On Transitive Systems of Subspaces in a Hilbert Space
Methods of *-representations in Hilbert space are applied to study of systems of n subspaces in a linear space. It is proved that the problem of description of n-transitive subspaces in a finite-dimensional linear space is *-wild for n ≥ 5.
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| Cite this: | On Transitive Systems of Subspaces in a Hilbert Space / Y.P. Moskaleva, Y.S. Samoilenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 12 назв. — англ. |
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| citation_txt | On Transitive Systems of Subspaces in a Hilbert Space / Y.P. Moskaleva, Y.S. Samoilenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 12 назв. — англ. |
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| description | Methods of *-representations in Hilbert space are applied to study of systems of n subspaces in a linear space. It is proved that the problem of description of n-transitive subspaces in a finite-dimensional linear space is *-wild for n ≥ 5.
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 042, 19 pages
On Transitive Systems of Subspaces in a Hilbert Space
Yuliya P. MOSKALEVA † and Yurii S. SAMOǏLENKO ‡
† Taurida National University, 4 Vernads’kyi Str., Simferopol, 95007 Ukraine
E-mail: YulMosk@mail.ru
‡ Institute of Mathematics, National Academy of Sciences of Ukraine,
3 Tereshchenkivs’ka Str., Kyiv-4, 01601 Ukraine
E-mail: yurii sam@imath.kiev.ua
Received February 27, 2006; Published online April 12, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper042/
Submitted by Anatoly Klimyk
Abstract. Methods of ∗-representations in Hilbert space are applied to study of systems
of n subspaces in a linear space. It is proved that the problem of description of n-transitive
subspaces in a finite-dimensional linear space is ∗-wild for n ≥ 5.
Key words: algebras generated by projections; irreducible inequivalent representations; tran-
sitive nonisomorphic systems of subspaces
2000 Mathematics Subject Classification: 47A62; 16G20
1 Introduction
Systems of n subspaces H1,H2, . . . ,Hn of a Hilbert space H, denoted in the sequel by S =
(H;H1,H2, . . . ,Hn), is a mathematical object that traditionally draws an interest both by it-
self [1, 4, 5, 6] and in connection with the discussion on whether there exists a deeper connection
between this object and the famous H. Weyl problem, the Coxeter groups, singularity theory,
and physical applications.
Systems of subspaces that can be regarded as candidates for being the simplest building
blocks for arbitrary systems of subspaces are those that are indecomposable or transitive [4, 5, 6].
A description of transitive and indecomposable systems is carried out up to an isomorphism of
the systems of subspaces. For a description of transitive and indecomposable systems of two
subspaces of a Hilbert space, as well as for transitive and indecomposable triples of a finite
dimensional linear space, see, e.g., [6]. For an infinite dimensional space, not only the problem
of description but even the problem of existence of transitive and indecomposable triples of sub-
spaces is an unsolved problem [2]. For a finite dimensional linear space, transitive quadruples
of subspaces are described in [3], and [4, 5] give indecomposable quadruples. Examples of non-
isomorphic transitive and indecomposable systems of four subspaces in an infinite dimensional
space can be found, e.g., in [6].
In [6] the authors make a conjecture that there is a connection between systems of n subspaces
and representations of ∗-algebras that are generated by the projections, — “There seems to be
interesting relations of systems of n-subspaces with the study of representations of ∗-algebras
generated by idempotents by S. Kruglyak, V. Ostrovskyi, V. Rabanovich, Yu. Samǒılenko and
other. But we do not know the exact implication . . . ”. The present article deals with this
implication.
Let us consider systems of subspaces of the form Sπ = (H;P1H,P2H, . . . , PnH), where the
orthogonal projections P1, P2, . . . , Pn make a ∗-representation π of the ∗-algebra generated by the
mailto:YulMosk@mail.ru
mailto:yurii_sam@imath.kiev.ua
http://www.emis.de/journals/SIGMA/2006/Paper042/
2 Yu.P. Moskaleva and Yu.S. Samǒılenko
projections, and H is the representation space. For the ∗-algebras P4,com = C〈p1, p2, p3, p4 | p2
k =
p∗k = pk,
[ 4∑
k=1
pk, pi
]
= 0, ∀ i = 1, 2, 3, 4〉, it was proved in [11] that irreducible inequivalent
∗-representations π of the ∗-algebra P4,com make a complete list of nonisomorphic transitive
quadruples of subspaces Sπ of a finite dimensional linear space.
In this paper, we make an analysis of complexity of the description problem for transitive
systems of subspaces S = (H;H1,H2, . . . ,Hn) for n ≥ 5. In Section 3, we prove that it is
an extremely difficult problem to describe nonisomorphic transitive quintuples of subspaces
S = (H;P1H,P2H, . . . , P5H) even under the assumption that the sum of the corresponding five
projections equals 2I; in other words, the problem of describing inequivalent ∗-representations
of the ∗-algebras that give rise to nonisomorphic transitive systems, is ∗-wild.
Since the problem of describing the system of n subspaces up to an isomorphism is compli-
cated, it seems natural to describe transitive systems that correspond to ∗-representations of
various algebras generated by projections (Sections 4 and 5).
In Section 4, we consider transitive systems Sπ of n subspaces, where π ∈ Rep Pn,α, Pn,α =
C〈p1, p2, . . . , pn | p1 + p2 + · · · + pn = αe, p2
j = pj , p
∗
j = pj , ∀ j = 1, . . . , n〉, and α takes values
in a fixed set. In Section 5, using nonisomorphic transitive systems Sπ of n subspaces, where
π belongs to Rep Pn,α, we construct nonisomorphic transitive systems Sπ̂ of n + 1 subspaces,
where π̂ is in Rep Pn,abo,τ , Pn,abo,τ = C〈q1, q2, . . . , qn, p | q1 + q2 + · · · + qn = e, qjpqj = τqj ,
q2
j = qj , q∗j = qj ,∀ j = 1, . . . , n, p2 = p, p∗ = p〉.
2 Definitions and main properties
In this section we make necessary definitions and recall known facts; the proofs can be found
in [6, 9]. Let H be a Hilbert space and H1,H2, . . . ,Hn be n subspaces of H. Denote by
S = (H;H1,H2, . . . ,Hn) the system of n subspaces of the space H. Let S = (H;H1,H2, . . . ,Hn)
be a system of n subspaces of a Hilbert space H and S̃ = (H̃; H̃1, H̃2, . . . , H̃n) a system of n
subspaces of a Hilbert space H̃. A linear map R : H → H̃ from the space H to the space H̃
is called a homomorphism of the system S into the system S̃ and denoted by R : S → S̃, if
R(Hi) ⊂ H̃i, i = 1, . . . , n. A homomorphism R : S → S̃ of a system S into a system S̃ is
called an isomorphism, R : S → S̃, if the mapping R : H → H̃ is a bijection and R(Hi) = H̃i,
∀ i = 1, . . . , n. Systems S and S̃ will be called isomorphic, denoted by S ∼= S̃, if there exists an
isomorphism R : S → S̃.
Denote by Hom(S, S̃) the set of homomorphisms of a system S into a system S̃ and by
End(S) := Hom(S, S) the algebra of endomorphisms of S into S, that is,
End(S) = {R ∈ B(H) |R(Hi) ⊂ Hi, i = 1, . . . , n}.
A system S = (H;H1,H2, . . . ,Hn) of n subspaces of a Hilbert space H is called transitive, if
End(S) = CIH .
Denote
Idem(S) = {R ∈ B(H) |R(Hi) ⊂ Hi, i = 1, . . . , n, R2 = R}.
A system S = (H;H1,H2, . . . ,Hn) of n subspaces of a space H is called indecomposable, if
Idem(S) = {0, IH}.
Isomorphic systems are either simultaneously transitive or intransitive, decomposable or inde-
composable. We say that S ∼= S̃ up to permutation of subspaces, if there exists a permutation σ ∈
Sn such that the systems σ(S) and S̃ are isomorphic, where σ(S) = (H;Hσ(1),Hσ(2), . . . ,Hσ(n)),
so that there exists an invertible operator R : H → H̃ such that R(Hσ(i)) = H̃i, ∀ i = 1, . . . , n.
On Transitive Systems of Subspaces in a Hilbert Space 3
Let us now recall the notion of unitary equivalence for systems and collections of orthogonal
projections. Systems S and S̃ are called unitary equivalent, or simply equivalent, if S ∼= S̃ and
it is possible to choose the isomorphism R : S → S̃ to be a unitary operator.
To every system S = (H;H1,H2, . . . ,Hn) of n subspaces of a Hilbert space H, one can nat-
urally associate a system of orthogonal projections P1, P2, . . . , Pn, where Pi is the orthogonal
projection operator onto the space Hi, i = 1, . . . , n. A system of projections P1, P2, . . . , Pn
on a Hilbert space H such that Im Pi = Hi for i = 1, . . . , n is called a system of orthogo-
nal projections associated to the system of subspaces, S = (H;H1,H2, . . . ,Hn). Conversely,
to each system of projections there naturally corresponds a system of subspaces. A system
S = (H;P1H,P2H, . . . , PnH) is called a system corresponding to the system of projections
P1, P2, . . . , Pn.
A system of orthogonal projections P1, P2, . . . , Pn on a Hilbert space H is called unitary
equivalent to a system P̃1, P̃2, . . . , P̃n on a Hilbert space H̃, if there exists a unitary operator
R : H → H̃ such that RPi = P̃iR, i = 1, . . . , n. Systems S and S̃ are unitary equivalent if and
only if the corresponding systems of orthogonal projections are unitary equivalent.
A system of orthogonal projections P1, P2, . . . , Pn on a Hilbert space H is called irreducible
if zero and H are the only invariant subspaces. Unitary equivalent systems of orthogonal pro-
jections are both either reducible or irreducible.
If systems S and S̃ are unitary equivalent, then S ∼= S̃. The converse is not true.
Example 1. Let S = (C2; C(1, 0), C(cos θ, sin θ)), θ ∈ (0, π/2), and S̃ = (C2; C(1, 0), C(0, 1)).
The decomposable system S that corresponds to an irreducible pair of orthogonal projections,
is isomorphic but not unitary equivalent to the decomposable system S̃ that corresponds to a
reducible pair of orthogonal projections.
Finally, let us mention the relationship between the notions of transitivity, indecomposability,
and irreducibility. If a system of subspaces is transitive, then it is indecomposable, but not vice
versa. Indecomposability of a system of subspaces implies irreducibility of the corresponding
system of orthogonal projections, but not conversely.
3 On ∗-wildness of the description problem
for transitive systems of n subspaces for n ≥ 5
3.1 On ∗-wildness of the description problem for transitive systems
that correspond to orthogonal projections
A description of transitive quadruples of subspaces of a finite dimensional linear space is given
in [3]. We will show that such a problem for n subspaces, n ≥ 5, is extremely complicated
(∗-wild).
Consider a system of five subspaces, which corresponds to the five orthogonal projections
P1 =
(
I 0
0 0
)
, P2 =
(
0 0
0 I
)
, P3 =
1
2
(
I I
I I
)
,
P4 =
1
2
(
I U
U∗ I
)
, P5 =
1
2
(
I V
V ∗ I
)
that act on the space H = H ⊕ H, where H is a Hilbert space and U and V are unitary
operators. Denote this system of subspaces by SU,V . So, SU,V = (H;P1H, P2H, P3H, P4H, P5H).
Consider the system SŨ ,Ṽ = (H̃; P̃1H̃, P̃2H̃, P̃3H̃, P̃4H̃, P̃5H̃) that corresponds to the collection
of orthogonal projections P̃1, P̃2, P̃3, P̃4, P̃5 that have the above type and act on the space
H̃ = H̃ ⊕ H̃; here H̃ is a Hilbert space and Ũ , Ṽ is a pair of unitary operators.
4 Yu.P. Moskaleva and Yu.S. Samǒılenko
Theorem 1. The system SU,V is transitive if and only if the unitary operators U , V are irre-
ducible. Also, SU,V
∼= SŨ ,Ṽ if and only if the pair of unitary operators U , V is unitary equivalent
to the pair of unitary operators Ũ , Ṽ .
Proof. Denote Hi = PiH, i = 1, . . . , 5. For H1 and H2, we have
H1 = H ⊕ 0, H2 = 0⊕H.
For H3, H4, and H5, respectively,
H3 = {(x, x) |x ∈ H}, H4 = {(Ux, x) |x ∈ H}, H5 = {(V x, x)|x ∈ H}.
Let us prove an auxiliary identity
{R ∈ B(H, H̃) |R(Hi) ⊂ H̃i, i = 1, . . . , 5}
= {R⊕R ∈ B(H, H̃) |R ∈ B(H, H̃), RU = ŨR,RV = Ṽ R}. (1)
The first three inclusions, R(Hi) ⊂ H̃i, i = 1, 2, 3, imply that any operator R in B(H, H̃) can
be represented as R = R⊕R, where R ∈ B(H, H̃). The fourth inclusion, R(H4) ⊂ H̃4, implies
RU = ŨR, and the fifth one, R(H5) ⊂ H̃5, gives RV = Ṽ R. The converse implications finish
the proof of (1).
It directly follows from (1) that SU,V
∼= SŨ ,Ṽ if and only if the pair of unitary operators U ,
V is similar to the pair of unitary operators Ũ , Ṽ . By [9], a pair of unitary operators U , V is
similar to a pair of unitary operators Ũ , Ṽ if and only if the pair of unitary operators U , V is
unitary equivalent to the pair of unitary operators Ũ , Ṽ .
Now, setting SŨ ,Ṽ = SU,V , rewrite the identity (1) as follows:
End(SU,V ) = {R ∈ B(H) |R(Hi) ⊂ Hi, i = 1, . . . , 5}
= {R⊕R ∈ B(H) |R ∈ B(H), RU = UR,RV = V R}.
The latter identity immediately implies that the system SU,V is transitive if and only if the
unitary operators U , V are irreducible. �
Theorem 1 allows to identify the description problem for nonisomorphic transitive quintuples
that correspond to five orthogonal projections of a special type with that for inequivalent irre-
ducible pairs of unitary operators. The latter problem is ∗-wild in the theory of ∗-representations
of ∗-algebras [8, 9].
3.2 On ∗-wildness of the description problem for transitive systems
corresponding to orthogonal projections with an additional relation
Let P1, P2, P3 be orthogonal projections on a Hilbert space H, and P2, P3 be mutually orthogo-
nal. Introduce a system of five subspaces of the space H corresponding to the collection of
orthogonal projections P1, P⊥
1 ,P2, P3,(P2 + P3)⊥. Denote
SP1,P2⊥P3 = (H; Im P1, Im P⊥
1 , Im P2, Im P3, Im (P2 + P3)⊥).
Theorem 2. Let P1, P2, P3 be orthogonal projections on a Hilbert space H such that P2 and P3
are mutually orthogonal, and P̃1, P̃2, P̃3 be orthogonal projections on a Hilbert space H̃ such
that P̃2 and P̃3 are mutually orthogonal. Then the system SP1,P2⊥P3 is transitive if and only if
the projections P1, P2, P3 are irreducible. Also, SP1,P2⊥P3
∼= SP̃1,P̃2⊥P̃3
if and only if the triple
of the orthogonal projections P1, P2, P3 is unitary equivalent to the triple of the orthogonal
projections P̃1, P̃2, P̃3.
On Transitive Systems of Subspaces in a Hilbert Space 5
Proof. Denote H1 = Im P1, H2 = Im P⊥
1 , H3 = Im P2, H4 = Im P3, H5 = Im (P2 + P3)⊥, and
let H̃1 = Im P̃1, H̃2 = Im P̃⊥
1 , H̃3 = Im P̃2, H̃4 = Im P̃3, H̃5 = Im (P̃2 + P̃3)⊥.
The proof of the theorem directly follows from the identity
{R ∈ B(H, H̃) |R(Hi) ⊂ H̃i, i = 1, . . . , 5} = {R ∈ B(H, H̃) |RPi = P̃iR, i = 1, 2, 3}. �
Theorem 2 identifies the description problem for nonisomorphic transitive quintuples of sub-
spaces corresponding to quintuples of orthogonal projections of a special type, the ones such
that their sum equals 2IH , with that for inequivalent irreducible triples P1, P2, P3 of orthogo-
nal projections satisfying the condition P2⊥P3. The latter problem is ∗-wild in the theory of
∗-representations of ∗-algebras [8, 9].
4 Transitive systems of subspaces corresponding to Rep Pn,com
4.1 On ∗-representations of the ∗-algebra Pn,com
Denote by Σn (n ∈ N) the set α ∈ R+ such that there exists at least one ∗-representation of
the ∗-algebra Pn,α = C〈p1, p2, . . . , pn | p2
k = p∗k = pk,
n∑
k=1
pk = αe〉, i.e., the set of real numbers
α such that there exist n orthogonal projections P1, P2, . . . , Pn on a Hilbert space H satisfying
n∑
k=1
Pk = αIH . It follows from the definition of the algebra Pn,com = C〈p1, p2, . . . , pn | p2
k = p∗k =
pk, [
n∑
k=1
pk, pi] = 0, ∀ i = 1, . . . , n〉 that all irreducible ∗-representations of Pn,com coincide with
the union of irreducible ∗-representations of Pn,α taken over all α ∈ Σn.
A description of the set Σn for all n ∈ N is obtained by S.A. Kruglyak, V.I. Rabanovich, and
Yu.S. Samǒılenko in [7], and is given by
Σ2 = {0, 1, 2}, Σ3 =
{
0, 1, 3
2 , 2, 3
}
,
Σn =
{
Λ0
n,Λ1
n,
[
n−
√
n2−4n
2 , n+
√
n2−4n
2
]
, n− Λ1
n, n− Λ0
n
}
for n ≥ 4,
Λ0
n =
{
0, 1 + 1
n−1 , 1 + 1
(n−2)− 1
n−1
, . . . , 1 + 1
(n−2)− 1
(n−2)− 1
. . .− 1
n−1
, . . .
}
,
Λ1
n =
{
1, 1 + 1
n−2 , 1 + 1
(n−2)− 1
n−2
, . . . , 1 + 1
(n−2)− 1
(n−2)− 1
. . .− 1
n−2
, . . .
}
.
Here, the elements of the sets Λ0
n, Λ1
n, n−Λ1
n, n−Λ0
n, in what follows, will be called points of the
discrete spectrum of the description problem for unitary representations of the algebra Pn,com,
whereas the elements of the line segment
[
n−
√
n2−4n
2 , n+
√
n2−4n
2
]
are called point of the conti-
nuous spectrum. For each point α in the sets Λ0
n, n−Λ0
n there exists, up to unitary equivalence,
a unique irreducible ∗-representation of the ∗-algebra Pn,α and, hence, that of Pn,com. For each
point α in the sets Λ1
n, n − Λ1
n there exist n inequivalent irreducible ∗-representations of the
∗-algebra Pn,α and, hence, those of Pn,com.
An important instrument for describing the set Σn and representations of Pn,com is use of
Coxeter functors, constructed in [7], between the categories of ∗-representations of Pn,α for
different values of the parameters.
Define a functor T : Rep Pn,α → Rep Pn,n−α, see [7]. Let π be a representation of the
algebra Pn,α, and π(pi) = Pi, i = 1, . . . , n, be orthogonal projections on a representation space H.
6 Yu.P. Moskaleva and Yu.S. Samǒılenko
Then the representation π̂ = T(π) in Rep Pn,n−α is defined by the identities π̂(pi) = (I − Pi)
that give orthogonal projections on H. We leave out a description of the action of the functor T
on morphisms of the category Rep Pn,α, since it is not used in the sequel. Let us now define
a functor S : Rep Pn,α → Rep Pn, α
α−1
, see [7]. Again, let π denote a representation in Rep Pn,α,
and by P1, P2, . . . , Pn denote the corresponding orthogonal projections on the representation
space H. Consider the subspaces Hi = Im Pi (i = 1, . . . , n). Let Γi : Hi → H, i = 1, . . . , n, be
the natural isometries. Then
Γ∗i Γi = IHi , ΓiΓ∗i = Pi, i = 1, . . . , n. (2)
Let an operator Γ be defined by the matrix Γ = [Γ1,Γ2, . . . ,Γn] : H = H1⊕H2⊕ · · ·⊕Hn →
H. Then the natural isometry
√
α−1
α ∆∗ that acts from the orthogonal complement Ĥ to the
subspace Im Γ∗ into the space H defines the isometries ∆k = ∆|Im Pk
: Hk → Ĥ, k = 1, . . . , n.
The orthogonal projections Qi = ∆i∆∗
i , i = 1, . . . , n, on the space Ĥ make the corresponding
representation in S(Rep Pn,α), i.e. the representation π̂ = S(π) in Rep Pn, α
α−1
is given by the
identities π̂(pi) = Qi. Write down the relations satisfied by the operators {∆i}n
i ,
∆∗
i ∆i = IHi , ∆i∆∗
i = Qi, i = 1, . . . , n. (3)
We will not describe the action of the functor S on morphisms of the category Rep Pn,α, since
we will not use it in the sequel.
Following [7], introduce a functor Φ+ : Rep Pn,α → Rep Pn,1+ 1
n−1−α
defined by Φ+(π) =
S(T(π)) for α < n− 1. Denote by πk (k = 0, 1, . . . , n) the following representations in Rep Pn,α:
π0(pi) = 0, i = 1, . . . , n, where the space of representation is C; πk(pi) = 0 if i 6= k and
πk(pk) = 1, k = 1, . . . , n, with C as the representation space. For an arbitrary irreducible
representation π of the algebra Pn,α in the case of points of the discrete spectrum, one can
assert that either π or T(π) is unitary equivalent to a representation of the form Φ+s(π̌), where
the representation π̌ is one of the simplest representations πk, k = 0, n, and s is a natural
number.
4.2 Transitive systems of n subspaces corresponding to Rep Pn,α
The systems of subspaces, Sπk
, k = 0, 1, . . . , n, are clearly nonisomorphic transitive systems
of n subspaces of the space C. I.M. Gel’fand and V.A. Ponamarev in [4], by using the functor
technique, construct from the systems Sπk
, k = 0, 1, . . . , n, infinite series of indecomposable
systems, which turn out to be are transitive, of n subspaces. In this section we show that
the Coxeter functors in [7], as the functors in [4], transform nonisomorphic transitive systems
into nonisomorphic transitive systems and, consequently, all systems of the form SΦ+s(π̌) and
S⊥
Φ+s(π̌), where the representation π̌ is one of the simplest representations πk, k = 0, 1, . . . , n, and
s is a natural number, will be nonisomorphic transitive systems. Hence, we have the following
theorem.
Theorem 3. Systems of n subspaces Sπ constructed from irreducible inequivalent representations
π ∈ Rep Pn,α, for α in the discrete spectrum, are nonisomorphic and transitive.
To prove the theorem, by using the Coxeter functors T and S in [7], we construct auxiliary
functors T′ and S′. The action of the functors T′ : Rep Pn,α → Rep Pn,n−α and S′ : Rep Pn,α →
Rep Pn, α
α−1
on the objects of the category is defined to coincide with the actions of T and S,
that is, T(π) = T′(π) and S(π) = S′(π) ∀π ∈ Rep Pn,α. The morphisms of the category of
representations are defined differently. Let π ∈ Rep (Pn,α,H) and π̃ ∈ Rep (Pn,α, H̃). A linear
On Transitive Systems of Subspaces in a Hilbert Space 7
operator C ∈ B(H, H̃) is called a morphism of the category of representations, C ∈ Mor(π, π̃),
if Cπ(pi) = π̃(pi)Cπ(pi), i = 1, . . . , n, that is,
CPi = P̃iCPi, i = 1, . . . , n. (4)
The restrictions C|Hi , i = 1, . . . , n, are denoted by Ci. Let us show that the operators Ci
map Hi into H̃i, that is,
Ci(Hi) ⊂ H̃i, i = 1, . . . , n. (5)
Indeed, for x ∈ Hi, we have Cix = Cx = CPix = P̃iCPix and, consequently, Cix ∈ H̃i.
If x ∈ Hi, then (4) and (5) give
CΓix = Cx = CPix = P̃iCPix = P̃iCix = Cix = Γ̃iCix,
so that
CΓi = Γ̃iCi, i = 1, . . . , n. (6)
The identities (4) are equivalent to the inclusions C(Hi) ⊂ H̃i, i = 1, . . . , n, which imme-
diately gives the following relations:
Ci = Γ̃∗i CΓi, i = 1, . . . , n. (7)
Formula (6) allows to represent C as
C =
1
α
n∑
i=1
Γ̃iCiΓ∗i . (8)
Indeed, 1
α
n∑
i=1
Γ̃iCiΓ∗i = 1
α
n∑
i=1
CΓiΓ∗i = C( 1
α
n∑
i=1
Pi) = C.
Consider an operator Ĉ : Ĥ → ˆ̃H defined by
Ĉ =
α− 1
α
n∑
i=1
∆̃iCi∆∗
i . (9)
Using the following properties of the operators [7] {Γi}n
i=1, {Γ∗i }n
i=1, {∆i}n
i=1, {∆∗
i }n
i=1:
n∑
i=1
Γi∆∗
i = 0, (10)
∆∗
i ∆j = − 1
α− 1
Γ∗i Γj , i 6= j, (11)
let us prove that
∆̃∗
kĈ = Ck∆∗
k, k = 1, . . . , n. (12)
Indeed,
∆̃∗
kĈ = ∆̃∗
k
(
α− 1
α
n∑
i=1
∆̃iCi∆∗
i
)
=
α− 1
α
n∑
i=1
(∆̃∗
k∆̃i)Ci∆∗
i
=
α− 1
α
(∆̃∗
k∆̃k)Ck∆∗
k +
α− 1
α
n∑
i=1
i6=k
(− 1
α− 1
)Γ̃∗k(Γ̃iCi)∆∗
i
8 Yu.P. Moskaleva and Yu.S. Samǒılenko
=
α− 1
α
Ck∆∗
k −
1
α
n∑
i=1
i6=k
Γ̃∗k(Γ̃iCi)∆∗
i =
α− 1
α
Ck∆∗
k −
1
α
Γ̃∗kC
n∑
i=1
i6=k
Γi∆∗
i
=
α− 1
α
Ck∆∗
k −
1
α
Γ̃∗kC
(
n∑
i=1
Γi∆∗
i − Γk∆∗
k
)
=
α− 1
α
Ck∆∗
k +
1
α
(Γ̃∗k)CΓk∆∗
k = Ck∆∗
k.
Now, let us show that
Ck = ∆̃∗
kĈ∆k, k = 1, . . . , n. (13)
Using (2), (3), (7), (8), (9), and (11) we get
∆̃∗
kĈ∆k = ∆̃∗
k
(
α− 1
α
n∑
i=1
∆̃iCi∆∗
i
)
∆k =
α− 1
α
n∑
i=1
∆̃∗
k∆̃iCi∆∗
i ∆k
=
α− 1
α
∆̃∗
k∆̃kCk∆∗
k∆k +
α− 1
α
n∑
i=1
i6=k
∆̃∗
k∆̃iCi∆∗
i ∆k
=
α− 1
α
Ck +
1
α(α− 1)
n∑
i=1
i6=k
Γ̃∗kΓ̃iCiΓ∗i Γk
=
α− 1
α
Ck +
1
α− 1
Γ̃∗k
1
α
n∑
i=1
i6=k
Γ̃iCiΓ∗i
Γk
=
α− 1
α
Ck +
1
α− 1
Γ̃∗kCΓk −
1
α(α− 1)
Γ̃∗kΓ̃kCkΓ∗kΓk = Ck.
Now, it follows from (12) and (13) that Q̃kĈ = ∆̃k∆̃∗
kĈ = ∆̃kCk∆∗
k = Q̃kĈQk, that is,
Q̃kĈ = Q̃kĈQk, k = 1, . . . , n. Whence,
Ĉ∗Q̃k = QkĈ
∗Q̃k, k = 1, . . . , n. (14)
The latter identities mean that Ĉ∗ ∈ Mor(S′(π̃), S′(π)). The action of the auxiliary functors T′
and S′ on morphisms of the category Rep Pn,α are defined by T′(C) = C∗ and S′(C) = Ĉ∗ for
any C ∈ Mor(π, π̃). This completes the construction of the auxiliary functors.
Lemma 1. The functors T′ and S′ are category equivalences.
Proof. It is easy to check by using the definition that the functor T′ is univalent and complete.
T2 = Id and T′(π) = T(π) for any π ∈ Rep Pn,α. Consequently, the functor T′ is an equivalence
between the categories Rep Pn,α and Rep Pn,n−α.
Now, let us prove the lemma for the functor S′. Let us show that the functor S′ is univalent.
Let C,D ∈ Mor(π, π̃) and C 6= D, and show that S′(C) 6= S′(D). Indeed, if S′(C) = S′(D), then
Ĉ∗ = D̂∗ and Ĉ = D̂. By (13), we have
Ci = ∆̃∗
i Ĉ∆i = ∆̃∗
i D̂∆i = Di, i = 1, . . . , n.
Using the decomposition (8) we get
C =
1
α
n∑
i=1
Γ̃iCiΓ∗i , D =
1
α
n∑
i=1
Γ̃iDiΓ∗i .
Then C = D and, hence, the functor S′ is univalent.
On Transitive Systems of Subspaces in a Hilbert Space 9
Let us now show that S′ is complete. Let R ∈ Mor(S′(π̃), S′(π)). To prove the completeness,
construct a linear operator from the set Mor(π, π̃) such that the functor takes on this morphism
the value R. Since R ∈ Mor(S′(π̃), S′(π)), the operator R : ˆ̃H → Ĥ satisfies
RQ̃k = QkRQ̃k, k = 1, . . . , n.
Consider an operator r̂ in B(Ĥ, ˆ̃H) such that r̂∗ = R. Then the former identities can be
written as
r̂∗Q̃k = Qkr̂
∗Q̃k, k = 1, . . . , n,
and, consequently,
Q̃kr̂ = Q̃kr̂Qk, k = 1, . . . , n. (15)
Denote by rk the operators rk = ∆̃∗
kr̂∆k : Hk → H̃k, k = 1, . . . , n, and show that r̂ can be
represented as
r̂ =
α− 1
α
n∑
k=1
∆̃krk∆∗
k. (16)
Indeed,
α− 1
α
n∑
k=1
∆̃krk∆∗
k =
α− 1
α
n∑
k=1
∆̃k∆̃∗
kr̂∆k∆∗
k =
α− 1
α
n∑
k=1
Q̃kr̂Qk =
(
α− 1
α
n∑
k=1
Q̃k
)
r̂ = r̂.
It follows from the definition of rk and identities (3), (15) that
rk∆∗
k = (∆̃∗
kr̂∆k)∆∗
k = ∆̃∗
kr̂(∆k∆∗
k) = ∆̃∗
kr̂Qk = IH̃k
∆̃∗
kr̂Qk = (∆̃∗
k∆̃k)∆̃∗
kr̂Qk
= ∆̃∗
k(∆̃k∆̃∗
k)r̂Qk = ∆̃∗
kQ̃kr̂Qk = ∆̃∗
kQ̃kr̂ = ∆̃∗
k(∆̃k∆̃∗
k)r̂ = (∆̃∗
k∆̃k)∆̃∗
kr̂ = ∆̃∗
kr̂.
Hence, we have
rk∆∗
k = ∆̃∗
kr̂, k = 1, . . . , n. (17)
Consider the operator
r =
1
α
n∑
i=1
Γ̃iriΓ∗i . (18)
Using (2), (10), (11), (17) we get
rΓk = Γ̃krk, k = 1, . . . , n, (19)
rk = Γ̃∗krΓk, k = 1, . . . , n. (20)
Indeed,
rΓk =
1
α
n∑
i=1
Γ̃iriΓ∗i Γk =
1
α
Γ̃krk +
1
α
n∑
i=1
i6=j
Γ̃iri(Γ∗i Γk) =
1
α
Γ̃krk −
α− 1
α
n∑
i=1
i6=j
Γ̃i(ri∆∗
i )∆k
=
1
α
Γ̃krk −
α− 1
α
n∑
i=1
i6=j
Γ̃i(∆̃∗
i r̂)∆k =
1
α
Γ̃krk +
α− 1
α
Γ̃k∆∗
kr̂∆k = Γ̃krk
10 Yu.P. Moskaleva and Yu.S. Samǒılenko
and
Γ̃∗krΓk =
1
α
Γ̃∗k
(
n∑
i=1
Γ̃iriΓ∗i
)
Γk =
1
α
rk +
1
α
n∑
i=1
i6=j
Γ̃∗kΓ̃iriΓ∗i Γk
=
1
α
rk +
(α− 1)2
α
n∑
i=1
i6=j
∆̃∗
k∆̃iri∆∗
i ∆k =
1
α
rk + (α− 1)∆̃∗
kr̂∆k −
(α− 1)2
α
rk = rk.
It follows from (19) and (20) that rPk = rΓkΓ∗k = Γ̃krkΓ∗k = Γ̃kΓ̃∗krkΓkΓ∗k = P̃krPk, which means
that r ∈ Mor(π, π̃).
It is easy to check that S′(r) = R and, consequently, the functor S′ is complete. So the
univalence and completeness properties of the functor S′ are checked, S2 = Id and S′(π) = S(π)
for any π ∈ Rep Pn,α. Consequently, the functor S′ is an equivalence between the categories
Rep Pn,α and Rep Pn, α
α−1
. �
Lemma 2. If a system Sπ, π ∈ Rep Pn,com, of subspaces is transitive, then the system SΦ+(π)
of subspaces is transitive. Here, Sπ
∼= Sπ̃ if and only if SΦ+(π)
∼= SΦ+(π̃).
Proof. For the functors T and S, we have T(π) = T′(π) and S(π) = S′(π) for any π ∈ Rep Pn,α.
Consequently, ST(π) = ST′(π) and SS(π) = SS′(π). By Lemma 1, T′ is an equivalence of the
categories that shows that if a system Sπ, π ∈ Rep Pn,com, of subspaces is transitive, then the
system ST(π) of subspaces is transitive. We also have that Sπ
∼= Sπ̃ if and only if ST(π)
∼= ST(π̃).
Let us now consider the systems SS(π), π ∈ Rep Pn,α of subspaces. Let π, π̃ ∈ Rep Pn,α.
Consider the systems of subspaces Sπ = (H;H1,H2, . . . ,Hn) and Sπ̃ = (H̃; H̃1, H̃2, . . . , H̃n),
that, respectively, correspond to the representations π and π̃. Let the systems of subspaces be
isomorphic, that is, Sπ
∼= Sπ̃. By the definition of isomorphic systems, there exists a linear
operator T ∈ B(H, H̃) such that T−1 ∈ B(H̃,H) and T (Hi) = H̃i, i = 1, . . . , n. It follows
from T (Hi) = H̃i, i = 1, . . . , n, that T (Hi) ⊂ H̃i, i = 1, . . . , n, and, consequently, we get
the relations TPi = P̃iTPi, i = 1, . . . , n. The latter relations mean that T ∈ Mor(π, π̃) if
T̂ ∗ ∈ Mor(S′(π̃), S′(π)), and
T̂ ∗(Im Q̃i) ⊂ (Im Qi), i = 1, . . . , n. (21)
Again, using T (Hi) = H̃i, i = 1, . . . , n, we get T (Hi) ⊃ H̃i, i = 1, . . . , n, so that T−1(H̃i) ⊂
Hi, i = 1, . . . , n, and, respectively, T−1P̃i = PiT
−1P̃i, i = 1, . . . , n. This means that T−1 ∈
Mor(π̃, π), hence, T̂−1
∗
∈ Mor(S′(π), S′(π̃)), and using T̂−1
∗
= (T̂−1)∗ = (T̂ ∗)−1 we get
Im Q̃i ⊃ (T̂ ∗)−1(Im Qi), i = 1, . . . , n,
so that
T̂ ∗(Im Q̃i) ⊃ Im Qi, i = 1, . . . , n. (22)
It follows from (21) and (22) that
T̂ ∗(Im Q̃i) = Im Qi, i = 1, . . . , n,
i.e., it is an isomorphism of the systems corresponding to the representations S′(π) and S′(π̃)
and, since the functors S′ and S coincide on the objects of the categories, it is an isomorphism
of the systems corresponding to the representations S(π) and S(π̃).
Since S′ is complete, using similar reasonings it is easy to show that the functor S′ and, hence,
S takes the representations corresponding to nonisomorphic systems to representations that also
correspond to nonisomorphic systems.
On Transitive Systems of Subspaces in a Hilbert Space 11
Let again π be a representation of the algebra Pn,α, and π(pi) = Pi, i = 1, . . . , n, be orthogonal
projections on a representation space H. Assume that the system of projections P1, P2, . . . , Pn
gives rise to a transitive system of subspaces Sπ = (H;H1,H2, . . . ,Hn), where Hi = PiH,
i = 1, . . . , n, that is,
End(Sπ) = {r ∈ B(H) | r(Hi) ⊂ Hi, i = 1, . . . , n} = Mor(π, π) = CI.
Consider S′(π) = π̂, where π̂(qi) = Qi, i = 1, . . . , n, and the corresponding system of sub-
spaces Sπ̂. Let now R ∈ End(Sπ̂). Since End(Sπ̂) = Mor(S′(π), S′(π)) and the functor S′
is complete, we see that S′(r) = R, where r ∈ Mor(π, π) is constructed from the operator
R∗ = α−1
α
∑n
k=1 ∆̃krk∆∗
k, ri = ∆̃∗
i R
∗∆i : Hi → H̃i, i = 1, . . . , n, as follows:
r =
1
α
n∑
i=1
ΓiriΓ∗i . (23)
By using R ∈ Mor(S′(π), S′(π)), we obtain, similarly to (20), that
ri = Γ∗i rΓi, i = 1, . . . , n. (24)
Since the system Sπ is transitive, the operator r is a scalar, that is, r = λIH . Using that
Γ∗i Γi = IHi , i = 1, . . . , n, and (24) we get
ri = λIHi , i = 1, . . . , n.
Then R∗ is a scalar operator and, consequently, R is also a scalar operator that means that the
system SS′(π) is transitive and such is SS(π). �
The statement of Theorem 3 follows directly from Lemma 2.
5 Transitive systems of subspaces corresponding to Rep Pn,abo,τ
5.1 Equivalence of the categories Rep Pn,α and Rep Pn,abo,τ
Let us examine the equivalence F, constructed in [10], between the categories of ∗-representa-
tions Pn,α and Pn,abo, 1
α
, α 6= 0. Theorem 3 allows to consider nonisomorphic transitive systems
of n subspaces of the form Sπ, constructed from representations of the algebras Pn,α for α lying
in the discrete spectrum. The equivalence F, in its turn, allows to construct nonisomorphic
transitive systems SF(π) of n + 1 subspaces starting with nonisomorphic transitive systems Sπ,
π ∈ Pn,α, of n subspaces.
Let us describe the equivalence F. Let π be a representation of the algebra Pn,α, and π(pi) =
Pi, i = 1, . . . , n, be orthogonal projections on a representation space H. As it was done in
Section 4, let us introduce the spaces Hi = Im Pi and the natural isometries Γi : Hi → H. Let
H = H1 ⊕ H2 ⊕ · · · ⊕ Hn. Define a linear operator Γ : H1 ⊕ H2 ⊕ · · · ⊕ Hn → H in terms of
the matrix Γ = (Γ1 Γ2 . . .Γn) of the dimension n× 1. Let Qi denote n orthogonal projections,
Qi = diag(0, . . . , 0, IHi , 0, . . . , 0), i = 1, . . . , n, and P : H → H an orthogonal projection defined
by P = 1
αΓ∗Γ with the block matrix P = 1
α ||Γ
∗
i Γj ||ni,j=1 on the space H1 ⊕H2 ⊕ · · · ⊕Hn.
Let a functor F : Rep Pn,α → Rep Pn,abo, 1
α
, α 6= 0, be defined on objects of the category
of representations as follows: F(π) = π̂, where π̂(qi) = Qi, i = 1, . . . , n, and π̂(p) = P . The
identities
n∑
i=1
Qi = I and QiPQi = 1
αQi, i = 1, . . . , n, are easily checked. We do not describe the
action of the functor F on morphisms of the category Rep Pn,α, since we will not use it.
12 Yu.P. Moskaleva and Yu.S. Samǒılenko
Theorem 4. Systems of n+1 subspaces, SF(π), constructed from irreducible inequivalent repre-
sentations π ∈ Rep Pn,α, where α is in the discrete spectrum, are nonisomorphic and transitive.
To prove the theorem, construct an auxiliary functor F′ : Rep Pn,α → Rep Pn,abo, 1
α
, α 6= 0,
the action of which on objects coincides with the action of F, that is, F′(π) = F(π) for all π ∈
Rep Pn,α. Morphisms are defined as in Section 4. Let π ∈ Rep (Pn,α,H) and π̃ ∈ Rep (Pn,α, H̃).
A linear operator C ∈ B(H, H̃) will be called a morphism of the category of representations,
written C ∈ Mor(π, π̃), if Cπ(pi) = π̃(pi)Cπ(pi), that is,
CPi = P̃iCPi, i = 1, . . . , n. (25)
As it was for the functors in Section 4, denote the restrictions C|Hi , i = 1, . . . , n, by Ci.
Then, as in Section 4, the operators Ci map Hi into H̃i, that is,
Ci(Hi) ⊂ H̃i, i = 1, . . . , n. (26)
It follows from (25) and (26) that
CΓi = Γ̃iCi, i = 1, . . . , n. (27)
The identities (25) are equivalent to the inclusions C(Hi) ⊂ H̃i, i = 1, . . . , n, whence it follows
that
Ci = Γ̃∗i CΓi, i = 1, . . . , n. (28)
Similarly to Section 4, identities (27) allow to represent C as
C =
1
α
n∑
i=1
Γ̃iCiΓ∗i . (29)
The above presents all the similarities with the calculations performed in Section 4; the
operator Ĉ is now defined differently. For the operator Ĉ = diag(C1, C2, . . . , Cn) : H → H̃, it
is easy to check that ĈQi = Q̃iĈ, i = 1, . . . , n. Then QiĈ
∗ = Ĉ∗Q̃i, i = 1, . . . , n. The latter
allows to conclude that Ĉ∗(Im Q̃i) ⊂ Im Qi and, consequently,
Ĉ∗Q̃i = QiĈ
∗Q̃i, i = 1, . . . , n. (30)
Denote by (P̃ ĈP )ij the elements of the block matrix of the operator P̃ ĈP : H1⊕H2⊕ · · · ⊕
Hn → H̃1 ⊕ H̃2 ⊕ · · · ⊕ H̃n. Then (P̃ ĈP )ij = 1
α2
n∑
k=1
Γ̃∗i Γ̃kCkΓ∗kΓj = 1
α2 Γ̃∗i (
n∑
k=1
Γ̃kCkΓ∗k)Γj =
1
α Γ̃∗i CΓj = 1
α Γ̃∗i Γ̃jCj = (P̃ Ĉ)ij , that is, P̃ ĈP = P̃ Ĉ and, consequently,
Ĉ∗P̃ = PĈ∗P̃ . (31)
Identities (30) and (31) mean that Ĉ∗ ∈ Mor(F′(π̃),F′(π)). Define F′(C) = Ĉ∗, and this
finishes the construction of the functor F′.
Lemma 3. The functor F′ is an equivalence between the categories.
Proof. Let us show that the functor is univalent. Let C,D ∈ Mor(π, π̃) and C 6= D, and show
that F′(C) 6= F′(D). Indeed, if F′(C) = F′(D), i.e., Ĉ∗ = D̂∗, then Ci = Di, ∀ i = 1, . . . , n. Let
us use (29),
C =
1
α
n∑
i=1
Γ̃iCiΓ∗i , D =
1
α
n∑
i=1
Γ̃iDiΓ∗i .
On Transitive Systems of Subspaces in a Hilbert Space 13
It follows from Ci = Di, i = 1, . . . , n, and the form of the representation operators C and D
that C = D and, hence, the functor F′ is univalent.
Let us show that F′ is complete. Let R ∈ Mor(F′(π̃),F′(π)) and construct a linear operator
in the set Mor(π, π̃) such that the value of this functor on the morphism is R. It follows from
R ∈ Mor(F′(π̃),F′(π)) that the operator R : H̃ → H satisfies
QiRQ̃i = RQ̃i, i = 1, . . . , n, PRP̃ = RP̃ .
Denote by r̂ an operator in B(H, H̃) such that r̂∗ = R. Then the latter identities can be
rewritten as follows:
Qir̂
∗Q̃i = r̂∗Q̃i, i = 1, . . . , n, P r̂∗P̃ = r̂∗P̃ ,
and, consequently,
Q̃ir̂Qi = Q̃ir̂, i = 1, . . . , n (32)
and
P̃ r̂P = P̃ r̂. (33)
Let now rij be elements of the block matrix of the operator r̂ from H1 ⊕H2 ⊕ · · · ⊕Hn into
H̃1⊕H̃2⊕· · ·⊕H̃n. Identities (32) imply that if i 6= j, then rij = 0. Denote ri = rii, i = 1, . . . , n.
Then ri : Hi → H̃i, i = 1, . . . , n, and r̂ = diag(r1, r2, . . . , rn). Consider r : H → H̃ defined by
r =
1
α
Γ̃r̂Γ∗. (34)
Identity (33) and definition (34) imply that 1
α Γ̃∗rΓ = P̃ r̂P = P̃ r̂, then comparing the elements
on the main diagonal of the corresponding block matrices gives
ri = Γ̃∗i rΓi, i = 1, . . . , n. (35)
Using the relation 1
α Γ̃Γ̃∗ = IH̃ we get rΓ = IH̃rΓ = ( 1
α Γ̃Γ̃∗)rΓ = Γ̃( 1
α Γ̃∗rΓ) = Γ̃(P̃ r̂P ) =
Γ̃(P̃ r̂) = Γ̃( 1
α Γ̃∗Γ̃)r̂ = ( 1
α Γ̃Γ̃∗)Γ̃r̂ = IH̃ Γ̃r̂ = Γ̃r̂. Rewrite the identity rΓ = Γ̃r̂ in the matrix
form,
(rΓ1 rΓ2 . . . rΓn) = (Γ̃1r1 Γ̃2r2 . . . Γ̃nrn)
that gives
rΓi = Γ̃iri, i = 1, . . . , n. (36)
Using identities (35) and (36) we get
rPi = P̃irPi, i = 1, . . . , n. (37)
Indeed, rPi = rΓiΓ∗i = Γ̃iriΓ∗i = Γ̃iΓ̃∗i rΓiΓ∗i = P̃irPi. Identities (37) mean that r ∈ Mor(π, π̃).
Let us check that F′(r) = r̂∗ = R. Denote by C the constructed morphism r and find F′(C).
Since F′(C) = Ĉ∗, where Ĉ = diag(C1, C2, . . . , Cn) : H → H̃, let us find Ci = C|Hi = r|Hi ,
i = 1, . . . , n. Since C ∈ Mor(π, π̃), it follows from (28) and (35) that Ci = Γ̃∗i CΓi = Γ̃∗i rΓi = ri.
Then Ĉ = r̂, Ĉ∗ = r̂∗ and F′(r) = F′(C) = Ĉ∗ = r̂∗ = R. This proves that the functor F′ is
complete.
Since F′(π) = F(π) for any π ∈ Rep Pn,α and F2 = Id , we see that F′ is an equivalence of
the categories. �
14 Yu.P. Moskaleva and Yu.S. Samǒılenko
Lemma 4. If a system Sπ, π ∈ Rep Pn,com, of n subspaces is transitive, then the system SF(π)
of n + 1 subspaces is transitive. Also, Sπ
∼= Sπ̃ if and only if SF(π)
∼= SF(π̃).
Proof. Since the functors F and F′ coincide on the objects of the categories, the representations
constructed using the functors and the corresponding systems of subspaces will coincide, SF(π) =
SF′(π) for ∀π ∈ Rep Pn,α. Let π, π̃ ∈ Rep Pn,α, α 6= 0, and the systems of subspaces Sπ =
(H;H1,H2, . . . ,Hn) and Sπ̃ = (H̃; H̃1, H̃2, . . . , H̃n), which correspond to the representations π
and π̃, be isomorphic, that is, Sπ
∼= Sπ̃. By the definition of isomorphic systems, there exists a
linear operator T ∈ B(H, H̃) such that T−1 ∈ B(H̃, H) and T (Hi) = H̃i, i = 1, . . . , n. It follows
from T (Hi) = H̃i, i = 1, . . . , n, that T (Hi) ⊂ H̃i, i = 1, . . . , n, and, consequently, TPi = P̃iTPi,
i = 1, . . . , n. The latter identities mean that T ∈ Mor(π, π̃). Then T̂ ∗ ∈ Mor(F′(π̃),F′(π)) and
T̂ ∗(Im Q̃i) ⊂ (Im Qi) (i = 1, . . . , n) and T̂ ∗(Im P̃ ) ⊂ (Im P ). (38)
Again, considering the identities T (Hi) = H̃i, i = 1, . . . , n, we conclude that T (Hi) ⊃ H̃i, i =
1, . . . , n, that is, T−1(H̃i) ⊂ Hi, i = 1, . . . , n, and, respectively, T−1P̃i = PiT
−1P̃i, i = 1, . . . , n.
These identities imply that T−1 ∈ Mor(π̃, π). Then T̂−1
∗
∈ Mor(F′(π),F′(π̃)), whence using
T̂−1
∗
= (T̂−1)∗ = (T̂ ∗)−1 we have
Im Q̃i ⊃ (T̂ ∗)−1(Im Qi), i = 1, . . . , n and Im P̃ ⊃ (T̂ ∗)−1(Im P ),
and, consequently,
T̂ ∗(Im Q̃i) ⊃ Im Qi, i = 1, . . . , n and T̂ ∗(Im P̃ ) ⊃ Im P. (39)
It follows from (38) and (39) that
T̂ ∗(Im Q̃i) = Im Qi, i = 1, . . . , n and T̂ ∗(Im P̃ ) = Im P
that shows that it is an isomorphism of the systems that correspond to the representations F′(π)
and F′(π̃) and, since the functors F′ and F coincide on the objects of the category, it is an
isomorphism of the systems corresponding to the representations F(π) and F(π̃).
Since the functor F′ is complete, similar reasonings show that the representations that cor-
respond to nonisomorphic systems are mapped by the functor F′, and hence the functor F, into
representations that give rise to nonisomorphic systems.
Let us now prove the first part of the proposition. Let π be a representation of the algebra Pn,α
and π(pi) = Pi, i = 1, . . . , n, be orthogonal projections on a representation space H. And let
the system of orthogonal projections P1, P2, . . . , Pn induce a transitive system of subspaces
Sπ = (H;H1,H2, . . . ,Hn), where Hi = PiH, i = 1, . . . , n, that is,
End(Sπ) = {r ∈ B(H) | r(Hi) ⊂ Hi, i = 1, . . . , n} = Mor(π, π) = CI.
Consider F′(π) = π̂, where π̂(qi) = Qi, i = 1, . . . , n, and π̂(p) = P , and the corresponding
system Sπ̂ of subspaces. Let now R ∈ End(Sπ̂). Using End(Sπ̂) = Mor(F′(π),F′(π)) and since
the functor F′ is complete, we see that F′(r) = R, where r ∈ Mor(π, π) is constructed from the
diagonal operator R∗ = diag(r1, r2, . . . , rn) on the space H1 ⊕H2 ⊕ · · · ⊕Hn as follows:
r =
1
α
ΓR∗Γ∗. (40)
Using the inclusion R ∈ Mor(F′(π),F′(π)), which is similar to identity (35), we get
ri = Γ∗i rΓi, i = 1, . . . , n. (41)
On Transitive Systems of Subspaces in a Hilbert Space 15
Since the system Sπ is transitive, the operator r is scalar, that is, r = λIH . Using Γ∗i Γi = IHi ,
i = 1, . . . , n, and identities (41) we get
ri = λIHi , i = 1, . . . , n.
Then R∗ is a scalar operator and, consequently, R is also a scalar operator that means that the
system SF′(π) is transitive and such is SF(π). �
The claim of Theorem 4 follows from Theorem 3 and Lemma 4.
5.2 Transitive quintuples of subspaces
By Theorem 4, the functor F maps known nonisomorphic transitive quadruples of subspaces of
the form Sπ, where π ∈ Rep P4,com, into nonisomorphic transitive quintuples SF(π) [11, 12]. In
this section, we give inequivalent irreducible ∗-representations of the ∗-algebras P4,abo,τ , τ ∈ Σ̃4,
where Σ̃4 that is the set of τ ∈ R+ such that there exists at least one ∗-representation of
the ∗-algebra P4,abo,τ , is related to Σ4, the set α ∈ R+ such that there exists at least one
∗-representation of the ∗-algebra P4,α, via the following relation [10]:
Σ̃4 = {0} ∪
{
1
α
|α 6= 0, α ∈ Σ4
}
.
Here, by [7], Σ4 =
{
0, 1, 2 − 2
2k+1 (k = 1, 2, . . .), 2 − 1
n (n = 2, 3, . . .), 2, 2 + 1
n (n = 2, 3, . . .), 2 +
2
2k+1 (k = 1, 2, . . .), 3, 4
}
. For these representations, the corresponding systems of subspaces are
nonisomorphic and transitive.
Let er×s
i,j denote an (r × s)-matrix that has 1 at the intersection of the ith row and the jth
column, with other elements being zero.
1) The ∗-algebra P4,abo,0 has 4 irreducible inequivalent one-dimensional representations, Q1 =
· · · = Qk−1 = Qk+1 = · · · = Q4 = P = 0, Qk = 1.
2) The ∗-algebra P4,abo,1 has 4 irreducible inequivalent one-dimensional representations, Q1 =
· · · = Qk−1 = Qk+1 = · · · = Q4 = 0, P = Qk = 1.
3) The ∗-algebra P4,abo, 1
3
has 4 irreducible inequivalent three-dimensional representations
that are unitary equivalent, up to a permutation, to
Q1 = 1⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ 1, P =
1
3
3∑
i,j=1
e3×3
i,j ,
Q2 = 0⊕ 1⊕ 0, Q4 = 0⊕ 0⊕ 0, H = C⊕ C⊕ C.
4) The ∗-algebra P4,abo, 1
4
has a unique irreducible four dimensional representation,
Q1 = 1⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ 1⊕ 0, P =
1
4
4∑
i,j=1
e4×4
i,j ,
Q2 = 0⊕ 1⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ 1, H = C⊕ C⊕ C⊕ C.
5) The ∗-algebra P4,abo, 1
2
has 6 irreducible two-dimensional representations that are unitary
equivalent, up to a permutation, to
Q1 = 1⊕ 0, Q2 = 0⊕ 1, Q3 = 0⊕ 0, Q4 = 0⊕ 0, P =
1
2
(
1 1
1 1
)
,
16 Yu.P. Moskaleva and Yu.S. Samǒılenko
where the representation space is H = C ⊕ C, and the following inequivalent four dimensional
representations that depend on the points of the set Ω = {(a, b, c) ∈ R|a2 + b2 + c2 = 1, a > 0,
b > 0, c ∈ (−1, 1); or a = 0, b2 + c2 = 1, b > 0, c > 0; or b = 0, a2 + c2 = 1, a > 0, c > 0}:
Q1 = 1⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ 1⊕ 0,
Q2 = 0⊕ 1⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ 1,
P =
1
2
1 c(c−ib)√
1−a2
b(b+ic)√
1−a2
a
c(c+ib)√
1−a2
1 −a b(b−ic)√
1−a2
b(b−ic)√
1−a2
−a 1 c(c+ib)√
1−a2
a b(b+ic)√
1−a2
c(c−ib)√
1−a2
1
,
where the representation space is H = C⊕ C⊕ C⊕ C.
6) The ∗-algebras P4,abo, 1
α
, for α = 2 − 2
2k+1 , k = 1, 2, . . ., have unique irreducible represen-
tations
Q1 = I ⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ I ⊕ 0,
Q2 = 0⊕ I ⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ I,
P =
1
α
(
A B
Bt C
)
, where A =
(
I A1
A1 I
)
, C =
(
I C1
C1 I
)
, B =
(
B00 B01
B10 B11
)
,
A1 =
1
2k + 1
k∑
i=1
(2k + 3− 4i)ek×k
i,i , C1 =
1
2k + 1
k∑
i=1
(2k + 1− 4i)ek×k
i,i ,
Blm =
(−1)`
2k + 1
k∑
i=1
√
(2k − 2i + 1)(2i− 1) ek×k
i,i +
(−1)m
2k + 1
k−1∑
i=1
√
(2k − 2i)2i ek×k
i+1,i,
and the representation space is
H = Ck ⊕ Ck ⊕ Ck ⊕ Ck.
7) The ∗-algebras P4,abo, 1
α
, for α = 2 − 1
2k+1 , k = 1, 2, . . ., have unique irreducible represen-
tations
Q1 = I ⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ I ⊕ 0,
Q2 = 0⊕ I ⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ I,
P =
1
α
(
A B
Bt C
)
, where A =
(
I A1
At
1 I
)
, C =
(
I C1
C1 I
)
, B =
η η
B00 B10
B01 B11
,
η = (
√
k
2k+1
, 0, 0, . . . , 0︸ ︷︷ ︸
k−1
),
A1 = − 1
2k + 1
k∑
i=1
2ie
(k+1)×k
i+1,i , C1 = − 1
2k + 1
k∑
i=1
(2i− 1)ek×k
i,i ,
Blm =
(−1)`
4k + 2
k∑
i=1
√
(2k − 2i + 1)(2k + 2i) ek×k
i,i +
(−1)m
4k + 2
k−1∑
i=1
√
(2k − 2i)(2k + 2i + 1) ek×k
i,i+1,
and the representation space is
H = Ck+1 ⊕ Ck ⊕ Ck ⊕ Ck.
On Transitive Systems of Subspaces in a Hilbert Space 17
8) The ∗-algebras P4,abo, 1
α
, for α = 2− 1
2k , k = 1, 2, . . ., have unique irreducible representations
Q1 = I ⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ I ⊕ 0,
Q2 = 0⊕ I ⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ I,
P =
1
α
(
A B
Bt C
)
, where A =
(
I A1
At
1 I
)
, C =
(
I C1
C1 I
)
, B =
B00 B10
η η
B01 B11
,
η = (
√
2k−1
4k
, 0, 0, . . . , 0︸ ︷︷ ︸
k−1
),
A1 = −1
k
k−1∑
i=1
ie
(k−1)×k
i,i+1 , C1 = − 1
2k
k∑
i=1
(2i− 1)ek×k
i,i ,
Blm =
(−1)`
4k
k−1∑
i=1
√
(2k − 2i)(2k + 2i− 1) e
(k−1)×k
i,i
+
(−1)m
4k
k−1∑
i=1
√
(2k − 2i− 1)(2k + 2i) e
(k−1)×k
i,i+1 ,
and the representation space is
H = Ck−1 ⊕ Ck ⊕ Ck ⊕ Ck.
9) The ∗-algebras P4,abo, 1
α
, for α = 2+ 1
2k , k = 1, 2, . . ., have unique irreducible representations
Q1 = I ⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ I ⊕ 0,
Q2 = 0⊕ I ⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ I,
P =
1
α
(
A B
Bt C
)
, where A =
(
I A1
At
1 I
)
, C =
(
I C1
C1 I
)
, B =
η η
B11 B01
B10 B00
,
η = (
√
2k+1
4k
, 0, 0, . . . , 0︸ ︷︷ ︸
k−1
),
A1 =
1
k
k∑
i=1
ie
(k+1)×k
i+1,i , C1 =
1
2k
k∑
i=1
(2i− 1)ek×k
i,i ,
Blm =
(−1)`
4k
k∑
i=1
√
(2k + 2i)(2k − 2i + 1) ek×k
i,i
+
(−1)m
4k
k−1∑
i=1
√
(2k + 2i + 1)(2k − 2i) ek×k
i,i+1,
and the representation space is
H = Ck+1 ⊕ Ck ⊕ Ck ⊕ Ck.
10) The ∗-algebras P4,abo, 1
α
, for α = 2 + 1
2k+1 , k = 1, 2, . . ., have unique irreducible represen-
tations
Q1 = I ⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ I ⊕ 0,
Q2 = 0⊕ I ⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ I,
18 Yu.P. Moskaleva and Yu.S. Samǒılenko
P =
1
α
(
A B
Bt C
)
, where A =
(
I A1
At
1 I
)
, C =
(
I C1
C1 I
)
, B =
B11 B01
η η
B10 B00
,
η = (
√
k+1
2k+1
, 0, 0, . . . , 0︸ ︷︷ ︸
k
),
A1 =
1
2k + 1
k∑
i=1
2ie
k×(k+1)
i,i+1 , C1 =
1
2k + 1
k+1∑
i=1
(2i− 1)e(k+1)×(k+1)
i,i ,
Blm =
(−1)`
4k + 2
k∑
i=1
√
(2k − 2i + 2)(2k + 2i + 1) e
k×(k+1)
i,i
+
(−1)m
4k + 2
k∑
i=1
√
(2k + 2i− 1)(2k + 2i + 2) e
k×(k+1)
i,i+1 ,
and the representation space is
H = Ck ⊕ Ck+1 ⊕ Ck+1 ⊕ Ck+1.
11) The ∗-algebras P4,abo, 1
α
, for α = 2 + 2
2k+1 , k = 1, 2, . . ., have unique irreducible represen-
tations
Q1 = I ⊕ 0⊕ 0⊕ 0, Q3 = 0⊕ 0⊕ I ⊕ 0,
Q2 = 0⊕ I ⊕ 0⊕ 0, Q4 = 0⊕ 0⊕ 0⊕ I,
P =
1
α
(
A B
Bt C
)
, where A =
(
I A1
A1 I
)
, C =
(
I C1
C1 I
)
, B =
(
B11 B01
B10 B00
)
,
A1 = − 1
2k + 1
k+1∑
i=1
(2k + 3− 4i)e(k+1)×(k+1)
i,i ,
C1 = e
(k+1)×(k+1)
1,1 − 1
2k + 1
k+1∑
i=2
(2k + 5− 4i)e(k+1)×(k+1)
i,i ,
Blm =
1√
2k + 1
e
(k+1)×(k+1)
1,1 +
(−1)`
2k + 1
k+1∑
i=2
√
(2k − 2i + 3)(2i− 1) e
(k+1)×(k+1)
i,i
+
(−1)m
2k + 1
k∑
i=1
√
(2k − 2i + 2)2i e
(k+1)×(k+1)
i,i+1 ,
and the representation space is
H = Ck+1 ⊕ Ck+1 ⊕ Ck+1 ⊕ Ck+1.
Acknowledgments
The authors are grateful to S.A. Kruglyak for useful remarks and suggestions.
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[2] Halmos P.R., Ten problems in Hilbert space, Bull. Amer. Math. Soc., 1970, V.76, 887–933.
[3] Brenner S., Endomorphism algebras of vector spaces with distinguished sets of subspaces, J. Algebra, 1967,
V.6, 100–114.
[4] Gel’fand I.M., Ponomarev V.A., Problems of linear algebra and classification of quadruples of subspaces in
a finite-dimensional vector space, Coll. Math. Spc. Bolyai 5, Tihany, 1970, 163–237.
On Transitive Systems of Subspaces in a Hilbert Space 19
[5] Nazarova L.A., Representations of a quadruple, Izv. AN SSSR, 1967, V.31, N 6, 1361–1377 (in Russian).
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2002, V.36, N 3, 30–35 (English transl.: Funct. Anal. Appl., 2002, V.36, N 3, 182–195).
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by idempotents, Proc. Amer. Math. Soc., 2000, V.128, 1655–1664.
[9] Ostrovskyi V.L., Samǒılenko Yu.S., Introduction to the theory of representations of finitely presented ∗-al-
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[10] Popovich S.V., Samǒılenko Yu.S., On homomorphisms of algebras generated by projections, and the Coxeter
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1 Introduction
2 Definitions and main properties
3 On *-wildness of the description problem for transitive systems of n subspaces for n5
3.1 On *-wildness of the description problem for transitive systems that correspond to orthogonal projections
3.2 On *-wildness of the description problem for transitive systems corresponding to orthogonal projections with an additional relation
4 Transitive systems of subspaces corresponding to `39`42`"613A``45`47`"603ARepPn,com
4.1 On *-representations of the *-algebra Pn,com
4.2 Transitive systems of n subspaces corresponding to `39`42`"613A``45`47`"603ARepPn,
5 Transitive systems of subspaces corresponding to `39`42`"613A``45`47`"603ARepPn,abo,
5.1 Equivalence of the categories `39`42`"613A``45`47`"603ARepPn, and `39`42`"613A``45`47`"603ARepPn,abo,
5.2 Transitive quintuples of subspaces
|
| id | nasplib_isofts_kiev_ua-123456789-146166 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-28T17:33:22Z |
| publishDate | 2006 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Moskaleva, Y.P. Samoilenko, Y.S. 2019-02-07T20:02:45Z 2019-02-07T20:02:45Z 2006 On Transitive Systems of Subspaces in a Hilbert Space / Y.P. Moskaleva, Y.S. Samoilenko // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 12 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 47A62; 16G20 https://nasplib.isofts.kiev.ua/handle/123456789/146166 Methods of *-representations in Hilbert space are applied to study of systems of n subspaces in a linear space. It is proved that the problem of description of n-transitive subspaces in a finite-dimensional linear space is *-wild for n ≥ 5. The authors are grateful to S.A. Kruglyak for useful remarks and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Transitive Systems of Subspaces in a Hilbert Space Article published earlier |
| spellingShingle | On Transitive Systems of Subspaces in a Hilbert Space Moskaleva, Y.P. Samoilenko, Y.S. |
| title | On Transitive Systems of Subspaces in a Hilbert Space |
| title_full | On Transitive Systems of Subspaces in a Hilbert Space |
| title_fullStr | On Transitive Systems of Subspaces in a Hilbert Space |
| title_full_unstemmed | On Transitive Systems of Subspaces in a Hilbert Space |
| title_short | On Transitive Systems of Subspaces in a Hilbert Space |
| title_sort | on transitive systems of subspaces in a hilbert space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146166 |
| work_keys_str_mv | AT moskalevayp ontransitivesystemsofsubspacesinahilbertspace AT samoilenkoys ontransitivesystemsofsubspacesinahilbertspace |