On Projections in the Noncommutative 2-Torus Algebra
We investigate a set of functional equations defining a projection in the noncommutative 2-torus algebra Aθ. The exact solutions of these provide various generalisations of the Powers-Rieffel projection. By identifying the corresponding K₀(Aθ) classes we get an insight into the structure of projecti...
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irk-123456789-1468292019-02-12T01:23:07Z On Projections in the Noncommutative 2-Torus Algebra Eckstein, M. We investigate a set of functional equations defining a projection in the noncommutative 2-torus algebra Aθ. The exact solutions of these provide various generalisations of the Powers-Rieffel projection. By identifying the corresponding K₀(Aθ) classes we get an insight into the structure of projections in Aθ. 2014 Article On Projections in the Noncommutative 2-Torus Algebra / M. Eckstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 21 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 46L80; 19A13; 19K14; 46L87 DOI:10.3842/SIGMA.2014.029 http://dspace.nbuv.gov.ua/handle/123456789/146829 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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We investigate a set of functional equations defining a projection in the noncommutative 2-torus algebra Aθ. The exact solutions of these provide various generalisations of the Powers-Rieffel projection. By identifying the corresponding K₀(Aθ) classes we get an insight into the structure of projections in Aθ. |
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Eckstein, M. |
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Eckstein, M. On Projections in the Noncommutative 2-Torus Algebra Symmetry, Integrability and Geometry: Methods and Applications |
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Eckstein, M. |
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Eckstein, M. |
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On Projections in the Noncommutative 2-Torus Algebra |
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On Projections in the Noncommutative 2-Torus Algebra |
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On Projections in the Noncommutative 2-Torus Algebra |
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On Projections in the Noncommutative 2-Torus Algebra |
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On Projections in the Noncommutative 2-Torus Algebra |
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on projections in the noncommutative 2-torus algebra |
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Інститут математики НАН України |
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On Projections in the Noncommutative 2-Torus Algebra / M. Eckstein // Symmetry, Integrability and Geometry: Methods and Applications. — 2014. — Т. 10. — Бібліогр.: 21 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT ecksteinm onprojectionsinthenoncommutative2torusalgebra |
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2025-07-11T00:43:02Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 029, 14 pages
On Projections in the Noncommutative
2-Torus Algebra?
Micha l ECKSTEIN
Faculty of Mathematics and Computer Science, Jagellonian University,
ul. Lojasiewicza 6, 30-348 Kraków, Poland
E-mail: michal.eckstein@uj.edu.pl
Received December 09, 2013, in final form March 16, 2014; Published online March 23, 2014
http://dx.doi.org/10.3842/SIGMA.2014.029
Abstract. We investigate a set of functional equations defining a projection in the noncom-
mutative 2-torus algebra Aθ. The exact solutions of these provide various generalisations of
the Powers–Rieffel projection. By identifying the corresponding K0(Aθ) classes we get an
insight into the structure of projections in Aθ.
Key words: noncommutative torus; projections; noncommutative solitons
2010 Mathematics Subject Classification: 46L80; 19A13; 19K14; 46L87
1 Introduction
Projections (i.e. selfadjoint, idempotent elements) in associative ∗-algebras are the main building
blocks of the algebraic K-theory. Commutative C∗-algebras, which by Gelfand–Naimark theo-
rem are equivalent to locally compact Hausdorff spaces, do not contain non-trivial projections,
when the corresponding space is connected. To determine the K0 group of a unital C∗-algebra A
one thus has to study the equivalence classes of projections in the matrix algebra M∞(A). How-
ever, when one abandons the assumption of commutativity of the algebra one may encounter
various non-trivial projections in the algebra itself which, in some cases, are sufficient to fully
determine the group K0(A).
TheK-theory of the noncommutative 2-torus algebra Aθ, known also as the irrational rotation
algebra, has been thoroughly investigated in the 1980’s. From the works of Pimsner, Voiculescu
and Rieffel (see [16, 18] and references therein) we know that K0(Aθ) ∼= Z ⊕ θZ ∼= Z ⊕ Z. In
the case of noncommutative tori it turns out that projections in the algebra Aθ itself generate
the whole group K0(Aθ) (see [19, Corollary 7.10]). The K0 class of a projection is uniquely
determined by its algebraic trace, so any two projections with the same trace must be unitarily
equivalent in M∞(Aθ) (see [18, Corollary 2.5]). On the other hand, it has been already pointed
out by Rieffel in [19] that the structure of projections in Aθ is more robust than it would appear
from the K-theory level.
The purpose of this paper is to look closer into the structure of projections in Aθ itself.
Our main results are summarised in Theorems 1 and 2 in Section 4. Proposition 3 addresses
the problem of the existence of projections invariant under the flip automorphism (23). The
statements are proven by an explicit construction of the relevant projections. The latter may
be useful in the applications where explicit formulae for projections are needed. For an example
see [8], where projections in M2(C(T2)) were constructed using Rieffel’s method, which we
generalise here1.
?This paper is a contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in
honor of Marc A. Rieffel. The full collection is available at http://www.emis.de/journals/SIGMA/Rieffel.html
1We thank the anonymous referee for pointing out this possible application to us.
mailto:michal.eckstein@uj.edu.pl
http://dx.doi.org/10.3842/SIGMA.2014.029
http://www.emis.de/journals/SIGMA/Rieffel.html
2 M. Eckstein
The uses of noncommutative tori in physics are multifarious. A most natural one concerns
gauge theories developed in terms of finitely generated projective modules, which are noncom-
mutative counterparts of vector bundles [4, 21]. Recently, the projections in the noncommutative
torus algebra Aθ gained more interest in the context of string theory [5, 20]. They turned out
to be extrema of the tachyonic potential providing solitonic field solutions interpreted in terms
of D-branes [1, 12, 13]. Moreover, the projections in Aθ are extensively used in the context of
quantum anomalies [6, 15], knot theory [9] or theoretical engineering [14].
The paper is organised as follows: Below we recall some basic definitions to fix notation
and make the paper self-contained. In the next section we present a set of functional equations
defining a projection in Aθ and comment on the adopted method of solving these. We also
compute the Chern class of a projection satisfying these equations. In Section 3 we investigate
some special solutions – the Powers–Rieffel type projections. These will serve us firstly to provide
an alternative proof of the important Corollary 7.10 from [19]. Secondly, we will use them as
a starting point for generalisations to come in Section 4. Section 5 contains the discussion of the
flip-invariance (23). Finally, in Section 6, we make conclusions and discuss some open questions
that arose during the presented analysis.
The algebra of noncommutative 2-torus Aθ is the universal C∗-algebra generated by two
unitaries U , V satisfying the following commutation relation
V U = e2πiθUV (1)
for some real parameter θ ∈ [0, 1[, which we assume to be irrational.
We shall work with Aθ, a dense ∗-subalgebra [11, 21] of Aθ which is made up of “smooth”
elements of the form
Aθ ⊃ Aθ 3 a =
∑
(m,n)∈Z2
am,nU
mV n, with {am,n} ∈ S
(
Z2
)
, (2)
where S(Z2) denotes the space of Schwartz sequences on Z2, i.e.
{am,n} ∈ S
(
Z2
)
⇐⇒ sup
m,n∈Z
(
1 +m2 + n2
)k|am,n| <∞, for all k ∈ N.
Let us note that Aθ is a Fréchet pre-C∗-algebra [21], hence K0(Aθ) ∼= K0(Aθ) as Abelian
groups [10, Theorem 3.44].
For the purposes of this paper we will only consider elements of Aθ with am,n 6= 0 for a finite
range of the index m. These can be written as
b =
∑
m-finite
∑
n∈Z
bm,nU
mV n, with {bm,n}n∈Z ∈ S(Z) for any m. (3)
Recall that any function f ∈ C∞(S1), regarded as a periodic function on R with period 1, has
a Fourier series presentation
f(x) =
∑
n∈Z
fne
2πixn, with {fn} ∈ S(Z).
Thus, via the funcional calculus, we may uniquely define an element from Aθ by
f(V ) =
∑
n∈Z
fnV
n, for f ∈ C∞(S1)
and the expression (3) is conveniently rewritten as
b =
∑
m-finite
Umbm(V ), with bm ∈ C∞(S1) for any m. (4)
On Projections in the Noncommutative 2-Torus Algebra 3
The noncommutative 2-torus algebra is equipped with a canonical trace [11, 21], which on
the elements of the form (4) can be expressed as (compare with [17, p. 415])
τ(b) = b0,0 =
∫ 1
0
b0(x) dx. (5)
We shall use it to determine the K0 class of a projection on the strength of Corollary 2.5 in [18].
To make a connection with the original framework of Rieffel’s construction [17] we shall
describe Aθ as the crossed product algebra C(S1) o Z. Here Z acts on S1 by rotations by 2πθ
and thus induces an action of Z as automorphisms of C(S1). A convenient concrete realisation
of Aθ as bounded operators on L2(S1) is obtained with
(Uf)(x) = f(x+ θ), (V f)(x) = e2πixf(x).
Note that the elements of the form (4) fall in the dense ∗-subalgebra of Aθ considered by Rieffel
in [17]. The only difference is that we have chosen to work with the smooth functions on S1
rather than continuous ones. This is necessary as we want to compute the Chern number [3] of
the relevant projections, which requires bn functions to be differentiable.
Let δ1, δ2 be the basic unbounded derivations of Aθ, which act on the generators as
δ1U = 2πiU, δ1V = 0, δ2U = 0, δ2V = 2πiV.
Then the Chern number of a projection reads
c1(p) =
1
2πi
τ
(
p(δ1pδ2p− δ2pδ1p)
)
.
The Chern number is related to the index of a Fredholm operator and thus it is always an integer
(see [3, Theorem 11]).
2 Equations for a projection in Aθ
Having recalled the basic features of the noncommutative 2-torus algebra we are ready to inves-
tigate the structure of projections in it.
Let us consider the following element of Aθ:
p =
M∑
n=−M
Unpn(V ), for some M ∈ N. (6)
The conditions for p to be a projection yield a set of functional equations for the functions
pi ∈ C∞(R/Z)
pk(x) = p−k(x+ kθ), for k = −M, . . . ,M, (7)
pk(x) =
M∑
m,a=−M
pm(x+ aθ)pa(x)δm+a,k, for k = −M, . . . ,M, (8)
0 =
M∑
m,a=−M
pm(x+ aθ)pa(x)δm+a,k, for k < −M and k > M. (9)
Some of the above equations are redundant and the number of independent ones is 3M + 2. It
can be easily seen by noticing that equations (7)–(9) with k < 0 are equivalent to those with
k > 0, because the functions pk with negative indices are actually defined by (7) with k > 0.
4 M. Eckstein
For M = 0 formulae (7)–(9) imply p0(x) ≡ 1 as one may expect. When M = 1 one obtains
the familiar Powers–Rieffel equations [17]. However, for M ≥ 2 the equations become more
and more involved and even the existence of a solution is not obvious. In [6] we found four
particular solutions to (7)–(9) with M = 2, which represent different classes of K0(Aθ). In the
next sections we present a generalisation of the construction given in [6]. Before we start solving
the equations (7)–(9) let us adopt the following definition.
Definition 1. We say a projection in Aθ is of order M if it is of the form (6) and pM 6= 0.
We shall not attempt to provide a general solution to (7)–(9), but rather present a class of
special solutions. Nevertheless, this class turns out to be large enough to accommodate the
known projections as well as a number of new ones.
We will consider only real-valued functions although (7) requires only p0 to be real. Moreover,
we have already noted that (7) defines the functions pk for k < 0 and it is convenient to get
rid of the functions pk with negative index k in the equations (8) and (9) before solving them.
Our special solutions will be such that each summand on the r.h.s. of (9) is equal to zero
independently. The same should hold for summands of (8) with k > 0 excluding those with
m = 0 or a = 0, these are combined to form equations
pk(x)
(
p0(x) + p0(x+ kθ)− 1
)
= 0, for k = 1, . . . ,M, (10)
which we also require to be satisfied independently. The equations (8) with k < 0 are redundant
and the case k = 0 cannot be split into independent equations. After (7) is substituted into (8)
for k = 0 we obtain
p2M (x−Mθ) + p2M (x) + p2M−1(x− (M − 1)θ) + p2M−1(x)
+ · · ·+ p21(x− θ) + p21(x) + p0(x)
(
p0(x)− 1
)
= 0. (11)
In the forthcoming sections we provide a systematic method of constructing projections of a given
trace (5) and order, that will satisfy the equations (7)–(9) refined according to the above-listed
conditions.
Before we start, let us compute the Chern number of a projection of order M , as this quantity
might prove useful in the task of classification.
Proposition 1. The Chern number of a projection p of order M reads
c1(p) =
M∑
n=1
M−n∑
k=−M
∫ 1
0
dx
(
n pk+n(x)
[
pn(x+ kθ)p′k(x)− pn(x)p′k(x+ nθ)
]
+
(
{k, n} ←→ {−k,−n}
))
. (12)
Moreover, for any projection constructed with the method adopted in this paper, the formula (12)
simplifies to
c1(p) = 6
∫ 1
0
dx
M∑
n=1
n pn(x)2p′0(x). (13)
Proof. The application of the derivations δ1, δ2 to a projection p of the form (6) yields
δ1p = 2πi
M∑
n=−M
nUnpn(V ), δ2p =
M∑
n=−M
Unp′n(V ).
On Projections in the Noncommutative 2-Torus Algebra 5
Let us denote pk := 0 for |k| > M . Then using (1), (5) and (7) we obtain
c1(p) =
M∑
j,k,n=−M
τ
(
U jpj(V )
[
kUkpk(V )Unp′n(V )− nUkp′k(V )Unpn(V )
])
=
M∑
j,k,n=−M
τ
(
U j+k+npj(e
2πiθ(k+n)V )
[
kpk(e
2πiθnV )p′n(V )− np′k(e2πiθnV )pn(V )
])
=
M∑
k,n=−M
∫ 1
0
dx p−k−n(x+ (k + n)θ)
[
kpk(x+ θn)p′n(x)− np′k(x+ nθ)pn(x)
]
=
M∑
k,n=−M
n
∫ 1
0
dx pk+n(x)
[
pn(x+ θk)p′k(x)− p′k(x+ nθ)pn(x)
]
.
In the last equality we have relabelled the indices k ↔ n in the first term in the square bracket.
The formula (12) is just the above expression, with the fact pk = 0 for |k| > M taken into
account.
Now, in the method adopted in this paper we have assumed that the summands of the r.h.s.
of (8) and (9) are equal to zero independently, excluding those with m = 0 or a = 0. This means
that for the considered projections only the terms k = 0 and k = −n will contribute to the sum
in (12) and we have
c1(p) =
M∑
n=1
∫ 1
0
dxn
(
pn(x)2p′0(x)− p2n(x)p′0(x+ nθ) + 2p0(x)pn(x− nθ)p′−n(x)
− 2p0(x)pn(x)p′−n(x+ nθ) + pn(x− nθ)2p′0(x− nθ)− p2n(x− nθ)p′0(x)
)
= 2
M∑
n=1
∫ 1
0
dxn
(
pn(x)2p′0(x)− p2n(x)p′0(x+ nθ)
+ p0(x)pn(x− nθ)p′n(x− nθ)− p0(x)pn(x)p′n(x)
)
,
We have used the formula (7) together with the assumption of all pn being real and the fact that
shifting the integration variable does not change the value of the integral. Now, let us make use
of the periodicity of the integrant to integrate by parts one of the terms and shift the variable
in another:
c1(p) = 2
M∑
n=1
∫ 1
0
dxn
(
pn(x)2p′0(x) + 2pn(x)p′n(x)p0(x+ nθ)
+ p0(x+ nθ)pn(x)p′n(x)− p0(x)pn(x)p′n(x)
)
.
Let us note that equations (10) imply∫ 1
0
dx p′n(x)pn(x)p0(x+ nθ) =
∫ 1
0
dx p′n(x)pn(x)−
∫ 1
0
dx p′n(x)pn(x)p0(x)
= −
∫ 1
0
dx p′n(x)pn(x)p0(x),
since p′n(x)pn(x) is a total derivative. Finally, we obtain
c1(p) = 2
M∑
n=1
∫ 1
0
dxn
(
pn(x)2p′0(x)− 4p0(x)pn(x)p′n(x)
)
= 6
M∑
n=1
∫ 1
0
dxn pn(x)2p′0(x).
�
6 M. Eckstein
Figure 1. Depiction of functions constituing a Powers–Rieffel type projection. We have 0 < εM ≤ θ′,
εM + θ′ ≤ 1.
3 Powers–Rieffel type projections
We start with recalling the construction of the Powers–Rieffel projection in a slightly more
general framework. It will serve us as a starting point for generalisations to come in the next
section.
If one sets pk = 0 for all 1 ≤ k ≤M − 1 then (7)–(9) reduce to the Powers–Rieffel equations
with parameter Mθ [17, Theorem 1.1]
pM (x+Mθ)pM (x) = 0, (14)
p2M (x) + p2M (x−Mθ) + p0(x)(p0(x)− 1) = 0, (15)
pM (x)
(
1− p0(x)− p0(x+Mθ)
)
= 0. (16)
A standard solution to (14)–(16) is known as a Powers–Rieffel type projection [7, 13]
p0(x) =
dM (x), 0 ≤ x < εM ,
1, εM ≤ x ≤Mθ,
1− dM (x−Mθ), Mθ ≤ x < Mθ + εM ,
0, Mθ + εM ≤ x ≤ 1,
(17)
pM (x) =
{√
dM (x)(1− dM (x)), 0 ≤ x < εM ,
0, εM < x ≤ 1,
(18)
where θ′ = Mθ − bMθc and dM is a smooth function with dM (0) = 0, dM (εM ) = 1. The
functions p0 and p1 are depicted in Fig. 1.
Let us stress that we do not assume that the dM function starts growing directly at x = 0 as
shown on Fig. 1. We may take dM such that dM = 0 for x ∈ [0, δM ] with some δM < εM and
then smoothly growing to reach 1 at x = εM . This ensures that what we call here a Powers–
Rieffel type projection is sufficiently general to incorporate the existing definitions (see [13] for
instance).
Let us now discuss the properties of these projections. First of all, note that due to the
periodicity of pi functions, equations (14)–(16) are invariant with respect to the transformation
Mθ → Mθ + z for any z ∈ Z. This means that a Powers–Rieffel type projection of order M
has the algebraic trace (5) equal to θ′ = Mθ − bMθc. Since θ is irrational we have infinitely
many M such that
0 < Mθ − n < 1 ⇐⇒ n
M
< θ <
n+ 1
M
.
On Projections in the Noncommutative 2-Torus Algebra 7
Hence, the following proposition (which is also a consequence of the Corollary 7.10 in [19])
holds.
Proposition 2. The algebra Aθ contains projections representing infinitely many different
classes of K0(Aθ).
Another point of view one may adopt for the projection (17)–(18) is that for any fixed M it
is the standard Powers–Rieffel projection [17] in the subalgebra of Aθ generated by UM and V .
This fact may be used to construct an approximation of Aθ in terms of two algebras of matrix
valued functions on S1 [7, 13].
For p[M ] a Powers–Rieffel type projection of order M the formula (13) gives
c1(p
[M ]) = 6M
∫ εM
0
dx dM (x)(1− dM (x))d′M (x) = 6M
(
dM (x)2
2
− dM (x)3
3
) ∣∣∣εM
0
= M.
This is in accordance with the result of [3] stating that if τ(p) = |a− bθ| then c1(p) = ±b.
From the K-theoretic point of view, these projections are sufficient to understand the struc-
ture of the equivalence classes of projective modules over Aθ. On the other hand, the algebra Aθ
contains other interesting projections, which we shall present in next section.
4 More general projections in Aθ
Let us now see what kind of projections one can get by letting functions pk in (6) to be non-
zero for some of the indices k ∈ {1, . . . ,M − 1}. The results are summarised in the following
Theorems.
Theorem 1. A projection of order M may represent the K0(Aθ) class [nθ], as well as the class
[1− nθ], for all n = 1, 2, . . . , 12M(M + 1), provided that 0 < θ < 1/max(n,M).
By [nθ] ∈ K0(Aθ) we denote the K0 class represented by a projection p ∈ Aθ with τ(p) = nθ.
Theorem 2. The equations (7)–(9) for a projection of order M admit solutions with pk 6= 0 for
every k ∈ {0, . . . ,M} whenever 0 < θ < 1/M .
We shall start with the proof of Theorem 1 by showing how to use the functions pk to
increase or decrease the trace of a Powers–Rieffel type projection. Then we present a method
of including the remaining pk functions into the projections constructed in the previous proof
without changing its traces. In this way we will prove Theorem 2. Both proofs are constructive
so we are able to plot some examples of the p0 functions of the relevant projections which, as we
shall see, determine all of the other functions pk for k 6= 0. A brief discussion of the assumptions
limiting the θ parameter may be found in Section 6.
Proof of Theorem 1. Let us start with the case of τ(p) = nθ > Mθ. We shall begin with
a Powers–Rieffel type projection as defined in (17)–(18). First note that if Mθ < 1 then the
functions p0 and pM of the Powers–Rieffel type projection of order M vanish for x ≥Mθ+ εM .
If θ is small enough (i.e. (M + k)θ < 1) then we can “glue” a Powers–Rieffel type projection of
trace kθ to the previous one. Namely, let us keep the definition of p0 on [0,Mθ+ εM ] (see (17))
and set
p0(x) =
dk(x), Mθ + εM ≤ x < Mθ + εM + εk,
1, Mθ + εM + εk ≤ x ≤ (M + k)θ + εM ,
1− dk(x− kθ), (M + k)θ + εM ≤ x < (M + k)θ + εM + εk,
0, (M + k)θ + εM + εk ≤ x ≤ 1,
8 M. Eckstein
Figure 2. Examples of p0 functions for projections with traces (M + k)θ and (M + k + l)θ.
pk(x) =
{√
dk(x)(1− dk(x)), Mθ + εM ≤ x < Mθ + εM + εk,
0, elsewhere
for a smooth function dk with dk(Mθ + εM ) = 0, dk(Mθ + εM + εk) = 1 and a small parame-
ter εk. The summands of (8) and (9), which we have assumed to be equal to zero independently,
have the form pm(x + aθ)pa(x). This means that all of the non-zero functions pk for k 6= 0
shifted to the interval x ∈ [0, θ] must not intersect. The latter can be fulfilled by restricting the
parameters ε such that
0 < εM ≤Mθ, 0 < εk ≤ kθ, εM + εk + (M + k)θ ≤ 1 (19)
implying that equation (11) reduces to two equations of the form (15). Namely for x ∈ [0, εM ]∪
[Mθ,Mθ + εM ] and for x ∈ [Mθ + εM ,Mθ + εM + εk] ∪ [(M + k)θ + εM , (M + k)θ + εM + εk]
we have respectively
p0(x)(1− p0(x)) = p2M (x) + p2M (x−Mθ),
p0(x)(1− p0(x)) = p2k(x) + p2k(x− kθ).
These equations are satisfied by the construction of pk and pM . On the remaining part of the
interval [0, 1] the equation (11) is trivially satisfied, since both l.h.s. and r.h.s. are equal to 0.
By the same argument, equation (10) remains satisfied, as it is satisfied for both Powers–Rieffel
type projections independently. Thus, we have obtained a new projection with a trace (M+k)θ.
Examples of p0 functions defining such projections are depicted in Fig. 2.
If the parameter θ is small enough (i.e. nθ < 1) we can continue the process of “glueing”
Powers–Rieffel type projections to obtain a projection of trace nθ, with n ≥ M . If one makes
use of all of the functions pk with 1 ≤ k ≤ M − 1 to increase the trace, one will end with
a projection bearing the trace (1+2+ · · ·+M)θ = 1
2M(M+1)θ. The only thing one has to take
care of are the conditions satisfied by the parameters εk. The restrictions (19) may be easily
generalised to the case of non-vanishing pks functions with s ∈ [1,M − 1]:
0 < εkj ≤ kjθ, for 1 ≤ j ≤ s,
εk1 + · · ·+ εks + εM + nθ ≤ 1, with n = k1 + · · ·+ ks +M. (20)
Let us note, that the above construction can be obtained (for nθ < 1) by taking a sum of s
mutually orthogonal Powers–Rieffel type projections p[kj ] of respective orders kj . Indeed, one
can easily check that the functional equations resulting form the projection and orthogonality
conditions
(p[ki])2 = (p[ki])∗ = p[ki], p[ki]p[kj ] = p[kj ]p[ki] = 0, for 1 ≤ i 6= j ≤ s,
On Projections in the Noncommutative 2-Torus Algebra 9
Figure 3. Examples of p0 functions for projections of trace (M − k)θ and (M − k1 − k2 + l)θ.
coincide with the ones derived in Section 2. This process of “glueing” mutually orthogonal
Powers–Rieffel type projections appeared already in [7] and was extensively used therein. It has
also been presented in [1] in a more similar form to the one shown above.
Let us now consider the case of projections of order M and trace nθ with 1 ≤ n < M . Again,
we shall use as a starting point a Powers–Rieffel type projection (17)–(18), but now we will “cut
out” a part of it. Let us set
p0(x) =
dk(x), εM ≤ x < εM + εk,
0, εM + εk ≤ x ≤ kθ + εM ,
1− dk(x− kθ), kθ + εM ≤ x < kθ + εM + εk,
1, kθ + εM + εk ≤ x ≤Mθ,
(21)
pk(x) =
{√
dk(x)(1− dk(x)), εM ≤ x < εM + εk,
0, elsewhere
(22)
with a smooth function dk such that dk(εM ) = 1, dk(εM+εk) = 0. The conditions 0 < εM ≤Mθ,
0 < εk ≤ kθ and εM +Mθ ≤ 1 should be satisfied. The situation is now completely analogous
to the case of “glued” projections and the same arguments apply. A projection obtained in this
way bears the trace (M − k)θ for 1 ≤ k ≤M − 1 (see Fig. 3).
To end the proof of Theorem 1 it remains just to recall that if p is a projection then obviously
1− p is so. This means that all of the considerations hold for projections of traces (1−nθ), one
simply should take 1− p0 instead of p0 and leave pk for k 6= 0 as they are. �
The presented proof provides a great variety of possible projections with a given trace, which
have, in general, different orders. Let us notice that the two procedures of increasing and
decreasing the trace of a projection of a given order can be applied simultaneously and in
arbitrary sequence (see Figs. 3 and 5). One only has to choose well the parameters εk to have
the equations (20) satisfied. These equations guarantee that the functions dk do not superpose
and the equations (7)–(9) remain satisfied. This leads to an enormous number of projections if
the order M is big enough. Let us now pass on to the most general projections we were able to
construct with the adopted method.
Proof of Theorem 2. In fact one can let all pk functions to be non-zero by incorporating to p0
some “bump functions” dk. As a starting point, one should take an arbitrary projection defined
in Section 3 or 4. For sake of simplicity let us now denote by k a free index, i.e. we have pk = 0
in our starting point projection. Now, if one sets p0(x) = dk(x) for x ∈ [δk, δk + εk], with
dk(δk) = dk(δk + εk) = 1 or dk(δk) = dk(δk + εk) = 0 then, to fulfil the equation (10), one has to
set p0(x) = 1−dk(x−kθ) for x ∈ [kθ+ δk, kθ+ δk + εk]. The function pk should then be defined
10 M. Eckstein
Figure 4. Examples of p0 functions for projections of traces Mθ and (M − k)θ.
as previously by
√
dk(x)(1− dk(x)) for x ∈ [δk, δk + εk] and 0 elsewhere, so that (11) remains
fulfilled. The only task to accomplish is to choose well the parameters εk and δk to avoid the
possible intersection of dk functions. The parameters εk should be such that the equations (20)
remain satisfied, and δk = nθ + εk1 + · · · + εks for n, s ∈ Z which depend on the concrete
projection one has chosen as a starting point. �
Examples of p0 functions of the described above projections are shown in Fig. 4.
By giving constructive proofs of Theorems 1 and 2 we have exhausted all of the possibilities
of constructing projections in Aθ with the method described in Section 2. To end this section
let us note that the computation of the Chern number of the newly constructed projections does
not provide any new information. Indeed, it is straightforward either from direct computations
of the formula (13), either from an application of the results of [3] that if we have a projection p
of trace nθ, then c1(p) = n. In particular, the process of adding “bump” functions described in
the proof of Theorem 2 does not change the Chern class of a projection.
5 Flip-symmetric projections
The Powers–Rieffel type projection can be made invariant under the flip automorphism
σ ∈ AutAθ, σ(U) = U−1, σ(V ) = V −1. (23)
This is accomplished by setting εM = 1 −Mθ and requiring that dM (x) + dM (εM − x) = 1
for x ∈ [0, εM ] in formulae (17)–(18). It is interesting to check for which of the more general
projections presented in this paper the flip symmetry can be imposed2.
Requiring σ(p) = p for projections of order M translates to the following constraints on
functions pk:
p0(x) = p0(1− x), pk(1− x) = pk(x− kθ), for k = 1, . . . ,M. (24)
The fact that p0 is periodic with period 1 implies that for a flip-symmetric projection, p0 should
be symmetric around 1
2 .
Let us first note that the flip symmetry cannot be imposed on a projection, which is “glued”
from two or more segments (see Fig. 2). This is because the segments will necessarily have
different lengths and thus p0 cannot be symmetric around 1
2 . It is also clear (see Fig. 4) that
the inclusion of a “bump function” breaks the flip invariance of a projection.
On the other hand, the “cutting out” procedure (see Fig. 3) can be performed in such a way
that the invariance under the flip automorphism is preserved. Moreover, provided that the θ
2We are grateful to the anonymous referee for suggesting this interesting problem to us.
On Projections in the Noncommutative 2-Torus Algebra 11
Figure 5. Examples of p0 functions for flip-symmetric projections.
parameter is small enough, one can “glue” another Powers–Rieffel type projection inside the
“cut-out” region in a symmetric way. This process may be continued as long as the parameter θ
allows it. However, this requires a fine tuning of εk parameters. Examples of p0 functions of
such flip-symmetric projection are depicted in Fig. 5.
Let us now formulate the above considerations in a precise way.
Proposition 3. Let kj ∈ {1, . . . ,M − 1} for j = 1, 2, . . . , s, 1 ≤ s ≤M − 1 be such that
M > k1 > k2 > · · · > ks > 0 and θ ∈
]
c−1 , c
+
1
]
∩
[
c−2 , c
+
2
[
∩ · · · ∩
[
c−s , c
+
s
]
\ {c±s },
where ± = (−1)s and
c+j =
1
2(M − k1 + k2 − k3 + · · · − kj)
, for j odd,
1
2(M − k1 + k2 − k3 + · · · − kj−1) + kj
, for j even,
c−j =
1
2(M − k1 + k2 − k3 + · · ·+ kj−1)− kj
, for j odd,
1
2(M − k1 + k2 − k3 + · · ·+ kj)
, for j even.
Then, there exists a projection p ∈ Aθ of order M invariant under the flip automorphism (23)
σ(p) = p. Moreover, p is a representant of the K0 class [(M − k1 + k2 − k3 + · · ·+ (−1)sks)θ].
Proof. The construction of the projection p goes as follows: We start with a Powers–Rieffel
type projection of order M (17)–(18). We make it flip symmetric, i.e. we set εM = 1 −Mθ
and require that dM be such that dM (x) + dM (εM − x) = 1 for x ∈ [0, εM ]. Then, we “cut
out” a projection of trace k1θ (see formulae (21)–(22)) also in symmetry-preserving way. This
requires setting (compare the left plots in Figs. 3 and 5)
εk1 = (2M − k1)θ − 1 and dk1(x) + dk1(εk1 − x) = 1, for x ∈ [εM , εM + εk1 ].
This choice guarantees that constraints (24) are fulfilled for both functions p0 and pk1 .
Note however, that we need to have 0 < εk1 ≤ k1θ for p to be a projection, which is equivalent
to
1
2M − k1
< θ ≤ 1
2(M − k1)
⇐⇒ θ ∈
]
c−1 , c
+
1
]
.
Now, we “glue” a Powers–Rieffel type projection of trace k2θ in the middle of the “cut-out”
region. To preserve the flip symmetry we have to set
εk2 = 1− 2(M − k1)θ + k2θ
12 M. Eckstein
and
dk2(x) + dk2(εk2 − x) = 1, for x ∈ [εM + εk1 , εM + εk1 + εk2 ].
But since 0 < εk2 ≤ k2θ, the equation on εk2 can be fulfilled only if
1
2(M − k1 + k2)
≤ θ < 1
2(M − k1) + k2
⇐⇒ θ ∈
[
c−2 , c
+
2
[
.
By performing further consecutive “cut-outs” and “glueings” we obtain the following condi-
tions for all j ∈ {1, . . . , s}
εkj =
{
2(M − k1 + k2 − k3 + · · · − kj−1)θ − kjθ − 1, for j odd,
1− 2(M − k1 + k2 − k3 + · · · − kj−1)θ − kj , for j even,
dkj (x) + dkj (εkj − x) = 1, for x ∈ [εM + εk1 + · · ·+ εkj−1
, εM + εk1 + · · ·+ εkj ],
which can be met only if θ ∈ [c−j , c
+
j ] \ {c±j }, with ± = (−1)j .
To conclude the proof let us remind the reader that the procedure of “glueing” a projection
of trace kθ increases the trace of the overall projection by kθ and “cutting-out” decreases it
by kθ. �
We end this section with a remark, that some of the presented symmetric projections cannot
be constructed in any Aθ with θ ∈ [0, 1[. Let us take for instance M = 6, k1 = 5, k2 = 4, k3 = 1.
Then c−1 = 1
7 , c+1 = 1
2 , c−3 = 1
9 , c+3 = 1
8 , so ]c−1 , c
+
1 ]∩ ]c−3 , c
+
3 ] = ∅.
6 Conclusion and open questions
Let us now summarise the obtained results and outline the directions of possible further inves-
tigations.
We have presented many projections, which generalise the standard Powers–Rieffel projec-
tion. Some represented the same K0(Aθ) class, but had different orders. The others, conversely,
had the same order, but different traces. A natural question one can ask is what are the relations
between the presented projections? The answer is provided by Theorem 8.13 in [19]. It states
that if two projections in Aθ represent the same K0(Aθ) class (hence have the same trace),
then, not only they are unitarily equivalent in M∞(Aθ), but they are actually in the same path
component of the set of projections in Aθ itself. This means that there exists a homotopy of
projections in Aθ for any two projections which have the same trace. Indeed, if, for instance, one
takes dk(t, x) := tdk(x) + (1− t) instead of dk(x) with dk(δk) = dk(δk + εk) = 1 for the “bump
function” used in the proof of Theorem 2, then one would obtain a projection for all t ∈ [0, 1].
In consequence, from the topological point of view it is sufficient to consider Powers–Rieffel
type projections, since they are the generators of the K0(Aθ) group. On the other hand, the
richness of the structure of projections may show up in applications. In the proof of the The-
orem 1 it has already been mentioned that the procedure of “gluing” the Powers–Rieffel type
projections is in fact equivalent to taking sums of mutually orthogonal projections. However,
the “cutting out” described subsequently does not admit an interpretation in terms of subtract-
ing the projections. Indeed, it is straightforward to see that if one expresses a projection p of
order M and trace (M−k)θ as p = q−r, where q is a Powers–Rieffel type projection of order M ,
then r would not be a projection. This shows that the newly found projections are not just
linear combinations of the K0 generators.
The method adopted in this paper clearly does not pretend to cover every possible projection
in Aθ. For a most general projection in Aθ one would need to allow the order M of a projection
On Projections in the Noncommutative 2-Torus Algebra 13
go to infinity. This would imply the need of working directly with the elements of the form (2)
and would require completely different methods (see [2] for an example).
The puzzling thing about the newly found projections is that their existence in Aθ seems to
depend on the noncommutativity parameter θ as stated in the theorems in Section 4. Unfortu-
nately, the solutions presented there cannot be adapted to the case nθ > 1, as it was done for
the Powers–Rieffel type projections in Section 3. It is so because the translational symmetry
of (14)–(16), used in the proof of Proposition 2 is absent in general equations (7)–(9). Note
that the discussed symmetry is also broken whenever we introduce the mentioned “bump func-
tions”. What is more, the existence of flip-symmetric projections in Aθ seems to depend on θ
in an even more peculiar way. Whether there is a true difference in the structure of projections
in Aθ depending on the noncommutativity parameter θ or is it just an artefact of our method
of solving the equations (7)–(9) remains an open question.
To conclude the paper, let us comment on the possible applications of the obtained results to
the D-brane scenario in Type II string theories. As mentioned in the Introduction, projections
in Aθ correspond to solitonic field configurations which are identified with D-branes [1, 12, 13].
On one hand, unitarly equivalent projections yield gauge equivalent field configurations [13, Sec-
tion 3.1], hence the knowledge of K0(Aθ) alone seems to be sufficient. On the other hand, projec-
tions which cannot be written as linear combinations of K0 generators provide non-perturbative
field configurations. Moreover, the homotopy equivalence of projections may be exploited to
study the soliton dynamics. An example is provided in [13, Section 6.2], where the Boca pro-
jection [2], which is homotopy equivalent to the standard Powers–Rieffel projection, is used.
The possibility of adding “bump functions” to a projection as described at the end of Section 4
indicates the existence of an additional degree of freedom of the D-branes. It would also be
interesting to investigate the consequences for D-branes of a projection being invariant under
the flip symmetry. Finally, let us note that the D-brane point of view suggests that the num-
ber of projections in Aθ indeed depends on the value of the deformation parameter θ (see [12,
Section 4] or [1, Section V]).
Acknowledgements
We would like to thank Andrzej Sitarz for his illuminating remarks. Project operated within the
Foundation for Polish Science IPP Programme “Geometry and Topology in Physical Models”
co-financed by the EU European Regional Development Fund, Operational Program Innovative
Economy 2007-2013.
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1 Introduction
2 Equations for a projection in A
3 Powers–Rieffel type projections
4 More general projections in A
5 Flip-symmetric projections
6 Conclusion and open questions
References
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