера, конечная мера, аффилированная с алгеброй, спектральное разложение.
INTRODUCTION
Let us assure that H is a complex Gilbert space, given the algebra of all finite operators defined in B(N) - N. Von Neumann algebra is such a set of M that is a
partial algebra of B(N) (that is a e M, if iris), if it is closed to the top operator of the unit, including the operator.
The commutant of M algebra is a set of M' such a e M that this set consists of all b e the complementary elements in B(H). M'is also a von Neumann algebra and is relevant. All orthogonal projectorsn in M von Neumann's algebra form a grid (logic). We define it V through us. From now on we call the elements projectors.
Two projectors p, q eV are called equivalents, u e M if any u * u = p, uu* = q. Specify the p ~ q equivalent projectors. If p the projector is q
equivalent to a part of the projector, then we use the designation p<q. Throughp1 the projector 1-p. Through p v q the projection grid and upper limit(s) of the projector,
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f
14ж|
\ i -, i 11 /
and the lower bound (infimum). For M von Neumann's algebra p, q eV it is shown
p v q - q ~ p - p a q [1-8].
Definition 1. By a subadditive measure von Neuman algebra M we mean a mapping m: V ^ [0, <x>] wich the following properties:
a) m(0) = 0; m(p) = 0 because ofp=0;
b) p < q because of m(p) < m(q);
c) p~q because of m(p)=m(q);
d) m(p v q) < m(p) + m(q)
e) p„ T p because m(pn) T m(p) if the origin
the scale is called the limit, if m(1) < ro any. In relation to above p v q - q ~ p - p a q, we can substitute the condition m(p v q) = m(p + q) < m(p) + m(q) for projectors that are (d) p± q instead of condition (d'). Definition 2. The linear part space of the H Gilbert space DcH (operator a, repectively defined in H Gilbert space) is called M if u' e M it is for for a unitary operator u'(D) c D (as appropriate au' = u'a). In this case the DjM marking (as appropriate ajM) is used [9-17].
i
For any closed defined operator a, it is defined as a module | a |= (a * a)2. If
| a \=^Adex the spectral distribution is, then a jj M it is exeV necessary and
0
sufficient for a a M and for u e M that element in the polar distribution of operator a = u | a |.
The following statement provides the main features of the m subadditive measure.
Theorem. For the subadditive measure m, the following properties are executed:
1. P<q because m(p)<m(q);
2. m
Pr l<X m(P);
V -'1 J -=1
3. If p(H) c D(a), || ap\\<A so m(ej) < m(p1), on this | a |= J Mex, a^ M densely defined operator;
a,
0
4. m(p) < m(q) because p1 a q ^ 0;
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5. The projector q eV for any s> 0, m(p1) < s and p л q = 0 if there is a satisfying p eV, then q=0.
6. Projector q, q2 eV, for any s> 0, m( p1) <s and q л p = q2 л p if there is a satisfying condition p eV, then qi=q2.
Proof. Attribution 1 comes directly from the conditions of definition 1(b) and
(c).
2 properties are derived from conditions 1 (d) and (e) of Definition 1. any n is appropriate for a natural number e1 L < p1.
n
e1+1 T e^ it follows m(e^) = sup m (e^+1 ) from the condition of definition 1 (e).
n n V n /
The follows from the properties of (a) m(e1) < m(p1) of the theorem.
4. q - q л p1 ~ p - q 1л p and because p 1л q = 0 and q ~ p - q 1л p < p is derived. From this p1 л q = 0 because m(q) < m(p) is derived.
5. We own q = q - p л q ~ p v q - p < p1. In this case m(q) < m(p1) < s . Since
s>0 is optional and m its net value is q=0.
6. Checking that (q - q л q2) л p = 0 is not difficult. It q - q л q = 0 follows
that this theorem is based on the property of (v). it can be shown that this is the case
q2<q1.
Example 4. For each m subadditive measure m: V^ [0,1], we define:
ч fm(pX if m(p) <1
m(p) = \л S / 4 1
[1, if m(p) > 1
It is not difficult to check m whether a subadditive measure is V. Example 5. Let iy(Ä), Ä>0 us assume, for example, that there is a decreasing and nonnegative function, that is, у/(Л) > 0, y/(0) = 0, semi-additive, that is 0, for the left, continuous
+ Я2) <ИЛ) +
For any m subadditive measure
mw (p) = W(m( p)), p eV
The fuction mw : V ^ [0;да) defined by the formula is a subadditive measure. REFERENCES
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