2-local isometries of non-commutative Lorentz spaces Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2019, Volume 21, Issue 4, P. 5-10

УДК 517.98

DOI 10.23671/VNC.2019.21.44595

2-LOCAL ISOMETRIES OF NON-COMMUTATIVE LORENTZ SPACES A. A. Alimov1, V. I. Chilin2

1 Tashkent Institute of Design, Construction and Maintenance of Automobile Roads, 20 Amir Temur Av., Tashkent 100060, Uzbekistan, 2 National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan

E-mail: al imovakrom63@yandex. ru, [email protected], [email protected]

Dedicated to E. I. Gordon on the occasion of his 70th birthday

Abstract. Let M be a von Neumann algebra equipped with a faithful normal finite trace t, and let S (M, t) be an *-algebra of all t-measurable operators affiliated with M. For x £ S (M, t) the generalized singular value function ^(x) : t ^ ^(i; x), t > 0, is defined by the equality t; x) = inf{||xp||M : p2 = p* = p £ M, t(1 — p) < i}. Let ^ be an increasing concave continuous function on [0, to) with ^(0) = 0, ^(to) = to, and let A^(M,t) = {x £ S (M,t) : ||x||^ = /0° ^(t; x)d^(t) < to} be the non-commutative Lorentz space. A surjective (not necessarily linear) mapping V : A^ (M, t) ^ A^ (M, t) is called a surjective 2-local isometry, if for any x,y £ A,p(M,t) there exists a surjective linear isometry Vx,y : A^(M,t) ^ A^(M,t) such that V(x) = Vx,y(x) and V(y) = Vx,y(y). It is proved that in the case when M is a factor, every surjective 2-local isometry V : A^(M,t) ^ A^(M,t) is a linear isometry.

Key words: measurable operator, Lorentz space, isometry. Mathematical Subject Classification (2010): 46L52, 46B04.

For citation: Alimov, A. A. and Chilin, V. I. 2-Local Isometries of Non-Commutative Lorentz Spaces, Vladikavkaz Math. J., 2019, vol. 21, no. 4, pp. 5-10. DOI: 10.23671/VNC.2019.21.44595.

1. Introduction

Let H be a complex separable infinite-dimensional Hilbert space, let (CE, || ■ ||cE) be a Banach ideal of compact linear operators in H generated by symmetric sequence space (E, || ■ ||e) C c0, and let V be a surjective 2-local isometry on CE, that is, V : CE — CE is a surjective (not necessarily linear) mapping such that for any x,y € CE there exists a surjective linear isometry Vx,y on CE for which V(x) = Vx,y(x) and V(y) = Vx,y(y). In the papers [1, 2] it is shown that in the case when CE is separable or has the Fatou property, CE = Ci2, every surjective 2-local isometry on CE is a linear isometry. In the proof of this statement is essentially used explicit description of all surjective linear isometries on CE [1, 3].

Banach ideals (CE, || ■ ||ce) of compact linear operators are examples of non-commutative symmetric spaces E(m,t) of measurable operators affiliated with a von Neumann algebra M

© 2019 Alimov, A. A. and Chilin, V. I.

equipped with a faithful normal semifinite trace t (see, for example, [4, Ch. 2, § 2.5]). It is natural to expect that for these non-commutative symmetric spaces with the Fatou property, every surjective 2-local isometry V : E(M,t) — E(M,t) is a linear map. Unfortunately, the method of proof of a similar statement for Banach ideals (CE, y ■ ||Ce) can not be applied here, since there is no description of surjective linear isometries V : E(M, t) — E(M,t). At the same time, in the case of non-commutative Lorentz and Marcinkiewicz spaces, such a description of surjective linear isometries was obtained in the paper [5]. Using this description, we obtain the following description of surjective 2-local isometries of non-commutative Lorentz spaces.

Theorem 1. Let M be an arbitrary factor with a faithful normal finite trace t, and let (A^(M, t), || ■ ) be a non-commutative Lorentz space. Then every surjective 2-local isometry V : A^(M,t) — A^(M,t) is a linear isometry.

2. Preliminaries

Let H be an infinite-dimensional complex Hilbert space, let B(H) be the C*-algebra of all bounded linear operators in H, and let 1 be the unit in B(H). Let M C B(H) be a von Neumann algebra on Hilbert space H equipped with a faithful normal semifinite trace t (see, for example, [6]). A linear operator x : D (x) — H, where the domain D (x) of x is a linear subspace of H, is said to be affiliated with M if yx C xy for all y € M', where M' is the commutant of M. A linear operator x : D (x) — H is termed measurable with respect to M if x is closed, densely defined, affiliated with M and there exists a sequence {pn}^=i in the lattice P (M) of all projections of M, such that pn t 1, pn(H) C D (x) and 1 — pn is a finite projection (with respect to M) for all n. The collection S (M) of all measurable operators with respect to M is a unital *-algebra with respect to strong sums and products.

Let x be a self-adjoint operator affiliated with M and let {ex} be a spectral measure of x. It is well known that if x is a closed operator affiliated with M with the polar decomposition x = u|x|, then u € M and e € M for all projections e € {e|x|}. Moreover, x € S(M) if and only if x is closed, densely defined, affiliated with M and e|x|(A, to) is a finite projection for some A > 0.

An operator x € S (M) is called t-measurable if there exists a sequence {pn}^L! in P (M) such that pn t 1, pn (H) C D (x) and t(1 — pn) < to for all n. The collection S (M,t) of all t-measurable operators is a unital *-subalgebra of S (M). It is well known that a linear operator x belongs to S (M,t) if and only if x € S(M) and there exists A = A(x) > 0 such that t(e|x|(A, to)) < to.

The generalized singular value function ^(x) : t — ^(t; x), t > 0, of the operator x € S (M,t) is defined by setting [7]

^(t; x) =inf{||xp|| : p € P (M) , t(1 — p) ^ t} = inf {s> 0: t(e|x|(s, to)) < t}.

A non-zero linear subspace E(M, t) C S (M, t) with the Banach norm || ■ (M>T) is called a symmetric space if the conditions

x € E(M, t), y € S (M, t), ^t(y) ^ ^t(x) for all t> 0,

imply that y € E(M,t) and |y|£(M,T) < ||x|£(m,t).

It is known that in the case t(1) < to it is true

S (M) = S (M, t ) and MCE (M,t ) C Li(M,t )

for each symmetric space E(M,t), where

Li(M,t) = < x € S (M, t): ||x||i = J ^t(x) dt < to >.

^ 0 '

In addition,

M - E(M,t) -M C E(M,t),

and

11axb|e(m,T) < || a||M ' ||b||M ' ||x||£(m,T)

for all a, b € M, x € E(M, t).

Let — be an increasing concave continuous function on [0, to) with -0(0) = 0, -(to) = lim 0(t) = to, and let

t—y^o

A^(M, t) = j x € S (M, t) : ||x||^ = J ^(t; x) d-(t) < to >

0

be the non-commutative Lorentz space. It is known that (A^(M,t), || ■ ) is a symmetric space [8], and the norm || ■ has the Fatou property, that is, the conditions 0 ^ xk € A^(M, t) for all k, and supfc^1 ||xk< to, imply that there exists 0 ^ x € A^(M, t) such that xk t x and ||x||^ = supfc>i ||xfc.

Denote by M^(M, t) the set of all x € S (M, t) for which

t

\\x\\m^ = sup —— / /x(s; x) ds t>0 0(t) J 0

is finite. The set M^(M,t) with the norm || ■ is a symmetric space which is called a Marcinkiewicz space.

Denote by M°(M,t) the closure of M in M^(M,t). It is known [9] that the conjugate space of (A^(M,t), || ■ ) is identified with (M^(M,t), || ■ ), and the conjugate space of (M°(A4,r), || • ||m,p), under the condition lim = 0, is identified with(A^,(A4, r), || •

The duality in these pairs of spaces is realized via the bilinear form (x,y) = t(xy). It should be pointed out that the spaces (A^(M, t), || ■ ), (M^(M, t), || ■ ) and (M°(M, t), || ■ ) are symmetric spaces [4, Ch. 2, § 2.6], [8].

3. Isometries of Non-Commutative Lorentz Spaces

Let M C B(H) be a von Neumann algebra on Hilbert space H. A linear bijective mapping $: M — M is called a Jordan isomorphism if $(x2) = ($(x))2 and $(x*) = ($(x))* for all x € M.

If $: M — M is a Jordan isomorphism, then there exists a central projection z € M such that $z(x) = $(x) ■ z, x € M, is an *-homomorphism, and $z±(x) = $(x) ■ (1 — z), x € M, is an *-antihomomorphism (see, for example, [10, Ch. 3, § 3.2.1]). Consequently, if M is a factor then a Jordan isomorphism $ : M — M is an *-homomorphism or *-antihomomorphism.

If t is a faithful normal finite trace on von Neumann algebra M then a Jordan isomorphism $: M — M is continuous with respect to measure topology tT generated by trace t (see, for

example, [11, ^Ch. 5, §3, Proposition 1]). Therefore, $ extends to a tT-continuous Jordan isomorphism $: S (M,t) — S (M,t). In addition, if t( $(x)) = t(x) for all x € M then ^(t;$(x)) = ^(t;x) for all x € S(M,t), in particular, $(E(M,t) = E(M,t)) and ||$(x)|E(m,t) = ||x||E(M,T) for all x € E(M,t), that is, $: E(M,t) — E(M,t) is a surjective linear isometry for any symmetric space (E(M,t), || ■ ||E(m,t)).

Thus, it is true the following

Proposition 1. Let M be an arbitrary von Neumann algebra with a faithful normal finite trace t, and let $: M — M be a Jordan isomorphism such that t( $(x)) = t(x) for all x € M. Then for every symmetric space (E(M,t), || ■ ||e(m,t)) the mapping V: E(M,t) — E(M,t) given by the equality V(x) = u ■ $(x) ■ v, x € E(M,t), u, v are unitary operators in M, is a surjective linear isometry.

We need the following description of surjective linear isometries of the spaces (A^(M,t), || ■ ) and (M°(M,t), || ■ ||m^) [5, Theorems 5.1, 6.1].

Theorem 2. Let M be an arbitrary von Neumann algebra with a faithful normal finite trace t, and let V : A^(M,t) — a/(M,t) (respectively, V : M°(M,t) — M°(M,t)) be a surjective linear isometry. Then there exist uniquely an unitary operator u € M and a Jordan isomorphism $ : M — M such that V(x) = u ■ $(x) and t( $(x)) = t(x) for all x € M.

4. Local Isometries of Non-Commutative Lorentz Spaces

Let (X, || ■ ||x) be an arbitrary Banach space over the field K of complex or real numbers. A surjective (not necessarily linear) mapping T: X — X is called a surjective 2-local isometry [2], if for any x,y € X there exists a surjective linear isometry VX,y: X — X such that T(x) = VX,y(x) and T(y) = Vx,y(y). It is clear that every surjective linear isometry on X is a surjective 2-local isometry on X. In addition,

T (Ax) = Vx)Ax(Ax) = AVx)Ax(x) = AT (x)

for any x € X and A € K.

Consequently, in order to establish linearity of a 2-local isometry T, it is sufficient to show that T(x + y) = T(x) + T(y) for all x, y € X.

Since

||T(x) - T(y)||x = ||Vx,y(x) - Vx,y(y)||x = ||x - y||x for all x,y € X,

it follows that T is continuous map on (X, || ■ ||X). In addition, in the case a real Banach space X (K = R), every surjective 2-local isometry on X is a linear map (see Mazur-Ulam Theorem [12, Ch. 1, § 1.3, Theorem 1.3.5.]). In the case a complex Banach space X (K = C), this fact is not true.

Using the description of all surjective linear isometries on a separable Banach symmetric ideal CE [3] (respectively, on a Banach symmetric ideal CE with Fatou property [1]), CE = Cl2, in the papers [1, 2] it is proved that every surjective 2-local isometry T : CE — CE is a linear isometry.

The following Theorem is a version of the above results for the spaces A^(M,t) and M (M, t ).

Theorem 3. Let M be an arbitrary factor with a faithful normal finite trace t, and let T : A^(M, t) — A^(M, t) (respectively, T : M°(M,t) — M°(M,t)) be a surjective 2-local isometry. Then T is a linear isometry.

< Fix x,y € M and let Vx,y : A^(M,t) — A^(M,t) be a surjective isometry such that T(x) = Vx,y(x) and T(y) = Vx,y(y). By Theorem 2, there exist uniquely an unitary operator u € M and a Jordan isomorphism $ : M — M such that Vx,y(a) = u ■ $(a) and t( $(a)) = t(a) for all a € M. Since M is a factor it follows then $ : M — M is an ^isomorphism or $ is an *-anti-isomorphism.

We assume that $ is an ^isomorphism (in the case when $ is an *-anti-isomorphism, the proof is similar). We have

t(T(x) ■ (T(y))*) = t(Vx,y(x) ■ (Vx,y(y))*) = t(u ■ $(x) ■ (u ■ $(y))*) = t(u ■ $(xy*) ■ u*) = t( $(xy*)) = t(xy*).

Consequently, t(T(x) ■ (T(y))*) = t(xy*) for all x,y € M. If x, y, z € M, then

t(T(x + y) ■ (T(z))*) = t((x + y)z*), t(T(x) ■ T(z)*) = t(xz*),

t(T(y) ■ T(z)*)= t(y ■ z*).

Therefore

t((T(x + y) - T(x) - T(y)) ■ (T(z))*) = 0 for all z € M. Taking z = x + y, z = x and z = y, we obtain

t((T(x + y) - T(x) - T(y) ■ ((T(x + y) - T(x) - T(y))*) = 0,

that is, T(x + y) = T(x) + T(y) for all x, y € M.

Since the Lorentz space A^(0, to) of measurable functions on a semi-axis [0, to) is separable space [13, Ch. 2I, §5], it follows that the non-commutative Lorentz (A^(M,t), || ■ ) has an order continuous norm [14, Proposition 3.6], that is, 11xn|^ X 0 whenever xn € A^(M,t) and xn X 0. Consequently, the factor M is dense in the space A^(M, t). Since T is a continuous mapping on A^(M,t) it follows that T(x + y) = T(x) + T(y) for all x,y € A^(M,t), that is, T is a surjective linear isometry.

For the space (M,t), the proof of the linearity of the surjective 2-local isometry T : M°(M,t) — Mo(M,t) repeats the previous proof. >

References

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6. Takesaki, M. Theory of Operator Algebras I, New York, Springer-Verlag, 1979.

7. Fack, T. and Kosaki, H. Generalized s-Numbers of T-Measurable Operators, Pacific Journal of Mathematics, 1986, vol. 123, no. 2, pp. 269-300. DOI: 10.2140/pjm.1986.123.269.

8. Chilin, V. I. and Sukochev, F. A. Weak Convergence in Non-Commutative Symmetric Spaces, Journal of Operator Theory, 1994, vol. 31, no. 1, pp. 35-55.

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13. Krein, M. G., Petunin, Ju. I. and Semenov, E. M. Interpolation of Linear Operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, 1982.

14. Dodds, P., Dodds. Th. K.-Y and Pagter, B. Noncommutative Kothe Duality, Transactions of the American Mathematical Society, 1993, vol. 339, no. 2, pp. 717-750. DOI: 10.1090/S0002-9947-1993-1113694-3.

Received 20 June, 2019 Akrom A. Alimov

Tashkent Institute of Design, Construction

and Maintenance of Automobile Roads,

20 Amir Temur Av., Tashkent 100060, Uzbekistan,

Associate Professor

E-mail: [email protected]

Vladimir I. Chilin National University of Uzbekistan, Vuzgorodok, Tashkent 100174, Uzbekistan Professor

E-mail: [email protected], [email protected]

Владикавказский математический журнал 2019, Том 21, Выпуск 4, С. 5-10

2-ЛОКАЛЬНЫЕ ИЗОМЕТРИИ НЕКОММУТАТИВНЫХ ПРОСТРАНСТВ ЛОРЕНЦА

Алимов А. А.1, Чилин В. И.2

1 Ташкентский институт по проектированию, строительству и эксплуатации автомобильных дорог, Узбекистан, 100060, Ташкент, пр. Амира Темура, 20 2 Национальный университет Узбекистана, Узбекистан, 100174, Ташкент, Вузгородок

E-mail: al imovakrom63@yandex. ru, [email protected], [email protected]

Аннотация. Пусть M алгебра фон Неймана с точным нормальным конечным следом т, и пусть S (М,т) инволютивная алгебра всех т-измеримых операторов, присоединенных к алгебре M. Для оператора x е S (М,т) невозрастающая перестановка ß(x) : t ^ ß(t; x), t > 0, определяется с помощью равенства ß(t; x) = inf{||xp|M : Р2 = Р* = P е M, т(1 — p) < t}. Пусть ф возрастающая вогнутая непрерывная функция на [0, то), для которой ф(0) = 0, ф(то) = то. Пусть Лф(М,т) = {х е S (М,т): ||х||ф = ß(t; x)d^(t) < то} некоммутативное пространство Лоренца. Сюръективное (не обязательно линейное) отображение V : Лф (M, т) ^ Лф (M, т) называется сюръективной 2-локаль-ной изометрией, если для любых x,y е Лф (М,т) существует такая сюръективная линейная изометрия

Vx, У : Лф (М,т) ^ Лф (M,т), что V (x) = Vx,y(x) и V (y) = Vx,y(y). Доказано, что в случае, когда M

есть фактор, каждая сюръективная 2-локальная изометрия V : Лф(М,т) ^ Лф(М,т) есть линейная изометрия.

Ключевые слова: измеримый оператор, пространство Лоренца, изометрия. Mathematical Subject Classification (2010): 46L52, 46B04.

Образец цитирования: Aminov B. R., Chilin V. I. Isometries of Real Subspaces of Self-Adjoint Operators in Banach Symmetric Ideals // Владикавк. мат. журн.—2019.—Т. 21, № 4.—C. 5-10 (in English). DOI: 10.23671/VNC.2019.21.44595.