Inverse semigroup from metrics on doubles III. Commutativity and (in)finiteness of idempotents
aa r X i v : . [ m a t h . M G ] J a n INVERSE SEMIGROUP FROM METRICS ON DOUBLES III.(IN)FINITENESS OF IDEMPOTENTS
V. MANUILOV
Abstract.
We have shown recently that, given a metric space X , the coarse equivalenceclasses of metrics on the two copies of X form an inverse semigroup M ( X ). Here westudy the property of idempotents in M ( X ) of being finite or infinite, which is similar tothis property for projections in C ∗ -algebras. We show that if X is a free group then theunit of M ( X ) is infinite, while if X is a free abelian group then it is finite. We also showthat the inverse semigroup M ( X ) is not a quasi-isometry invariant. More examples offinite and infinite idempotents are provided. Introduction
Given metric spaces X and Y , a metric d on X ⊔ Y that extends the metrics on X and Y , depends only on the values d ( x, y ), x ∈ X , y ∈ Y , and it may be not easy tocheck which functions d : X × Y → (0 , ∞ ) determine a metric on X ⊔ Y . The problemof description of all such extended metrics is difficult due to the lack of a nice algebraicstructure on the set of metrics, but, passing to coarse equivalence of metrics, we get analgebraic structure, namely, that of an inverse semigroup [3]. Coarse equivalence classes ofmetrics on X ⊔ Y can be considered as morphisms from X to Y [2], where the composition ρd of a metric d on X ⊔ Y and a metric ρ on Y ⊔ Z is given by the metric determined by( ρ ◦ d )( x, z ) = inf y ∈ Y [ d ( x, y ) + ρ ( y, z )], x ∈ X , z ∈ Z . The semigroup (with respect tothiscomposition) of coarse equivalence classes of metrics on X ⊔ X is an inverse semigroupwith the unit element and the zero element .In what follows we identify X ⊔ X with X × { , } , and write X for X × { } (resp., x for ( x, X ′ for X × { } (resp., x ′ for ( x, S is an inverse semigroup if for any s ∈ S there exists a unique t ∈ S (denoted by s ∗ ) such that s = sts and t = tst [1]. Philosophically, inverse semigroupsdescribe local symmetries in a similar way as groups describe global symmetries, andtechnically, the construction of the (reduced) group C ∗ -algebra of a group generalizes tothat of the (reduced) inverse semigroup C ∗ -algebra [5]. Our standard reference for inversesemigroups is [1].Any two idempotents of an inverse semigroup S commute, and the semilattice E ( S ) ofall idempotents of S generates a commutative C ∗ -algebra. There is a partial order on S defined by s ≤ t if s = ss ∗ t .Close relation between inverse semigroups and C ∗ -algebras allows to use classificationof projections in C ∗ -algebras for idempotents in inverse semigroups. Namely, as in C ∗ -algebra theory, we call two idempotents, e, f ∈ E ( S ) von Neumann equivalent (and write e ∼ f ) if there exists s ∈ S such that s ∗ s = e , ss ∗ = f . An idempotent e ∈ E ( S ) iscalled infinite if there exists f ∈ E ( S ) such that f (cid:22) e , f = e , and f ∼ e . Otherwise e is finite . An inverse semigroup is finite if every idempotent is finite, and is weakly finite if itis unital and the unit is finite.In [4] we gave a geometric description of idempotents in the inverse semigroup M ( X )(there are two types of idempotents, named type I and type II) and showed in Lemma 3.3that the type is invariant under the von Neumann equivalence. In this paper, we study the property of weak finiteness for M ( X ) (i.e. finiteness of the unit element) and discussits relation to geometric properties of X .We start with several examples of finite or infinite idempotents, and then show that if X is a free group then M ( X ) is not weakly finite, while if X is a free abelian group thenit is weakly finite. We also show that the inverse semigroup M ( X ) is not a quasi-isometryinvariant. The property of being weakly finite is also not a coarse invariant. We don’tknow if it is a quasi-isometry invariant.2. Some examples
The following example shows that in M ( X ), for an appropriate X , we can imitateexamples of partial isometries and projections in a Hilbert space. Example 2.1.
Let l ( N ) be the space of infinite l sequences, with the metric given bythe l -norm, and let X n = { (0 , . . . , , t, , . . . ) : t ∈ [0 , ∞ ) } with t at the n -th place, n ∈ N . Set X = ∪ n ∈ N X n ⊂ l ( N ).Let x = (0 , . . . , , t, , . . . ) ∈ X n , y = (0 , . . . , , s, , . . . ) ∈ X m . Define metrics d , e , f on X ⊔ X ′ by d ( x, y ′ ) = (cid:26) | s − t | + 1 , if m = n + 1; s + t + 1 , if m = n + 1 ,e ( x, y ′ ) = (cid:26) | s − t | + 1 , if m = n ; s + t + 1 , if m = n,f ( x, y ′ ) = (cid:26) | s − t | + 1 , if m = n ≥ s + t + 1 , if m = n or m = n = 1 . It is easy to see that d , e , f are metrics on X ⊔ X ′ , and that d ∗ d = e , dd ∗ = f are idempotents, and that e = = f . In particular, is infinite. Although d seemssimilar to a one-sided shift in a Hilbert space, it behaves differently: f is orthogonallycomplemented, i.e. there exists h such that f ∨ h = , f ∧ h = , but the complement isnot a minimal idempotent, i.e. there exists a lot of idempotents j ∈ E ( M ( X )) such that j ≤ h , j = h .On the other hand, if X ⊂ [0 , ∞ ) with the standard metric then the inverse semigroup M ( X ) is commutative (Prop. 7.1 in [3]), hence any idempotent can be equivalent only toitself, hence is finite.The next example shows that the picture may be more complicated. Theorem 2.2.
There exists an amenable space X of bounded geometry and s ∈ M ( X ) such that s ∗ s = , but ss ∗ = .Proof. Let x n = (log 2 , log 3 , . . . , log n, log( n + 1) , , , . . . ) ,x ′ n = (log 2 , log 2 , log 3 , log 3 , . . . , log n, log n, log( n + 1) , log( n + 1) , , , . . . ) , and let X = { x n : n ∈ N } , X ′ = { x ′ n : n ∈ N } ,X, X ′ ⊂ l ∞ ( N ) with the metric d ( x, y ) = sup k | x k − y k | , x = ( x , x , . . . ), y = ( y , y , . . . ).Take m > n , then d ( x n , x m ) = log m = d ( x ′ n , x ′ m ), hence the restriction of d onto the twocopies of X coincide (thus determining the metric d X on X ), and d ∈ M ( X ).We have, for n even, n = 2 k , d ( x n , X ′ ) = inf m ∈ N d ( x k , x ′ m ) = d ( x k , x ′ k ) = max i ≤ k ( | log( i + 1) − log(2 i + 1) | ≤ log 2 , NVERSE SEMIGROUP FROM METRICS ON DOUBLES III. (IN)FINITENESS OF IDEMPOTENTS 3 and for n odd, n = 2 k − d ( x k − , x ′ m ) ≥ log( k + 1)for any m ∈ N , hence d ( x n , X ′ ) = inf m ∈ N d ( x k − , x ′ m ) = d ( x k − , x ′ k ) = log( k + 1) , i.e. lim k →∞ d ( x k − , X ′ ) = ∞ . On the other hand, d ( x ′ n , X ) = inf m ∈ N d ( x ′ n , x m ) ≤ d ( x ′ n , x n ) = log(2 n + 1) − log( n + 1) ≤ log 2 . Let X + = { x k : k ∈ N } , X − = { x k − : k ∈ N } . Then d ∗ d ( x, x ′ ) = inf y ∈ X [ d ( x, y ′ ) + d ∗ ( y, x ′ )] = inf y ∈ X [ d ( x, y ′ ) + d ( x, y ′ )] = 2 d ( x, X ′ ) ≤ log 2for any x ∈ X + and lim x ∈ X − ; x →∞ d ∗ d ( x, x ′ ) = lim x ∈ X − ; x →∞ d ( x, X ′ ) = ∞ , while dd ∗ ( x, x ′ ) = 2 d ( x ′ , X ) ≤ log 2for any x ∈ X .Let d + , d − ∈ M ( X ) be the idempotent selfadjoint metrics defined by d ± ( x, y ′ ) = inf u ∈ X ± [ d X ( x, u ) + 1 + d X ( u, y )] . Then [ d ∗ d ] = [ d + ], [ dd ∗ ] = , and [ d + ] is strictly smaller than , hence M ( X ) is notfinite.Note that X is amenable. Set F n = { x , . . . , x n } ⊂ X . Let N r ( A ) denote the r -neighborhood of the set A . Then N r ( F n ) \ F n is empty when log( n + 1) > r , hence { F n } n ∈ N is a Følner sequence. For r = log m , the ball B r ( x n ) of radius r centered at x n contains either no other points besides x n (if n ≥ m + 1), or it consists of the points x , . . . , x m (if n ≤ m ), hence the metric on X is of bounded geometry. In fact, this spaceis of asymptotic dimension zero. (cid:3) Case of free groups
Let X = Γ be a finitely generated group with the word length metric d X . Consider thefollowing property (I):(i1) X = Y ⊔ Z , and for any D > z ∈ Z such that d X ( z, Y ) > D ;(i2) there exist g, h ∈ Γ such that gY ⊂ Y , hZ ⊂ Y and gY ∩ hZ = ∅ ;(i3) there exists C > | d X ( gy, hz ) − d X ( y, z ) | < C for any y ∈ Y , z ∈ Z .The property (I) is neither stronger nor weaker than non-amenability. If we requireadditionally that Y ∼ Z then it would imply non-amenability. Lemma 3.1.
The free group F on two generators satisfies the property (I).Proof. Let a and b be the generating elements of F , and let Y ⊂ X be the set of allreduced words in a , a − , b and b − that begin with a or a − , Z = X \ Y . Let g = ab , h = a . Clearly, gY ⊂ Y and hZ ⊂ Y .If z begins with a n , n > D , then d X ( z, Y ) ≥ n .If y ∈ Y , z ∈ Z then d X ( aby, a z ) = | y − b − a − a z | = | y − b − az | = | y − z | + 2 = d X ( y, z ) + 2 , V. MANUILOV as the word y − b − az cannot be reduced any further ( y − ends with a ± , and z eitherbegins with b ± , or is an empty word). (cid:3) Theorem 3.2.
Let X = Γ be a group with the property (I). Then X is not weakly finite.Proof. We shall prove that there exists d ∈ M ( X ) such that [ d ∗ d ] = and [ dd ∗ ] = .Let X = Y ⊔ Z , g, h ∈ Γ satisfy the conditions of the property (I). Define a map f : X → X by setting f ( x ) = (cid:26) gx, if x ∈ Y ; hx, if x ∈ Z. The maps f | Y and f | Z are left multiplications by g and h , respectively, hence areisometries. If y ∈ Y , z ∈ Z then (i3) holds for some C >
0, hence | d X ( f ( x ) , f ( y )) − d X ( x, y ) | < C holds for any x, y ∈ X .Set d ( x, y ′ ) = inf u ∈ X [ d X ( x, u ) + C + d X ( f ( u ) , y )]. It is easy to check that d satisfies alltriangle inequalities, hence d is a metric, d ∈ M ( X ). Then d ∗ d ( x, x ′ ) = 2 d X ( x, X ′ ) = 2 inf u,y ∈ X [ d X ( x, u ) + C + d X ( f ( u ) , y )] = 2 C for any x ∈ X , and dd ∗ ( x, x ′ ) = 2 d X ( x ′ , X ) = inf z,u ∈ X [ d X ( z, u ) + C + D X ( f ( u ) , x )] = 2( C + d X ( f ( X ) , x )) ≥ C + 2 d X ( Y, x )is not bounded. Thus, [ d ∗ d ] = , while [ dd ∗ ] = . (cid:3) Case of abelian groups
A positive result is given by the following Theorem.
Theorem 4.1.
Let X = R n , with a norm k · k , and let the metric d X be determined bythe norm k · k . If s ∈ M ( X ) , s ∗ s = then ss ∗ = .Proof. Let d ∈ M ( X ), [ d ] = s . As [ d ∗ d ] = , there exists C > d ( x, X ′ ) < C for any x ∈ X . It suffices to show that there exists D > d ( x ′ , X ) < D for any x ∈ X . Suppose the contrary: for any n ∈ N there exists x n ∈ X such that d ( x ′ n , X ) > n .Then d ( y ′ , X ) > n for any y ∈ X such that d X ( y, x n ) ≤ n .As 2 d ( x, X ′ ) < C for any x ∈ X , there is a (not continuous) map f : X → X such that d X ( x, f ( x ) ′ ) < C/ x ∈ X . This map satisfies | d X ( f ( x ) , f ( y )) − d X ( x, y ) | < C forany x, y ∈ X . Then there exists a continuous map g : X → X such that d X ( f ( x ) , g ( x ))
2, and d X ( y , y ) = 2 R n , the triangle inequality (for the trianlgewith the vertices y , y , z ′ ) is violated when 2 R n > C . This contradiction proves theclaim. NVERSE SEMIGROUP FROM METRICS ON DOUBLES III. (IN)FINITENESS OF IDEMPOTENTS 5 (cid:3)
Corollary 4.2.
Let X = Z n with an l p -metric, ≤ p ≤ ∞ , and let s ∈ M ( X ) . Then s ∗ s = implies ss ∗ = .Proof. By Proposition 9.2 of [3], M ( Z n ) = M ( R n ). (cid:3) M ( X ) doesn’t respect equivalences Proposition 5.1.
The inverse semigroup M ( X ) is not a coarse invariant.Proof. The space X from Theorem 2.2 is coarsely equivalent to the space Y = { n : n ∈ N } with the standard metric, which we denote by b X . Indeed, for n < m , we have b X ( x n , x m ) = m − n and d X ( x n , x m ) = log( m +1). As m − ( m − = 2 m − > log( m +1)for m >
1, we have d X ( x, y ) ≤ b X ( x, y ) for any x, y ∈ X , and taking f ( t ) = 2 e t , we have b X ( x, y ) ≤ f ( d X ( x, y )) for any x, y ∈ X .For the metric d X from Theorem 2.2, the inverse semigroup M ( X, d X ) is not commu-tative ([ d ∗ d ] = [ dd ∗ ]), while the inverse semigroup M ( X, b X ) is commutative by Prop. 7.1of [4]. (cid:3) Theorem 5.2.
The inverse semigroup M ( X ) is not a quasi-isometry invariant.Proof. Let X = N be endowed with the metric b X given by b X ( n, m ) = | n − m | , n, m ∈ N ,and let y n = s ( n )4 [ n ] , where s ( n ) = ( − [ n − ] and [ t ] is the greatest integer not exceeding t . Let d X be the metric on X given by d X ( n, m ) = | y n − y m | , n, m ∈ N . The two metricsare quasi-isometric. Indeed, suppose that n > m . If s ( n ) = − s ( m ) then d X ( n, m ) = 4 [ n ] + 4 [ m ] ≤ n +1 + 4 m +1 = 4( · n + 2 m ) ≤ b X ( n, m ); d X ( n, m ) = 4 [ n ] + 4 [ m ] ≥ n + 4 m ≥ n − m = b X ( n, m ) . We use here that r +12 r − ≤ r = n − m ∈ N . If s ( n ) = s ( m ) then d X ( n, m ) = 4 [ n ] − [ m ] ≤ n +1 − m = 4 · n − m ≤ b X ( n, m ) . We use here that · r − r − ≤ r = n − m ∈ N . To obtain an estimate in otherdirection, note that s ( n ) = s ( m ) implies that [ n/ ≥ [ m/
2] + 1, and that n − m = 2. If n = m + 1 then d X ( m + 1 , m ) = 3 · [ m/ ≥ · m = 32 b X ( m + 1 , m ) , If n ≥ m + 3 then d X ( n, m ) = 4 [ n ] − [ m ] ≥ n − m +1 = 2 n − · m ≥ b X ( n, m ) . We use here that r − r − ≥ for any r = n − m ≥
3. Thus,37 b X ( n, m ) ≤ d X ( n, m ) ≤ · b X ( n, m )for any n, m ∈ N , so the two metrics are quasi-isometric.We already know that M ( X, b X ) is commutative, so it remains to expose two non-commuting elements in M ( X, d X ).Let X = { ( y n ,
0) : n ∈ N } , X ′ = { ( − y n ,
1) : n ∈ N } , V. MANUILOV and let d be the metric on X ⊔ X ′ induced from the standard metric on the plane R , s = [ d ]. Note that − y n = y n − if y n > n >
1, and − y n = y n +1 if y n <
0. Hence, d ∗ = d and s = .Let A + = { y n : n ∈ N ; y n > } , A − = { y n : n ∈ N ; y n < } ,X = A + ⊔ A − , and let the metrics d + and d − on X ⊔ X ′ be given by d ± ( n, m ′ ) = inf k ∈ A ± [ d X ( n, k ) + 1 + d X ( k, m )] ,e = [ d + ], f = [ d − ]. Then es = , while se = f , i.e. e and s do not commute. (cid:3) References [1] M. V. Lawson.
Inverse Semigroups: The Theory of Partial Symmetries.
World Scientific, 1998.[2] V. Manuilov.
Roe bimodules as morphisms of discrete metric spaces.
Russian J. Math. Phys., (2019), 470–478.[3] V. Manuilov. Metrics on doubles as an inverse semigroup.
J. Geom. Anal., to appear.[4] V. Manuilov.
Metrics on doubles as an inverse semigroup II.
J. Math. Anal. Appl., to appear.[5] A. L. T. Paterson.
Groupoids, Inverse Semigroups, and their Operator Algebras.
Springer, 1998.
Moscow Center for Fundamental and Applied Mathematics and
Moscow State Univer-sity, Leninskie Gory 1, Moscow, 119991, Russia
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