Table Of ContentGr¨obner bases of ∗-algebras and faithful operator
6
0 representations.
0
2
n
S. Popovych, K. Vynogradov
a
J
2 February 1, 2008
2
]
A
Abstract
O
. We give sufficient conditions for existence of a faithful representation of a ∗-
h
t algebra in terms of its Go¨bner basis. In order to do this we propose a construction of
a
m a faithful representation. This construction applies to concrete examples: ∗-doubles,
monomial ∗-algebras, extension of a ∗-algebras allowing Wick ordering and others.
[
Several examples and counterexamples are also presented.
2
v
9
1 1 Introduction
4
7
0 We address a fundamental question in representation theory: When can an algebra
4 (with involution) built on generators and relations be represented by unbounded operators
0
/ on Hilbert space? And when is there a faithful representation? We make a case for the
h
usefulness of Gr¨obner basis techniques. Now, it has been known since the early days of the
t
a
algebraic approach to quantum problems (e.g., Heisenberg’s commutation relations) that
m
there are representation theoretic dichotomies: For example, the relations of the Bosons
:
v
and the Fermions display different features when represented in Hilbert space. This is the
i
X issue of unbounded operators vs bounded operators. Both examples fit the theme of the
r paper, viz., algebras A with involution over the complex field. In the special cases when A
a
is assumed abelian, or if A is a C∗ -algebra, we have the familiar theorems of Gelfand and
Naimark, but for algebras on generators and relations, the literature so far only consists of
isolated classes of examples. In particular, the question of representability by Hilbert space
operatorshas been studied for∗-algebrasallowing Wick ordering, monomial ∗-algebrasand
other (see [10, 5, 4, 3, 6]). Lance and Tapper [10, 5] undertook an attempt to treat C∗-
representability of monomial ∗-algebras with one defining relation. However there isn’t a
general theory yet.
Let us fix some notations. We will denote by E a pre-Hilbert space and by H a Hilbert
space. Let L(E) and L(H) denote the ∗-algebras of linear operators acting on these spaces.
We will study the question whether a ∗-algebra A can be embedded as a ∗-subalgebra in
L(E) or L(H). Note that in the latter case elements of A are represented by bounded
operators. If a ∗-algebra A is embedded in L(E) and every operator a ∈ A is bounded
1
then one can extend each a ∈ A to an operator acting on the completion H of E and thus
obtain an inclusion A ֒→ L(H). In the general case A will be represented by unbounded
operators on H such that the intersection of their domains is dense. Henceforth when
unbounded operators are involved the term representation of an algebra means that the
relations of the algebra are satisfied on the common dense invariant domain. Remind that
∗-algebra A is called C∗-representable if there is its faithful ∗-representation by bounded
operators acting on a Hilbert space or, equivalently, A can be embedded into C∗-algebra.
Thus we will decompose the question of C∗-representability into two parts: the first, of
algebraic nature, is to find a faithful representation of A in a pre-Hilbert space and the
second one, of analytic nature, is to find out whether this representation is in bounded
operators.
The question of C∗-representability of finitely-presented ∗-algebras has been investi-
gated in [6]. The main novelty of the approach used in [6] was the employment of Gr¨obner
basis technique. Here we will elaborate this approach.
In present work we find sufficient conditions for a ∗-algebra to be faithfull represented
in pre-Hilbert spaces. These conditions expressed in terms of its defining relations and can
be effectively verified.
For∗-algebra given by generators and relations if not the only then at least very natural
way to prove that a homomorphism is injective is to show that some linear basis is mapped
intoalinearbasisoftheimage. Toconstruct alinearbasisweusethemachinery ofGr¨obner
bases developed in [2]. A brief account of it is given in the appendix.
The main property of a ∗-algebra which enable us to construct a faithful representation
is the strongly non-expanding condition (see definition 2). But this property is hard to
verify in examples. So we gave several sufficient conditions (see definition 4 and theorem 4)
and apply them to many concrete examples: ∗-doubles, monomial ∗-algebras, extension of
a ∗-algebras with Wick ordering and others.
2 Unsrinkability type restrictions on Gr¨obner basis
of a *-algebra.
We will denoted by F the free associative algebra since the number of generators is not
important for further considerations. In this paper we will deal exclusively with finitely-
generated algebras. Let us denote by F the free associative algebra with generators
∗
x ,x ,...,x ,x∗,x∗,...,x∗ . Algebra F is a ∗-algebra with involution given on generators
1 2 m 1 2 m ∗
by (x )∗ = x∗ for all j = 1,...,m. This is just associative algebra with 2m generators
j j
F with a natural involution. Note that F is a semigroup algebra of a semigroup W
2m ∗
of all words in generators x ,x ,...,x ,x∗,x∗,...,x∗ . A set S ⊆ F of defining relations
1 2 m 1 2 m
of an associative algebra is called Gr¨obner basis if it is closed under compositions (see
Appendix). A Gr¨obner basis of a ∗-algebra A is a Gr¨obner basis of A considered as an
associative algebra. We need to put some extra requirements on a Gr¨obner basis to make
it ”compatible” with involution. The main requirement we impose is a generalization of
2
the notion of unshrinkability of the word. A set S ⊆ F is called symmetric if the ideal I
∗
generated by S in F is a ∗-subalgebra of F . In particular, S is symmetric if S∗ = S. The
∗ ∗
following definition is due to P.Tapper [10]:
Definition 1 The word w ∈ W is called unshrinkable if it can not be presented in the form
w = d∗du or w = ud∗d for some nonempty word d.
Recall that a ∗-algebra A is called proper if for every element x ∈ A condition x∗x = 0
implies x = 0. A ∗-algebra A is called completely proper if M (C) ⊗ A is proper for all
n
integer n. The importance of this notion follows from the fact that any bounded unital
simple ∗-algebra is C∗-representable if and only if it is completely proper [6].
P.Tapper has conjectured that ∗-algebra Chx,x∗|w = 0,w∗ = 0i is C∗-representable if
and only if word w is unshrinkable. The first author in [7] proved that monomial ∗-algebra
A is completely proper if and only if its defining relations are unshrinkable. Moreover,
in this case A can be faithfully represented by operators acting on pre-Hilbert space. A
much wider class of ∗-algebras for which we will prove similar results is defined below. For
the notations u ≺ w, R (w), BW and order on W used below we refer the reader to the
S
appendix.
Definition 2 A symmetric subset S ⊆ F closed under compositions will be called non-
∗
expanding if for every u,v,w ∈ BW such that u 6= v and ww∗ ≺ R (uv∗) the following
S
inequality holds w < sup(u,v). If in addition for every word d ∈ BW the word dd∗ also
belong to BW we will call S strictly non-expanding. A ∗-algebra A is called non-expanding
if it possesses a Gr¨obner basis GB which is non-expanding and A is strictly non-expanding
if some of its Gr¨obner bases is strictly non-expanding.
We will show (see theorem 2) that every strictly non-expanding algebra have a faithful
representation in a pre-Hilbert space.
3 Representation construction.
In this section we will show that strictly non-expanding ∗-algebra possesses a faithful
positive functional and thus is a pre-Hilbert ∗-algebra.
Let G ⊆ W be a subset. An enumeration of G is a map φ : G → N such that u > v
n
implies φ(u) > φ(v) which is bijection on N if |G| = ∞ and on some interval [1,n] ⊆ N if
|G| < ∞.
Definition 3 Let us define the Half-word operator H : F → F by the rule H(uu∗) = u
∗ ∗
for u ∈ W and H(v) = 0 if v is not positive word and extend by linearity to F . Let us fix
∗
a set S ⊆ F closed under compositions, an enumeration φ : BW → N of the corresponding
linear basis and a sequence of positive real numbers ξ = {ak}k∈N ⊂ R. Define a weight
functional Tφ : K → C by putting Tφ(u) = a for any word u ∈ BW, where K denote
ξ ξ φ(u)
the linear span of BW. Let n = |BW| ∈ N∪{∞}.
3
Let V be a vector space over C with basis {e }n . Let us define a form he ,e i on the
k k=1 i j ξ
basis elements and extend it by linearity in the first argument and conjugate-linearity in
the second one by the following rules. If i = j then he ,e i = a . If i 6= j then there are
i i ξ i
unique elements u,v ∈ BW such that i = φ(u), j = φ(v). Put he ,e i = Tφ◦H◦R (uv∗).
i j ξ ξ S
Theorem 1 If S is strictly non-expanding then there exist sequence ξ0 = {ak}k∈N ⊂ N
such that the sesquilinear form h·,·i is positively defined.
ξ0
Proof: That the form h·,·i is sesquilinear is obvious by definition. Let g = he ,e i
ξ ij i j ξ
for i, j ∈ N and G = (g ) denote the Gram matrix of the sesquilinear form. We
ij 1≤i,j≤∞
will use Silvester’s criterion and show by induction on m that a can be chosen such that
m
principal minor ∆ > 0. For m = 1 put a = 1 then ∆ = 1 > 0. Let a ,...,a be
m 1 1 1 m−1
already chosen such that ∆ > 0,...,∆ > 0.
1 m−1
By definition if u ∈ BW, then u∗u is also in BW. Thus by definition 3 we have
he ,e i = a . Take some i ≤ m and j ≤ m with i 6= j and find unique u,v ∈ BW
φ(u) φ(u) ξ φ(u)
such that i = φ(u), j = φ(v). Then uv∗ = α w for unique α ∈ C and w ∈ BW.
k k k k k
Clearly he ,e i is a sum α a whPere the sum is taken over those k for which
φ(u) φ(u) ξ k k φ(hk)
w is a positive word w = h hP∗. By definition we get that h < sup(u,v). Hence g is
k k k k k ij
a polynomial in variables a ,...,a . Now from the decomposition by the m-th row we
1 m−1
obtain ∆ = ∆ a + p (a ,...,a ), where p ∈ C[a ,...,a ] some polynomial.
m m−1 m m 1 m−1 m 1 m−1
Since ∆ > 0 it is clear that a can be chosen such that ∆ > 0. This completes
m−1 m m
inductive proof. △
The space K is isomorphic to V via the map u → e . Thus the inner product h·,·i
φ(u) ξ
on V give rise to an inner product on K which will be denoted by the same symbol. It
is routine to check that hu,vi = α(P(u⋄v∗)) where P : F → F is the projection on the
ξ
linear span of positive words W and α : K → C some linear functional. Let z → L
+ z
denote the right regular representation of A = F/I, L (f) = fz for any z,f ∈ A.
z
Theorem 2 Let S ⊆ F be strictly non-expanding and I the ideal generated by S in F .
∗
Then the right regular representation L of the ∗-algebra A = F /I on a pre-Hilbert space
∗
(K,h·,·i ) is a faithful ∗-representation.
ξ
Proof: The representation statedinthetheorem isassociated bytheGNSconstruction
with the positive functional α(P(·)) on A. Thus it is a ∗-representation. Indeed, as in the
GNS construction N = {a ∈ A|α(P(aa∗)) = 0} is a right ideal in A. We can define inner
product on A/N by usual rule ha+N,b+Ni = α(P(a∗b)). It is easy to verify that right
multiplication define ∗-representation of A on pre-Hilbert A/N. The only difference with
classical GNS construction is that this representation could not be, in general, extended
on completion of A/N.
Letusshowthat∗-representationisafaithful∗-representation. Takeanyf = c w ∈
1≤i≤n i i
A, where c ∈ C,w ∈ BW. Without loss of generality consider w to be the greaPtest word
i i 1
among w . Then L (w∗) contains element w w∗ with coefficient c . Hence L 6= 0. △
j f 1 1 1 1 f
4
Corollary 1 Every strictly non-expanding ∗-algebra has a faithful *-representation by un-
bounded operators.
4 Sufficient conditions. Examples.
We call a subset S ⊆ F reduced if for any s ∈ S and any word w ≺ s no word sˆ′ with s′ ∈ S
is contained in w as a subword. If the set S is closed under compositions then one can
obtain reduced set S′ closed under compositions generating the same ideal by replacing
each s ∈ S with R (s). The following, rather technical, modification of the notion of
S
appropriate ∗-algebra from [6] is the main source of examples.
Definition 4 Asymmetric reduced subsetS ⊆ F is called strictlyappropriate if itis closed
∗
under compositions and for every s ∈ S and every word u ≺ s such that |u| = deg(s) the
following conditions hold:
1. word u is unshrinkable;
2. if u 6= sˆ, sˆ = ab, and u = ac for some words a,b,c with a 6= ∅ then for any s ∈ S
1
such that there is word w ≺ s , w 6= sˆ ,|w| = |sˆ | either word sˆ does not contain u
1 1 1 1
as a subword or sˆ and u do not form a composition in such a way that sˆ = d ad
1 1 1 2
and u = ad d with some nonempty words d ,d ,d .
2 3 1 2 3
A ∗-algebra A is called strictly appropriate ∗-algebra if it possesses a strictly appropriate
Gr¨obner basis.
The following lemma shows that strictly appropriate ∗-algebras provide examples of
non-expanding ∗-algebras. Many concrete examples of strictly appropriate ∗-algebras will
be considered in the final section. In the following lemma for word w ∈ W of even length
w = w w ,|w | = |w | we will denote by H (w) the first half of w, H (w) = w .
1 2 1 2 0 0 1
Theorem 3 Every strictly appropriate set S ⊆ F is non-expanding. If in addition S = S∗
then S is strictly non-expanding.
Proof: Let u,v ∈ BW and u 6= v.
1. If uv∗ ∈ BW then ww∗ ≺ R (uv∗) implies ww∗ = uv∗. By lemma 2 [6] two cases are
S
possible (1) u = vdd∗ and w = vd or (2) v = udd∗ and w = vd where word d is nonempty.
Hence |w| = |d|+|v| = |u|−|d∗| < |u| in the first case and |w| = |d|+|u| = |v|−|d∗| < |v|
in the second one. Thus w < u or w < v.
2. Now let uv∗ 6∈ BW. There are words p,q ∈ BW and element s ∈ S such that
uv∗ = psˆq. Moreover, since u,v ∈ BW none of them can contain sˆ as a subword. Hence
sˆ = ab with nonempty words a and b such that u = pa and v∗ = bq. Write down
k
s = αsˆ+ w +f, where deg(f) < deg(s) and |sˆ| = |w | for all i ∈ {1,...,k}. Assume
i=1 i i
that for soPme integer i word pw q belongs to BW and pw q = ww∗ for some word w. If the
i i
middle of the word pw q comes across w , in other words max(|p|,|q|) < |w| then w = cd,
i i i
5
w = pc, and w∗ = dq with some nonempty words c,d. Hence pc = q∗d∗. If |c| ≤ |d| then
d∗ = gc for some word g and so w = cd = cc∗g∗ which contradicts unshrinkability of w . If
i i
|c| > |d| then pc = q∗d∗ implies c = gd∗ for some word g and we again see that w = gd∗d
i
is shrinkable. Thus max(|p|,|q|) ≥ |w|. If |p| > |w| then |u| = |p| + |a| > |w|, otherwise
|v| = |b|+|q| > |w|.
In the above-mentioned cases we have had |w| < max(|u|,|v|) which is a stronger
statement than that of the lemma. But on the second step of the decomposition process
which we now approaching this regularity breaks down.
3. Let uv∗ = psˆq and s = αsˆ+ w + f as above and ww∗ ≺ R (pw q). It is
i i S i
obvious that w ≤ sup(u,v). We need toPprove that w < sup(u,v). Suppose the contrary:
uu∗ ≺ R (uv∗) and u > v or vv∗ ≺ R (uv∗) and v > u. If |u| =6 |v| then ww∗ ≺ R (pw q)
S S S i
implies |w| < max(|u|,|v|) because d < c for all d ≺ R (c). If |sˆ| > |w | then again
S i
|w| < max(|u|,|v|). Hence we can assume that |u| = |v| and |sˆ| = |w |. Clearly u < v
i
implies uv∗ < vv∗. Hence relation vv∗ ≺ R (pw q) is impossible. Thus we are left with the
S i
only possibility u > v and uu∗ ≺ R (pw q). Since uu∗ < pw q < uv∗ word pw q begins with
S i i i
u. If sˆ= ab such that pa = u,bq = v∗ then w should also begin with a. Therefore sˆ and
i
w begin with the same generator. Since pw q 6∈ BW there is s = αsˆ + β u +g ∈ S
i i 1 1 j j j
with deg(g) < deg(s ) such that pw q = p sˆ q for some words p ,q . IfPwe assume that
1 i 1 1 1 1 1
for some j word uu∗ ≺ R (p u q ) then |sˆ | = |u | and H (p u q ) = u. Word sˆ could
S 1 j 1 1 j 0 1 j 1 1
not be a subword in the first half of the word pw q since H (p u q ) = H (pw q) = u and
i 0 1 j 1 0 i
assuming the contrary we see that sˆ and u are both subwords of u in the same position,
1 j
hence should be equal sˆ = u . Word sˆ could not contain subword w because of condition
1 j 1 i
1 in the definition of strictly appropriateness. Obviously, sˆ could not be a subword in q
1
because q ∈ BW. Thus either w and sˆ intersect or sˆ and w intersect in such a way
j 1 1 j
that sˆ = d ad and w = ad d . But this contradicts the strictly appropriateness of S. So
1 1 2 j 2 3
we have proved that w < sup(u,v). The fact that for any word g ∈ BW word gg∗ lies in
BW follows from lemma 2. [6]. △
The following is a convenient simplification of the preceding theorem:
Corollary 2 If a symmetric subset S ⊆ F is closed under compositions, for every s ∈ S
and every word u ≺ s such that |u| = deg(s) the word u is unshrinkable and words sˆ and
u begin with different generators then S is non-expanding. If in addition S = S∗ then S is
strictly non-expanding.
Example 1 Let L be a finite dimensional real Lie algebra with linear basis {e }n . Then
j j=1
its universal enveloping algebra U(L) (over C) is a ∗-algebra with involution given on
generators as e∗ = −e . We claim that this ∗-algebra is non-expanding. Indeed M = {e e −
j j i j
e e −[e ,e ],i < j} is a set of defining relations for U(L) it is closed under compositions
j i i j
(see example in [2] or use PBW theorem). Then the set S = {e∗ +e ,1 ≤ j ≤ n}∪M is
j j
also closed under compositions (we consider e∗ > e∗ > ... > e∗ > e > ... > e ) since e∗
1 2 1 1 n j
and e e do not intersect for any j, k, l. It is easy to see that S is symmetric. Thus S is
k l
non-expanding by corollary2.
6
Theorem 4 Let S ⊆ F be a symmetric subset of a free countably generated ∗-algebra
∗
F = F(X ∪X∗) such that the following conditions are satisfied:
∗
1. For every s ∈ S and every word w ≺ s with |w| = deg(s) is unshrinkable.
2. For every s , s ∈ S and every word u ≺ s with |u| = deg(s ) the words u and sˆ
1 2 1 1 2
do not form a composition. Then ∗-algebra A = ChX ∪X∗|Si is non-expanding. If
in addition S = S∗ then A is strictly non-expanding.
Proof: It is suffices to show that for any two basis words u, v ∈ BW such that |u| = |v|
and u > v and for any word p condition pp∗ ≺ R (uv∗) implies pp∗ < uu∗.
S
Indeed, if |u| =6 |v|then, clearly, |pp∗| ≤ |uv∗| < max(|uu∗|,|vv∗|). Hence p < sup(u,v).
If |u| = |v| and v > u then uv∗ < vv∗. Thus every word w such that w ≺ R (uv∗) is less
S
than vv∗.
Since pp∗ ≤ uv∗ it remains to prove that uu∗ 6≺ R (uv∗). Assume the contrary. Then
S
thereisasequence ofwords{q }n suchthatq = uv∗, q = uu∗ andforevery 1 ≤ i ≤ n−1
i i=1 1 n
thereis s ∈ S andwords c , d , u ∈ W such thatu ≺ s , u 6= sˆ, |u | = |sˆ|andq = c sˆd ,
i i i i i i i i i i i i i i
q = c u d .
i+1 i i i
Let j be the greatest with the property that sˆ intersects the middle of q . Such an
j j
index j exists because j = 1 satisfies this property and we choose within a finite set.
Clearly j < n since otherwise u would be a subword in uu∗ intersecting its middle and
n−1
thus would be shrinkable. Thus for every i ∈ {j +1,...,n−1} word sˆ does not intersect
i
the middle of the word c u d . But sˆ could not be situated in the first half of this
i−1 i−1 i−1 i
word because otherwise the first half of the word q would be strictly less than u and
i+1
consequently q < uu∗ which is a contradiction. Thus sˆ is a subword in the right half of
n i
the word q unlike that of q = uu∗ is intersected by the unshrinkable word u . Thus
j+1 n j
there is k ∈ {j+1,...,n−1} such that u and sˆ do form a composition. This contradicts
j k
to the conditions of the theorem. △
1. Let S = {w } be a symmetric set consisting of unsrinkable words. Since compo-
j j∈ℜ
sitions of any two words are always zero this set is closed under compositions. The other
conditions in the definition of strictly non-expanding set is obvious. Thus ∗-algebra
C x ,...,x ,x∗,...,x∗|w , j ∈ ℜ
1 n 1 n j
(cid:10) (cid:11)
has a faithful ∗-representation by unbounded operators.
2. Let us consider in more detail the simplest example of monomial ∗-algebras Ax2 =
Chx,x∗|x2 = 0,x∗2 = 0i. It was proved in [10] that ∗-algebra Chx,x∗|xp = 0,x∗p = 0i is
C∗-representable for every integer p. Wewill show that among the representations given by
theorem 2 there is a ∗-representation in bounded operators. We believe that this is true for
any monomial ∗-algebra with unshrinkable relations but even for Chx,x∗|x3 = 0,x∗3 = 0i
we could not make an explicit calculations as we do for Ax2.
It can be easily verified that BW consists of words u = x(x∗x)k, v = x∗(xx∗)k,
k k
a = (xx∗)m, b = (x∗x)m where k ≥ 0,m ≥ 1. Obviously BW consists of only words
m m +
a and b (m ≥ 1). If z ,w ∈ BW then zw∗ ∈ W only in the following cases
m m +
7
1.) z = u ,w = u ; 2.) z = v ,w = v ; 3.) z = a ,w = a ; 4.) z = b ,w = b . Moreover,
k t k t m n m n
u u∗ = a ,v v∗ = b ,a a∗ = a ,b b∗ = b .
k t k+t+1 k t k+t+1 m n n+m m n n+m
Consider the following ordering
u < u < ... < a < a < ... < v < v < ... < b < b < ....
0 1 1 2 0 1 1 2
Denote α(a ) = α , α(b ) = β then the Gram matrix of the inner product defined in
m m m m
theorem 1 is block diagonal
A 0 0 0
0 A′ 0 0
.
0 0 B 0
0 0 0 B′
Where A,A′,B,B′ are Gankel matrices A = (a ) , a = α , A′ = (a′ ) , a′ = α ,
ij ij ij i+j−1 ij ij ij i+j
B = (b ) , b = β , B′ = (b′ ) , b′ = β . Note that Y′ obtained from Y by
ij ij ij i+j−1 ij ij ij i+j
cancelling out the first column (here Y stands for A or B).
Thus the question of positivity of the form h·,·i is reduced to the question of simultane-
ous positivity of two Gankel matrices C and C′ where the second is obtained by cancelling
out the first column. We will show that such matrices A,A′,B,B′ could be chosen to be
positive and such that B = A and that the representation in theorem 2 is in bounded
operators.
Let f : [0,1] → [0,1] be continuous function f(x) > 0 for all x ∈ [0,1]. Let
1
α = tm+1f(t)dt
m Z
0
be the moments of the measure with density f(t). It is well known that then moment
matrix A = (a )n (where a = α ) is positively defined. But then A′ is the moment
ij i,j=1 ij i+j−1
matrix of the measure with density tf(t) and thus is also positively-defined. We can put
B = A.
To prove that the representation is in bounded operatorswe need only to verify that the
multiplication L by generator x is a bounded operator. Obviously, xu = 0 and xa = 0
x k m
for all k ≥ 0 and m ≥ 1. Compute ||xv ||2 = ha ,a i = α , ||v ||2 = α(b ) =
k k+1 k+1 2(k+1) k 2k+1
β = α . Analogously, ||xb ||2 = α and ||b ||2 = α . Thus L is bounded if there
2k+1 2k+1 k 2k+1 k 2k x
is a constant c ≥ 0 such that
α ≤ cα , α ≤ cα
2(k+1) 2k−1 2k+1 2k
for all k ≥ 1. But
1 1
α = t2k+1f(t)dt ≤ t2kf(t)dt = α
2k Z Z 2k−1
0 0
and
1 1
α = t2k+2f(t)dt ≤ t2k+1f(t)dt = α .
2k+1 Z Z 2k
0 0
8
Thus ||Lx|| ≤ 1. This proves that Ax2 is C∗-representable.
3. The ∗-algebra given by the generators and relations:
C a ,k = 1,n|a∗a = Tkla a∗;i 6= j ,
k i j ij l k
(cid:10) Xk6=l (cid:11)
with matrix of coefficient satisfying Tkl = T¯lk is strictly appropriate by corollary 2. since
ij ji
no two elements from defining relations form a composition and the greatest word of any
relationbegins withsome a andallother words beginwith somea∗. Hence it has afaithful
j k
unbounded representation. Note that with additional relations a∗a = 1+ Tkla a∗ we
i i k,l ii l k
obtain so called Wick’s algebras. P
4. If S ⊂ CW(x ,...,x ) is closed under compositions then the ∗-algebra
1 n
A = Chx ,...,x ,x∗,...,x∗| S ∪S∗i
1 n 1 n
(sometimes called ∗-double algebra) is non-expanding (see corollary 3).
If S satisfies additionally the property that the greatest word of any relation begins
with generator different from the beginnings of other longest words of this relation then
A is strictly appropriate since S ∪S∗ is closed under compositions. In particular, if B is
finite dimensional associative algebra then its table of multiplication form a set of relations
S with the greatest words of length 2 and others of length 1. Thus ∗-algebra A which
∗-double of B. In other words, A is a free product B ∗ B , where B ⋍ B ⋍ B and
1 2 1 2
involution is given on the generators b∗ = φ(b) for any b ∈ B and c∗ = φ−1(c) for any
1
c ∈ B with φ : B → B being any isomorphism.
2 1 2
Theorem 5 Let S ⊂ F be a symmetric subset of a free ∗-algebra in generators x ,...,x
1 n
and x∗,...,x∗ closed under compositions such that for any s ∈ S the following properties
1 n
holds:
1. sˆ∈ G = W(x ,...,x ) or sˆ∈ G∗ = W(x∗,...,x∗).
1 n 1 n
2. for any u ≺ s such that |u| = |sˆ| words u and sˆ both lie in the same semigroup G or
G∗.
Then S is non-expanding.
Proof: If some word w ∈ W (w = y ...y where y ∈ X ∪ X∗ are generators)
k1 kt kr
contains subword sˆ with s ∈ S. Then w = psˆq for some words p and q in W. After
substitution w → ps¯q = α pw q we see that all words w such that |w | = |sˆ| lie
i i i i i
in the same semigroup eithPer in G or in G∗. Since decomposition R (w) = β u ,u ∈
S j j j j
BW,u = z ...z ,z ∈ X∪X∗ canbeobtainedbyseveralstepsofsubstitutiPonconsidered
j i1 ik ir
above we see that for any j such that |u | = |w| for all r both generators z and y lie in
j ir kr
the same set X or X∗.
Let u,v ∈ BW, u 6= v. Let word w be such that ww∗ ≺ R (uv∗). If |u| =6 |v| then
S
|w| < max(|u|,|v|) hence w < sup(u,v). So assume that |u| = |v| then ww∗ ≤ uv∗
9
implies that w ≤ u. Let w = u. Write down u = z ...z ,z ∈ X ∪X∗. Without loss of
1 k r
generality assume z ∈ X. Then uu∗ = z ...z z∗...z∗ by the first part of the proof u
k 1 k k 1
ends with generator from X whereas v∗ begins with generator from X∗. Thus uv∗ ∈ BW
and R (uv∗) = uv∗. Clearly, uv∗ = uu∗ implies u = v. Obtained contradiction proves that
S
if ww∗ ≺ R (uv∗) then w < sup(u,v). Since S is appropriate for any d ∈ BW word dd∗
S
lies in BW by lemma 2 [6]. △
It could be shown using Zorn’s lemma that for any algebra A and any its set of gener-
ators S there is a Gr¨obner basis corresponding to S with any given inductive ordering of
the generators. Thus we have the following corollary.
Corollary 3 If B is a finitely generated associative algebra then its ∗-double A = B ∗B
is strictly non-expanding ∗-algebra. Hence A has a faithful ∗-representation in pre-Hilbert
space.
Below we give some known examples of *-doubles which have finite Gr¨obner bases.
5. Consider ∗-algebra:
Q = C q ,...,q ,q∗,...,q∗| q2 = q , q = α,q∗2 = q∗, q∗ = α .
4,α 1 4 1 4 j j j j j j
(cid:10) Xj Xj (cid:11)
It is a *-double of the algebra
B = Chq ,...,q | q2 = q , q = αi
n,α 1 4 j j j
X
j
which has the following Gr¨obner basis:
S = {q q −q ,q q −q ,−q q −2q −2q −2q +α+2αq +2αq +2αq −α2−q q −
1 1 1 2 2 2 3 2 1 2 3 1 2 3 1 2
q q −q q −q q −q q ,q q −q ,−q q q −3α+5α2−2α3+q (6−10α+4α2)+q (6−
1 3 2 1 2 3 3 1 3 3 3 3 1 2 2 3
10α+4α2)+q (8−13α+5α2)+(3−2α)q q +(6−4α)q q +(6−4α)q q +(6−4α)q q +
1 1 2 1 3 2 1 2 3
(3−2α)q q +q q q +q q q +q q q +q q q +q q q )}. More about this algebra can be
3 1 1 2 1 1 2 3 1 3 1 2 1 3 2 3 1
found in [9, 1]. Let us note that when α = 0 ∗-algebra Q = B ∗ B has only zero
4,0 4,0 4,0
representation in bounded operators (see [1]). Thus for this ∗-algebra unboundedness is
essential.
6. That the generators in the previous example are idempotents is not important, we
can consider other powers as well:
T = C q ,q ,q ,q∗,q∗,q∗| q3 = q , q = α,q∗3 = q∗, q∗ = α .
3,α 1 2 3 1 2 3 j j j j j j
(cid:10) Xj Xj (cid:11)
It is a *-double of the algebra Chq ,q ,q | q3 = q , q = αi. We can find its Gr¨obner
1 2 3 j j j j
basis. We have the following set of relations {q3 −Pq ,q3 − q ,q3 − q ,q + q + q − α}.
1 1 2 2 3 3 1 2 3
From these relations we see that this algebra is generated by q and q . That is why we
1 2
can consider the following set of relations: {q3−q ,q3−q ,(α−q −q )3−(α−q −q )}.
1 1 2 2 1 2 1 2
Consider the following order on generators q > q . All relations are already normalized.
2 1
The greatest words in this relations are q3, q3 and q2q . Thus we have no reductions. The
1 2 1 2
10
