CC BY
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УДК 517.9
ALGEBRAS ASSOCIATED WITH GRADED
SYSTEMS
Victor Arzumanian1 Suren Grigoryan
1Institute of Mathematics, National Academy of Sciences of RA, vicar@instmath.sci.am 2Kazan State Power Engineering University, gsuren@inbox.ru
The concept of group grading arises naturally in studying crossed products, especially in the context of irreversible dynamical systems. In the paper some general aspects concerning group graded systems are considered as well as some examples of such systems associated with some known constructions.
Keywords: C*-algebra, representation, bimodule, Hilbert module, graded system. 2010 Mathematical Subject Classification: 46L05, 46L08, 28D05
Introduction
In recent years, the attention of many specialists were focused on problems arising in constructions of algebras associated with irreversible dynamic systems. Since the concept of the group crossed product can not be directly transferred to the semigroup case, new methods are developed to avoid the known difficulties. Involvement into considerations the concept of group-graded system seems to be most promising in this regard.
We used in [1] a version of so-called Fell bundle (C*-algebraic bundle), which was developed in detail by Ruy Exel et al (see [2]-[5]) and presented in his recent remarkable book [7]. However, our purposes are a little differ from the accents of the book of Exel, who interested mainly on partial actions. In contrast, it seems that taking into account the tool of group-grading systems one can avoid the
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difficulties related to partial isometries accompanying the irreversible dynamical systems.
In the present paper we prefer consider the main object as an involutive semigroup structured in a special way with the help of a group. We present some properties of such systems based on its module structure, and on the existence of a natural action of the dual group. We are also interested on intensional examples of the graded systems associated with some known dynamical systems.
The work may be considered as an attempt to study semigroup dynamical systems in pursuit to replace partial actions on a given C*-algebra by the actions of a suitable group on a modified algebra.
Basically, in the stated part, we follow the contours of the paper [1] with some new details, clarifications and corrections. In addition, we have removed from consideration the concept of graded C*-algebra, the relationship with the graded systems, and most importantly, with the initial semigroup systems, leaving it to fuiture publications.
1. Definitions and elementary properties.
Let r be a discrete group with the unity e and A be a semigroup with zero. We say A is r-equipped with a system Ar = {AY , y G r} of the subsets of A if
(i) Ay is a Banach space for each y G r
(ii) uyer Ay = A
(iii) Aa n Ap = {0} for each a, P e r, a f p.
Definition 1. A r-equipped involutive semigroup A = (r, Ar) is called r-graded system if the operations of multiplication and involution on the semigroup are consistent with the operations on the Banach spaces (components of the system), and
(i) ab G Aap, for a G Aa, b G Ap;
(ii) a* G AY-1, for a G AY;
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(iii) llabll < lallbl, for a £ Aa, b £ Ap;
(iv) lla*al = llall2 = lla*ll2, for a £ Ay.
Obviously, the central algebra A = Ae is a C*-algebra as well as an involutive subsemigroup of the semigroup A.
In what follows we assume the semigroup A has an identity element (say, 1), so the central algebra A is unital.
Definition 2. We say that a Y-graded system B = (r, Br) is a subsystem of a Y-graded system A = (r, Ar) if B is a *-subsemigroup of A, and for each y £ r the Banach space BY is a subspace of the Banach space AY. A Y-graded subsystem I =(r, Ir) is called an ideal (two-sided) of the system A = (r, Ar) if it is an involutive ideal (two-sided) of the semigroup A.
Thus, the central algebras B = Be and I = Ie are, respectively, a C*-subalgebra and an ideal of the central C*-algebra A.
When I is an ideal of the system A, a standard equivalence relation can be defined on A in the following way: elements a and b are equivalent, a ~ b, if they are from the same AY, and a - b e IY.
The quotient semigroup A/I defined in the standard way is a r-graded system (r, Ar/Ir) composed of the quotient Banach spaces
(ay/iy, y e r}.
As a morphism of a graded system A = (r, Ar) into another system B = (G, BG) we mean the pair ^ = (p, 9), where p : r ^ G is a group homomorphism, and 9: A ^ B is a *-morphism of the semigroup A into the semigroup B such that 9(AY) e AP(Y) for each y from r, and the restriction of 9 on each AY is linear.
It is easy to prove that each morphism O = (p, 9) : A ^ B does not increase the norm and hence is continuous in this sense.
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Two graded systems are called isomorphic if the mappings p and 9 are bijective. The group of automorphisms of a r-graded system A is denoted by Aut (A).
In the case when the initial group is Abelian, the standard action of the dual group on the r-graded system is reasonable in the following way: if A is a T-graded system, and G = TA be its dual group then for any g G G the pair Og = (id, x(g)), where id is the identity map, and x(g) = g(y)a, a G AY is an automorphism of the system A. Moreover, the map t: G ^ Aut(A) is a faithful representation of the group G into the group of automorphisms of A, and each ideal of a T-graded system is invariant under the standard action of the dual group.
The following fact shows that some elements from different spaces (components of graded algebra) are not completely independent, but related to each other by means of an intertwining element from the central algebra.
An element u G AY, y G r, in a T-graded system A is called:
(i) of unitary type if u*u = uu* = 1,
(ii) of isometry type if u*u = 1, uu* ^ 1,
(iii) of partial isometry type if both u*u and uu* are projections in the central algebra.
Proposition. Let A be a r-graded system with the property: for each y G r there exists an element u G AY of unitary type. Then, for any a, p G r there is in the central algebra A a unique unitary w(a, p) such that uaup = w(a, p)uap.
Proof. It is easy to check that the element v = uaup is a unitary from the space Aap:
v*v = (uaup)*uaup = up*ua*uaup = 1 = vv*.
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Put w = vuap*. Then
w*w = uapv*v uap* = uapluap* = uapuap* = 1= ww*.
2. Modules.
Let A be a graded system, and C be a C* -algebra. We introduce a notion of C-module graded system limiting ourselves to bimodules.
Definition 2. Let A = (r, Ar) be a r-graded system, C be a unital C*-algebra. We say the system A is a (unital) C-module, if each AY is a unital C-module and the following consistency conditions satisfy for a, b £ C, ( £ A« and y £ Ap (modular multiplication is denoted as •)
(i) (b • 0n = b • tfn), tf • b)n = f(b • n)
(ii) (b • 0 • a = b • (^ • a), (^ • b) • a = • (ba)
(iii) (b • 0* = • b*, (^ • b)* = b*• f*.
Graded systems have the standard modular structure since each AY, Y £ r, is an A-module (more precisely, bimodule) with respect the operations
a • ^ = a ^ • a = ^ a
for a £ A, ^ £ Ay. Moreover, the system A is also an A-module.
The structure of one-sided module structure can be introduced in the obvious way.
In a graded system (r, Ar) each AY can be considered as a (right) Hilbert A-module HY using the following inner product
< n > = n*^, n £ Ay.
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Then a right Hilbert A-module structure H(A) can be associated the T-graded system A as a direct sum of the Hilbert modules on the fibers. It is easy to understand that the inner product in H(A) is determined by
< £ n > = 2
where the sum is taken over y G r, and n G Ay, ^ = n = { %)-
Remind that a system ^ G Ay belongs to H(A) if the series Zy^y is convergent in A. It certainly will be satisfied if the series 2yII^yII2 converges.
3. Representations.
We consider group graded systems as something like covariant systems. From this viewpoint, the following concept is an analog thereof.
Definition 4. Let A = (r, Ar) be a r-graded system, and H be a Hilbert space. We call a *-representation n of the semigroup A in H a representation of the r-graded system A linear on each AY, y G r.
Obviously, the restriction of the representation n on the central algebra is a representation of the central C*-algebra A. It is easy to verify that the kernel of any representation of a graded system is an ideal. It is easy to understand that each representation of a r-graded system is continuous in the sense that it does not increase the norm,
I a) I < I a I.
The uniformly closed involutive algebra C*(A, n) generated by rc(A) in B(H) we call the C*-algebra n-associated to the graded system A.
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For any Hilbert C-module H we denote by L(H) the algebra of all adjointable bounded C-linear operators on H.
Definition 5. Let A = (r, Ar) be a r-graded C-module, and H be a (right) Hilbert C-module. A *-representation n of the semigroup A in L(H) is called C-representation of the system A if it is a C-modular mapping on each space AY , y G r.
The uniformly closed involutive algebra C*(A, n) generated by n(A) in L(H) is called the C*-algebra associated to the convolution system (A, n).
Now we introduce a canonical representation of a graded system A in the associated Hilbert module H(A).
Theorem. The mapping specified on generators as
nr(a)£ = a£, for a, £ G A
determines a faithful representation of the system A in the associated Hilbert A-module H(A).
Proof. It is easy to verify that nr is a representation of the system A, and then inr(a)I < IaI. Let us show that it is faithful. For any a G AY, a ± 0 we have by (iv) of Definition 1
IInrI > Ia*I_1Inr(a)(a*)I = IaI'lIaa*I = IaI'lIaI2 = IaI.
Definition 3 The representation nr introduced via Theorem 1 is called (left) regular representation of the graded system A.
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The last result shows that a graded system can be realized as an operator system. We will show in the next publications that in the case of grading over Abelian groups, the functional realization is possible.
4. Examples.
It is easy to understand that if an algebra is graded with respect to a group, then it is possible in a obvious way to match to it a graded system with respect to the same group. This remark allows us to attract a huge array of examples, the flow of which does not run out (see, e.g. the recent work [8]). Among the examples given below, there are also systems of such kind, but not only.
1. Let r be a discrete (unimodular) group acting on a C*-algebra A by star automorphisms, t: r ^ Aut(A). Denote by Ay a copy of a the algebra A, indexed by a y £ r, and by A = UAy the disjunct union of the Banach spaces Ay with only common zero. Introduce the operations of product and involution in the following way.
For x £ Aa,y £ Ap put xy = xTa(y) £ Aap, and x* = Ta-1(x*).
Then the set A with these operations becomes an involutive semigroup, obviously T-equipped.
It is easy to check that conditions of Definition 1 are satisfied and then A is T-graded system with the central algebra A. Obviously, this graded system is associated with the crossed product of the algebra A and the group T with respect to the the dynamical system (A, T, t).
2. Let Ak be the Banach space T nVk for k > 0, and V*kT n for k<0 (we use the notations from [3]. Denote by A the disjoint union of this space with only common zero. Using the known relations between the generators S1, S2, ... one can define the product and involution in A turned this set into the Z-graded system which is associated to the Cuntz algebra 0n.
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3. Let A be a unital C*-algebra, and Ti, T2 be *-endomorphisms of the algebra A such that t1(A)t2(A) = {0}, and let r = F2 be the free group of rank 2 with the generators s, t. Denote by ro the subsemigroup of r generated by the generators.
Let y G r, and let the word Y1y2 ■" Y«, where each yt is a degree of one of the elements s, t, s"1, t"1 represent y after all possible cancellations. We say that the element y is regular if there is no neighbors of the type s'lt or t'ls in his expansion. The set of all regular elements we denote by rr. Obviously, if y G rr then it is the concatenation of the form y = y1y2 where y1, y2-1 e ro. We put Tst as T1°T2. Then, evidently ty is well defined for any y e ro.
Let Ay, y G rr be the linear subspace of A ®r generated by the elements of the form a®y, and Ay be the null-space for y g rr.
Note that each subspace Ay is a Banach space with the norm induced from lla®yll = Hal.
At last denote by A the disjoint union of the spaces AY with only common zero, and verify that it is an involutive semigroup with identity and zero.
The involution for an element a®y G Ay defined as the element of Ay-1 of the form
(a0y)* = a*0y-1.
The product of two elements a®a G Aa and b®p G Ap is defined as an element of Aap of the form
(i) If a, p, ap g rr we put (a®a)(b ®p) = 0.
(ii) If a e ro, and p, ap e r we put
(a0a)(b ®p) = aTa(b)0ap. (iii) If a G rr\ro, then, evidently p-1 G ro, and then we put
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(a®a)(b®p) = Tp-i(a)b®ap.
It is easy to check that all conditions of Definition 1 are satisfied. This example arose in the study of the work of Stacey [5] on the crossed product of C*-algebras by *-endomorphisms.
4. Let A be a unital C*-algebra, t be an isometric *-endomorphism of the algebra A, and t* be an isometric positive left inverse mapping to t, that is for a £ A the following conditions hold:
T*(A) c A, T*(a*a) > 0, T*T(a) = a, llT(a) I = lT*(a) l=lal.
We associate to such a dynamical system a Z-graded system. As the space An, n=1,2, ... we take the subspace {a0n, a £ A} in A 0 Z endowed with the norm Ia0nln =lal.
Now we define the operations of involution and product in (A, Z). For a0n put
(a0n)* = a*0(-n).
For a0n and b0m put the product (a0n)(b0m) as
(i) aTn(b)0(n+m) if n, m > 0
(ii) aTn(1)Tn+m(b)0(n+m) if n > -m > 0
(iii) t-(n+m)(a)T-m(1)b0(n+m) if -m > n > 0;
(iv) T*m(ab)0(n+m) if -n > m > 0;
(v) T*(-n)(ab)0(n+m) if m > -n > 0;
(vi) T(-m)(a)b0(n+m) if m, n < 0.
It is easy to check that all conditions of Definition 1 of the graded system are satisfying. This example is inspired by the work of
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W.Paschke [4] on the crossed product of a C*-algebra by an endomorphism.
Conclusion.
The concept of grading algebraic structures has existed for a long time and plays an important role in various fields of mathematics. The notion of Fell's bundle and the associated notion of graded system arose in operator algebras comparatively recently, but thanks to its potential it promises to become the universal tool in the study of irreversible dynamical systems. The basic idea is to structure the algebra in specified way with the help of a suitably selected group, to study the components of the obtained group graded system independently, and then to return back to the original algebra.
References
1. V. Arzumanian, S. Grigorian, Group-graded systems and algebras, Journal of Mathematical Sciences, 216:1 (2016), 1-7.
2. A. Buss, R. Exel, Fell bundles over inverse semigroups and twisted étale groupoids, Journal of Operator Theory, 67 (2012), 153-205.
3. A. Buss, R. Exel, Twisted actions and regular Fell bundles over inverse semigroups, Proceedings of London Mathematical Society, 103 (2011), 235-270.
4. A. Buss, R. Exel, R. Meyer, Reduced C*-algebras of Fell bundles over inverse semigroups, to appear in Israel J. of Math, arXiv: 1512.05570v2 (2015)
5. A. Buss, R. Exel, R. Meyer, Inverse semigroup actions as groupoid actions, Semigroup forum, 85 (2012), 227-243.
6. J. Cuntz, Simple C*-Algebras Generated by Isometries, Communications in Mathematical Physics, 57 (1977), 173-185.
7. R. Exel, Partial Dynamical Systems, Fell Bundles and Applications, http://mtm.ufsc.br/exel/papers/pdynsysfellbun.pdf, 2014.
8. A.Yu. Kuznetsova, On a class of operator algebras generated by a family of isometries, Journal of Mathematical Sciences, 216:1(2016), 84-93.
9. W. Paschke, The crossed product of a C*-algebra by an endomorphism. Proceeding of the American Mathematical Society, 80:1(1980), 113-118.
10. P. J. Stacey, Crossed product of C*-algebras by *-endomorphisms. Australian. Mathematical Society (Series A) 54(1993), 204-212.
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АЛГЕБРЫ, АССОЦИИРОВАНЫЕ С ГШРАДУИРОВАННЫМИ СИСТЕМАМИ
Арзуманян В., Григорян С.
Понятие градуирования по группе естественно возникает при изучении скрещенных произведений и особенно полезно в контексте необратимых динамических систем. В работе рассматриваются некоторые общие аспекты, касающиеся градуированных систем, а также некоторые примеры таких систем, связанных с известными конструкциями.
Ключевые слова: С*-алгебра, представление, бимодуль, гильбертов модуль, градуированная система.
Дата поступления 18.11.2016.
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