Dilation invariant Banach limits

Journal Pre-proof Dilation invariant Banach limits E. Semenov, F. Sukochev, A. Usachev, D. Zanin

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S0019-3577(19)30079-5 https://doi.org/10.1016/j.indag.2019.12.003 INDAG 682

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Indagationes Mathematicae

Please cite this article as: E. Semenov, F. Sukochev, A. Usachev et al., Dilation invariant Banach limits, Indagationes Mathematicae (2019), doi: https://doi.org/10.1016/j.indag.2019.12.003. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝

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Dilation invariant Banach limits E. Semenov, F. Sukochev, A. Usachev 1 , D. Zanin Abstract We study two subclasses of Banach limits: that consisting of Banach limits which are invariant with respect of the Ces` aro operator and that consisting of Banach limits which are invariant with respect to all dilations. We prove that the first is a proper subset of the second. We also show that these classes are at maximal distance from the set ext B of all extreme points of the set of all Banach limits. Keywords: Banach limits, Ces`aro operator, dilation operator, extreme points. MSC: Primary 46B45; Secondary 47B37

1

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To the memory of W. A. J. Luxemburg.

Introduction and Preliminaries Throughout the paper we denote by `∞ the space of all bounded sequences x = (x1 , x2 , . . .)

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equipped with the norm

kxkl∞ = sup |xn |, n∈N

and the usual partial order. Here N stands for the set of natural numbers.

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Definition 1. A linear functional B ∈ `∗∞ is said to be a Banach limit if 1. B > 0, that is Bx > 0 for every x > 0,

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2. B1I = 1, where 1I = (1, 1, . . .),

3. B(T x) = B(x) for every x ∈ `∞ , where T is a the translation operator, that is T (x1 , x2 , . . .) = (0, x1 , x2 , x3 , . . .).

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The existence of Banach limits was established by S. Mazur and then appeared in the book of S. Banach [4]. Since then, Banach limits proved to be a useful tool in functional analysis. Recent developments in the theory of operator ideals and singular traces [7, 5, 6, 17] reformulated important and difficult problems in these areas in terms of Banach limits. 1

Corresponding author

1

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We denote the set of all Banach limits by B. It is easy to see that B is a closed convex set of the unit sphere of `∗∞ . Hence, kB1 − B2 k`∗∞ 6 2 for every B1 , B2 ∈ B. It follows directly from the definition of Banach limits that

lim inf xn 6 Bx 6 lim sup xn n→∞

n→∞

for every x ∈ `∞ , B ∈ B. In particular, Bx = lim xn for every convergent sequence x ∈ `∞ . n→∞

Following G. G. Lorentz [10], we call a sequence x ∈ `∞ almost convergent to a number

a ∈ R, if B(x) = a for every B ∈ B.

We denote the set of all almost convergent sequences (resp., sequences almost convergent

to zero) by ac (resp., ac0 ). The following criterion for almost convergence was proved by Lorentz [10]. Theorem 2. A sequence x is almost convergent to a if and only if

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uniformly in m ∈ N.

f

m+n 1 X xk = a n→∞ n k=m+1

lim

rep

It is easy to see that Banach limits are not multiplicative functionals on `∞ . Indeed, for the sequence x = (1, 0, 1, 0, ...) we have x + T x = 1I and B(x) = 1/2. Hence, 1 0 = B(x · T x) 6= B(x) · B(T x) = . 4 This fact sets apart the set of all Banach limits and the set of all extreme points of the set of

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all extended limits (or, states on the algebra `∞ ), which coincides with the set of all bounded multiplicative functionals (or, pure states) on this algebra. Moreover, the distance between Banach limits and the bounded multiplicative functionals is 2. Indeed, for every multiplicative

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functional γ on `∞ and every x ∈ {0, 1}N (the set of all sequences of zeroes and ones) we have

γ(x) = γ(x2 ) = (γ(x))2 . Thus, γ(x) is either 0 or 1. Fix j ∈ N and take x = χjN ∈ {0, 1}N . P i i Since j−1 i=0 T x = 1I, it follows that γ(T x) = 1 for some 0 ≤ i ≤ j − 1. On the other hand, the sequence x is almost convergent and Bx = 1/j for every Banach limit B. Thus,

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1 kγ − Bk`∗∞ ≥ (γ − B)(2T i x − 1I) = 2(1 − ). j

Letting j → ∞ proves the claim.

However, some Banach limits are multiplicative on some subspace of `∞ . More precisely,

in [11], W. A. J. Luxemburg proved that every extreme point of the set B is multiplicative on the stabiliser of ac0 defined as follows: Stac0 := {z ∈ `∞ : z · ac0 ⊂ ac0 }. 2

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This set was further investigated in [1, 16, 2]. An analogue of this set for Banach limits with additional invariance properties was introduced in [3]. The paper [11] is also interesting for its approach to study Banach limits via non-standard analysis. It is also worth to mention the work [12] of W. A. J. Luxemburg and B. de Pagter, who introduced the notion of almost convergent elements with respect to an abelian semigroup of Markov operators on AM-spaces. The Lorentz’ result was strengthened by L. Sucheston [19] as follows: Theorem 3. For every x ∈ `∞ one has {Bx : B ∈ B} = [q(x), p(x)], where

m+n 1 X xk , n→∞ m∈N n k=m+1

m+n 1 X xk . n→∞ m∈N n k=m+1

q(x) = lim inf

p(x) = lim sup

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The first result concerning Banach limits with additional invariance properties is due to

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W. Eberlein [8], who established existence of Banach limits invariant under the Hausdorff transformations. Eberlein’s approach was further developed in [15] to study Banach limits invariant under an operator H ∈ Γ. Here, Γ stands for the set of all linear operators in `∞ ,

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which satisfy the following conditions: (i) H ≥ 0 and H1I = 1I; (ii) Hc0 ⊂ c0 ;

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(iii) lim sup(A(I − T )x)j ≥ 0 for all x ∈ l∞ , A ∈ R, where R = R(H) = conv{H n , n ∈ j→∞

N ∪ {0}}.

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Denote by B(H) the set of all Banach limits invariant with respect to the operator H. For every m ∈ N define a dilation operator σm : `∞ → `∞ as follows: σm (x1 , x2 , . . .) = (x1 , x1 , . . . , x1 , x2 , x2 , . . . , x2 , . . .). | {z } | {z }

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m

(1)

m

Denote by C the Ces`aro operator on `∞ given by n

(Cx)n =

1X xk , n ≥ 1. n k=1

It can be shown that C, σm ∈ Γ for every m ∈ N (see discussion after Theorem 8 in [2]

and [15]) and, so the sets B(σm ) and B(C) are non-empty. The sets B(σm ) were studied in [6, 2, 18]. In the present paper we investigate the set ∩m≥2 B(σm ). We also discuss properties of the set ext B of all extreme points of the convex set B. 3

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2

Relation to Ces` aro invariant Banach limits It was shown in [18, Theorem 3] that, for every m ∈ N, the following inclusion holds

B(C) ⊂ B(σm ). Hence,

B(C) ⊂

∞ \

B(σm ).

(2)

m=2

In [3, Theorem 4.8], it was shown that the inclusion B(C) ⊂ B(σm ) is also proper. In

this section, we show that the inclusion (2) is proper. T Set Σ := ∞ m=2 B(σm ). For a given x ∈ `∞ denote

B(C, x) = {B(x) : B ∈ B(C)} and B(Σ, x) = {B(x) : B ∈ Σ}.

Theorem 4. There exists B ∈ Σ such that B ∈ / B(C).

sup

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B(Σ, x) =

f

Proof. It was proved in [2, Theorem 7] that: "

lim inf (A(x + s))j , j→∞

A∈R(Σ),s∈S

inf

A∈R(Σ),s∈S

#

lim sup(A(x + s))j , j→∞

(3)

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where S := (I −T )`∞ and R(Σ) is a convex hull of all finite products of operators σm . Clearly,

conv{σn , n ∈ N} ⊂ R(Σ). On the other hand, for every n ∈ N and k1 , ..., kn ∈ N we have σk1 σk2 · · · σkn = σk1 k2 ···kn . Thus, R(Σ) = conv{σn , n ∈ N}. one has Bx = B(Cx) = 0.

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It is easy to see, that for every B ∈ B(C) and every x ∈ `∞ such that limj→∞ (Cx)j = 0 Taking these facts into account, in order to prove the assertion it is sufficient to construct

and

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x ∈ `∞ such that

lim (Cx)j = 0

j→∞

(4)

lim sup(A(x + s))j = 1 j→∞

for every A ∈ conv{σn , n ∈ N} and s ∈ S.

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n

n

For a given n ∈ N, denote Jn = [22 − n, 22 ) and let Jn,k be a subset of N, minimal with

respect to inclusion, such that σk Jn,k ⊃ Jn for all 1 ≤ k ≤ n. Here, σk Jn,k is understood as the set of all m ∈ N such that m = k · l for some l ∈ Jn,k . Set In = ∪nk=1 Jn,k . Since Jn,k is minimal with respect to inclusion, it follows that |Jn,k | ≤

1 n |Jn | + 2 = + 2. k k

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Hence,

n−1

and (22

n n X X n 1 |In | ≤ ( + 2) = n + 2n < n(n + 2) k k k=1 k=1

n

, 22 ] ⊃ In for all n ∈ N. Consider the sequence x = (x1 , x2 , . . . ) given by setting  1, m ∈ S∞ Ik k=1 xm = 0, otherwise.

If j ∈ (22

n−1

n

, 22 ] for some n ∈ N, then

n X

n−1

(Cx)j ≤ 2−2

k=1

and, thus,

n−1

|Ik | < 2−2

n X

k(k + 2)

k=1

lim (Cx)j = 0.

j→∞

We have

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Let now A ∈ conv{σn , 1 ≤ n ≤ m} for some m ∈ N and s = (I − T )y for some y ∈ `∞ . k+r X si = |yk+1 − yk+r | ≤ 2kyk`∞

for every k, r ∈ N. Hence,

rep

i=k+1

k+r X (σn s)i ≤ 2nkyk`∞ ≤ 2mkyk`∞ i=k+1

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for every n = 1, 2, . . . , m. Since A ∈ conv{σn , 1 ≤ n ≤ m}, it follows that k+r k+r X X (As) ≤ max (σ s) i n i ≤ 2mkyk`∞ 1≤n≤m i=k+1

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Therefore,

i=k+1

2mkyk`∞ r for every k, r ∈ N. Since the above inequality holds for arbitrary large r, it follows that max (As)i ≥ −

k
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max (As)i ≥ 0.

k
(5)

By construction, σk x|Jn = 1 for all 1 ≤ n ≤ m. Hence, Ax|Jn = 1. Next, if n ≥ 4mkyk`∞ ,

then by (5)

max(A(x + s))j ≥ 1 − 0 = 1. j∈Jn

This means that inf

A∈R(Σ),s∈S

lim sup(A(x + s))j ≥ 1. j→∞

The converse inequality is straightforward. This proves the assertion. 5

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3

Distance to extreme points Denote by ext B the set of extreme points of B. Since B is compact in the weak∗ topology,

it follows from Krein-Milman theorem that B = conv ext B, where the closure is taken in the weak∗ topology. The classical and recent studies of the set of extreme points of B demonstrate that this set has very interesting properties [1, 11, 13, 14, 16, 21]. In particular, it was shown in [14] that a subspace generated by any countable sequence in ext B is isometric to l1 . Hence, kB1 −B2 k = 2 for every B1 , B2 ∈ ext B, B1 6= B2 . It was shown in [2, Theorem 14] that the sets B(σn ), n ∈ N, n ≥ 2 and ext B are at

maximal distance from each other. This fact can be given in stronger form (see Corollary 6 below). We need some preparations to state the result. The kernel N (I − T ∗ ) of an operator

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I − T ∗ is a Riesz subspace in the Banach lattice `∗∞ (see e.g., [1]). So, it is an AL-subspace under the norm and the order induced from `∗∞ . Clearly, the positive part of the unit sphere

+ SN (I−T ∗ ) coincides with the set B.

As an AL-space, N (I − T ∗ ) is isometrically order isomorphic to the space L1 (Ω) of all

rep

functions integrable on some set Ω with a measure µ. The set Ω can be partitioned into Ωd and Ωc such that the restriction of µ to Ωd is discrete and the restriction of µ to Ωc is continuous. That is,

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N (I − T ∗ ) ≈ L1 (Ω) = L1 (Ωd ) ⊕ L1 (Ωc ). Atoms of Ωd correspond to ext B. So, L1 (Ωd ) is isometrically isomorphic to the closure of a convex hull of ext B in the norm topology. In particular, the closure of conv(ext B) in the

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norm topology coincides with the positive part SL+1 (Ωd ) of a unit sphere of the space L1 (Ωd ). We refer to [2] for more details.

Denote Bd := SL+1 (Ωd ) and Bc := SL+1 (Ωc ) . Since components L1 (Ωd ) and L1 (Ωc ) are disjoint in N (I − T ∗ ), it follows that kB1 − B2 k`∗∞ = 2 for every B1 ∈ Bd and B2 ∈ Bc .

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Theorem 5. For every n ∈ N, n ≥ 2, one has B(σn ) ⊂ Bc . Proof. For B1 ∈ B(σn ) and B2 ∈ ext B consider the function g(t) := kB1 − tB2 k`∗∞ , t ≥ 0.

It is easy to see that g is a convex function and g(t) ≤ 1 + t for all t ≥ 0. From [2, Theorem

14] we obtain that g(1) = 2. These three facts imply that g(t) = 1 + t for all t ≥ 0, that is, kB1 − tB2 k`∗∞ = 1 + t ≥ 1, t ≥ 0. 6

(6)

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On the contrary, suppose that B(σn ) is not a subset of Bc . Hence, there exist B3 , B4 ∈

B(σn ), Bi ∈ ext B, 5 ≤ i ≤ M ≤ ∞ and numbers α4 ≥ 0, αi > 0 for all 5 ≤ i ≤ M ≤ ∞ such that

B3 =

M X

αi Bi

and

i=4

We have

M X

αi = 1.

i=4

kB3 − α5 B5 k`∗∞ = k

M X i=4 i6=5

αi Bi k`∗∞ =

M X

αi < 1,

i=4 i6=5

which contradicts (6). Therefore, B(σn ) ⊂ Bc . Corollary 6. For every n ∈ N, n ≥ 2, and every B1 ∈ B(σn ), B2 ∈ convn ext B (the closure

is taken in the norm topology), one has kB1 − B2 k`∗∞ = 2.

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Proof. By Theorem 5, B(σn ) ⊂ Bc . The assertion follows from the discussion above Theorem

A special subclass of Banach limits

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4

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5.

Let γ : `∞ → C be an extended limit, that is a positive linear functional such that

γ(x) = limn→∞ xn for every convergent x ∈ `∞ . Consider the functional

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Bx = log 2 · γ

1 X n

k≥0

 xk 2−k/n , x ∈ `∞ .

It was proved in [20, Lemma 3.5] that for every extended limit γ the functional B is a Banach

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limit. Denote the set of all such Banach limits by Bζ . These are exactly the Banach limits which correspond (in sense of A. Pietsch [17]) to the extended ζ-function residues (see [20] and references therein).

Theorem 7. The sets Bζ and ext B are disjoint.

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Proof. It was shown in the proof of Lemma 3.7 in [20] that every B ∈ Bζ can be written in

the form B = B1 ◦ C for some Banach limit B1 . Since every Banach limit is an extended limit, it follows that

lim inf xk ≤ B1 x ≤ lim sup xk k→∞

for every x ∈ `∞ . Thus,

k→∞

lim inf (Cx)k ≤ Bx ≤ lim sup(Cx)k k→∞

k→∞

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for every x ∈ `∞ .

Consider the sequence x=

∞ X k=0

χ[4k ,2·4k ) ∈ `∞ .

It follows from [9, Proposition 2.4] that for every B ∈ ext B the value of Bx is either 0 or

1. Thus, it is sufficient to show that that 0 < Bx < 1. Direct computations yield n

2·4 n 1 X 1 X k 2 Bx ≤ lim sup(Cx)n = lim sup xk = lim sup 4 = n n 3 n→∞ n→∞ 2 · 4 n→∞ 2 · 4 k=0 k=0

and

n

4 −1 n−1 1 X 1 X k 1 xk = lim inf n 4 = . Bx ≥ lim inf (Cx)n = lim inf n n→∞ 4 − 1 n→∞ n→∞ 4 − 1 3 k=0 k=0

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This completes the proof.

Acknowledgment. The authors thank the anonymous referee for many valuable suggestions, which improved the manuscript. The work of the first and the third authors was

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supported by the Russian Science Foundation (grant 19-11-00197), the work of the second and the fourth authors was supported by the Australian Research Council.

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References

[1] E.A. Alekhno, Superposition operator on the space of sequences almost converging to zero,

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Cent. Eur. J. Math. 10 (2012), 619–645. [2] E. Alekhno, E. Semenov, F. Sukochev, A. Usachev, Order and geometric properties of the set of Banach limits, St. Petersburg Math. J. (2016), 3–35. [3] E. Alekhno, E. Semenov, F. Sukochev, A. Usachev Invariant Banach limits and their

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extreme points, Studia Math., 242:1 (2018), 79–107. ´ [4] Banach, S. Th´eorie des op´erations lin´eaires. Editions Jacques Gabay, Sceaux, 1993. Reprint of the 1932 original. [5] Dodds, P. G., de Pagter, B., Sedaev, A. A., Semenov, E. M., and Sukochev, F. A. Singular symmetric functionals. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 290, Issled. po Linein. Oper. i Teor. Funkts. 30 (2002), 42–71, 178.

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[6] Dodds, P. G., de Pagter, B., Sedaev, A. A., Semenov, E. M., and Sukochev, F. A. Singular symmetric functionals and Banach limits with additional invariance properties. Izv. Ross. Akad. Nauk Ser. Mat. 67, 6 (2003), 111–136. [7] Dodds, P. G., de Pagter, B., Semenov, E. M., and Sukochev, F. A. Symmetric functionals and singular traces. Positivity 2, 1 (1998), 47–75. [8] Eberlein, W. F. Banach-Hausdorff limits. Proc. Amer. Math. Soc. 1 (1950), 662–665. [9] Keller, G., and Moore, Jr., L. C. Invariant means on the group of integers. In Analysis and geometry. Bibliographisches Inst., Mannheim, 1992, pp. 1–18. [10] Lorentz, G. G. A contribution to the theory of divergent sequences. Acta Math. 80 (1948), 167–190.

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[11] W. A. J. Luxemburg, Nonstandard hulls, generalized limits and almost convergence, Analysis and Geometry, Bibliographisches Inst., Mannheim, (1992), 19–45. [12] W. A. J. Luxemburg, B. de Pagter, Invariant Means for Positive Operators and Semi-

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groups, Katholieke Universiteit Nijmegen, Subfaculteit Wiskunde, Nijmegen, (2005) 31– 55.

[13] Nillsen, R. Nets of extreme Banach limits. Proc. Amer. Math. Soc. 55, 2 (1976),

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[14] Semenov, E. M., and Sukochev, F. A. Extreme points of the set of Banach limits.

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Positivity 17, 1 (2013), 163–170.

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[16] E. M. Semenov, F. A. Sukochev, A. S. Usachev, Geometric properties of the set of

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Banach limits, Izv. Ross. Akad. Nauk Ser. Mat. 78 (2014), 177–204. [17] Semenov, E. M., Sukochev, F. A., Usachev, A. S., and Zanin, D. V. Banach limits and traces on L1,∞ . Adv. Math. 285 (2015), 568–628. [18] E. Semenov, F. Sukochev, A. Usachev, D. Zanin Invariant Banach limits and applications to noncommutative geometry, submitted manuscript. [19] Sucheston, L. Banach limits. Amer. Math. Monthly 74 (1967), 308–311. 9

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[20] Sukochev, F., Usachev, A., and Zanin, D. Singular traces and residues of the ζ-function. Indiana J. Math. 66, 4 (2017), 1107–1144. [21] Talagrand, M. Moyennes de Banach extr´emales. C. R. Acad. Sci. Paris S´er. A-B 282, 23 (1976), Aii, A1359–A1362.

Evgenii Semenov, Faculty of Mathematics, Voronezh State University, Voronezh, 394006 Russia. [email protected] Fedor Sukochev, School of Mathematics and Statistics, The University of New South Wales, 2052, NSW, Australia. [email protected] Alexandr Usachev, School of Mathematics and Statistics, Central South University, Hu-

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nan, 410085, China. [email protected]

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Dmitriy Zanin, School of Mathematics and Statistics, The University of New South Wales,

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2052, NSW, Australia. [email protected]

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