Nonstandard Quantum Complex Projective Line
In our attempt to explore how the quantum nonstandard complex projective spaces ℂⁿq,c studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to esta...
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| description | In our attempt to explore how the quantum nonstandard complex projective spaces ℂⁿq,c studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to establish the known identification of (ℂ¹q,c) with the pull-back of two copies of the Toeplitz *-algebra along the symbol map in a more direct way via an operator theoretic analysis, which also provides some interesting non-obvious details, such as a prominent generator of (ℂ¹q,c) being a concrete weighted double shift.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 073, 14 pages
Nonstandard Quantum Complex Projective Line
Nicola CICCOLI † and Albert Jeu-Liang SHEU ‡
† Dipartimento di Matematica e Informatica, University of Perugia, Italy
E-mail: nicola.ciccoli@unipg.it
‡ Department of Mathematics, University of Kansas, Lawrence, KS 66045, USA
E-mail: asheu@ku.edu
Received March 06, 2020, in final form July 24, 2020; Published online August 03, 2020
https://doi.org/10.3842/SIGMA.2020.073
Abstract. In our attempt to explore how the quantum nonstandard complex projective
spaces CPn
q,c studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those
arising from the geometrically constructed Bohr–Sommerfeld groupoids by Bonechi, Ciccoli,
Qiu, Staffolani, and Tarlini, we were led to establish the known identification of C
(
CP 1
q,c
)
with the pull-back of two copies of the Toeplitz C∗-algebra along the symbol map in a more
direct way via an operator theoretic analysis, which also provides some interesting non-
obvious details, such as a prominent generator of C
(
CP 1
q,c
)
being a concrete weighted double
shift.
Key words: quantum homogeneous space; Toeplitz algebra; weighted shift
2020 Mathematics Subject Classification: 58B32; 46L85
1 Introduction
In [9], the C∗-algebra C
(
CPnq,c
)
of nonstandard quantum complex projective spaces studied by
Korogodsky and Vaksman [5] and Dijkhuizen and Noumi [3] is embedded in a concrete groupoid
C∗-algebra, and then shown to have C
(
S2n−1q
)
as a quotient algebra, which reflects the geometric
observation [10] that the nonstandard SU(n+1)-covariant Poisson complex projective space CPn
contains a copy of the standard Poisson sphere S2n−1.
Although the work in [9] involves realizing C
(
CPnq,c
)
as part of a concrete groupoid C∗-algebra
in order to analyze the algebra structure and extract useful information, it is not clear whether
one can actually realize C
(
CPnq,c
)
as a groupoid C∗-algebra itself. However from a purely differ-
ential geometric consideration, an elegant program of constructing some quantum homogeneous
spaces as the groupoid C∗-algebras of geometrically constructed Bohr–Sommerfeld groupoids is
later successfully developed by Bonechi, Ciccoli, Qiu, Staffolani, Tarlini [1, 2]. Naturally, it is
of great interest to decide whether the quantum complex projective space arising from this new
program is indeed the same as the known version of C
(
CPnq,c
)
. Indeed they are the same for the
case of n = 1 because the underlying groupoids are shown to be isomorphic in Proposition 7.2
of [1]. But the higher dimensional cases are far from being settled.
It is hoped that by analyzing more carefully the embedding of C
(
CPnq,c
)
in a concrete groupoid
C∗-algebra found in [9] via a representation theoretic approach, one can see some direct con-
nection with the geometrically constructed Bohr–Sommerfeld groupoid and then possibly find
a way to identify these two different versions of quantum complex projective spaces. While
attempting this approach, we come to recognize the need of a more direct understanding of
the algebra structure of C
(
CPnq,c
)
based on some known representations of the ambient algebra
C(SUq(n+ 1)).
This paper is a contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in
honour of Giovanni Landi. The full collection is available at https://www.emis.de/journals/SIGMA/Landi.html
mailto:nicola.ciccoli@unipg.it
mailto:asheu@ku.edu
https://doi.org/10.3842/SIGMA.2020.073
https://www.emis.de/journals/SIGMA/Landi.html
2 N. Ciccoli and A.J.-L. Sheu
In particular, for n = 1, we want to directly derive the algebra structure of C
(
CP 1
q,c
)
from the
basic representations of C(SUq(2)), instead of via identifying C
(
CP 1
q,c
)
with the algebra C
(
S2µc
)
of the Podleś quantum 2-sphere [7] as indicated in [3, 5]. In this note, we show how to accomplish
it. Along the way, our detailed analysis reveals some nontrivial hidden structures, for example,
a distinguished generator x∗1x2 of C
(
CP 1
q,c
)
is a weighted double shift (on a core Hilbert space
that determines the C∗-algebra structure of C
(
CP 1
q,c
)
) with respect to an orthonormal basis,
and its weights are determined by a concrete formula.
2 Nonstandard quantum CP 1
q,c
We recall the description of C
(
CPnq,c
)
with c ∈ (0,∞) and q ∈ (1,∞) obtained by Dijkhuizen
and Noumi [3] as
C
(
CPnq,c
) ∼= C∗({x∗ixj | 1 ≤ i, j ≤ n+ 1}) ⊂ C(SUq(n+ 1)),
where
xi =
√
cu1,i + un+1,i
for the standard generators {ui,j}n+1
i,j=1 of C(SUq(n+ 1)).
In this paper, we focus on the case of n = 1, with the goal to directly show that C
(
CP 1
q,c
)
is
the pullback T ⊕C(T) T of two copies of the standard symbol map
σ : T → C(T)
for the Toeplitz algebra T that is the C∗-algebra generated by the (forward) unilateral shift S
on `2(Z≥), with ker(σ) = K
(
`2(Z≥)
)
the ideal of all compact operators.
For C(SUq(2)), consider the known faithful representation π of C(SUq(2)) determined by
π(u) ≡
(
π(u11) π(u12)
π(u21) π(u22)
)
:=
{(
t1α −q−1t1γ
t2γ t2α
∗
)}
t2=t1∈T
as a T-family of representations of C(SUq(2)) on `2(Z≥) with parameter t1 ≡ t2 ∈ T, where
α =
0
√
1− q−2 0
0 0
√
1− q−4 0
0 0 0
√
1− q−6 . . .
0 0 0
. . .
. . .
. . .
. . .
∈ B
(
`2(Z≥)
)
and
γ =
1 0 0
0 q−1 0 0
0 0 q−2 0
. . .
0 0 q−3
. . .
. . .
. . .
. . .
∈ B
(
`2(Z≥)
)
self-adjoint
satisfying
α∗α+ γγ∗ ≡ α∗α+ γ2 = I = αα∗ + q−2γ2 ≡ αα∗ + q−2γ∗γ
Nonstandard Quantum Complex Projective Line 3
and
γα∗ − q−1α∗γ = αγ∗ − q−1γ∗α = 0 ≡ αγ − q−1γα = γ∗α∗ − q−1α∗γ∗
which ensure the required condition π(u)π(u)∗ = I = π(u)∗π(u).
In this paper, we identify every element of C(SUq(2)) ⊃ C(CP 1
q,c) with a T-family of opera-
tors on `2(Z≥) via this faithful representation π, and we analyze such a T-family of operators
pointwise at each fixed t1 ∈ T.
More explicitly, the generators x∗1x2, x
∗
1x1, x
∗
2x2 of C
(
CP 1
q,c
)
are T-families of operators with
x1 :=
√
ct1α+ t2γ =
t2
√
ct1
√
1− q−2 0
0 t2q
−1 √
ct1
√
1− q−4 0
0 0 t2q
−2 √
ct1
√
1− q−6 . . .
0 0 t2q
−3 . . .
. . .
. . .
. . .
and
x2 := −q−1
√
ct1γ + t2α
∗ =
−q−1
√
ct1 0 0
t2
√
1− q−2 −q−2
√
ct1 0 0
0 t2
√
1− q−4 −q−3
√
ct1 0
. . .
0 t2
√
1− q−6 −q−4
√
ct1
. . .
. . .
. . .
. . .
.
At any fixed t1 ∈ T, it is easy to see that ker(x2) = 0 since
0 = x2
( ∞∑
n=0
znen
)
= −q−1
√
cz0e0 +
(√
1− q−2z0 − q−2
√
cz1
)
e1
+
(√
1− q−4z1 − q−3
√
cz2
)
e2 + · · ·
implies z0 = z1 = z2 = · · · = 0, and dim(coker(x2)) = 1 since x2 ≡ t2α
∗ ≡ t2S modulo K
is a Fredholm operator of index −1. On the other hand, dim(ker(x1)) = 1 by solving 0 =
x1
( ∞∑
n=0
znen
)
to get that if z0 = 1, then
zn = (−1)ntn2q
−n(n−1)
2
√
c
−n
t−n1
√
1− q−2
−1
· · ·
√
1− q−2n
−1
for all n ∈ N, and x1 is surjective since x1 ≡
√
ct1α ≡
√
ct1S∗ modulo K is a Fredholm
operator of index 1. So x∗1x2 is a Fredholm operator of index −2. Actually, ker(x∗1x2) = 0
(hence (x∗1x2)
∗(x∗1x2) is invertible) and dim(coker(x∗1x2)) = 2, and hence the partial isometry
(x∗1x2)|x∗1x2|−1 in the polar decomposition of x∗1x2 is S ⊕ S (up to a unitary direct summand)
after a suitable choice of orthonormal basis. This observation is consistent with our goal to show
that C
(
CP 1
q,c
)
is isomorphic to the pullback C∗-algebra T ⊕C(T) T , but is far from sufficient to
make such a conclusion. We need to do a much more detailed analysis which starts with the
following computation.
First we compute
x∗1x1 = cα∗α+
√
ct1
2
α∗γ +
√
ct21γα+ γ2
= c+ (1− c)γ2 +
√
ct1
2
α∗γ +
√
ct21γα ≡ c mod K,
4 N. Ciccoli and A.J.-L. Sheu
x∗2x2 = αα∗ − q−1
√
ct21αγ − q−1
√
ct1
2
γα∗ + q−2cγ2
= 1 + q−2(c− 1)γ2 − q−1
√
ct21αγ − q−1
√
ct1
2
γα∗
= 1 + q−2(c− 1)γ2 − q−2
√
ct21γα− q−2
√
ct1
2
α∗γ ≡ 1 mod K,
x∗1x2 =
√
ct1
2
(α∗)2 − cq−1α∗γ + γα∗ − q−1
√
ct21γ
2 ≡
√
ct1
2S2 mod K,
x∗2x1 =
√
ct21α
2 − cq−1γα+ αγ − q−1
√
ct1
2
γ2,
which imply
x∗1x1 + q2x∗2x2 = q2 + c.
So the C∗-algebra C
(
CP 1
q,c
)
is generated by x∗1x2 and x∗1x1, i.e.,
C
(
CP 1
q,c
)
= C∗({x∗1x2, x∗1x1}),
since x∗2x1 = (x∗1x2)
∗ and x∗2x2 = 1 + cq−2− q−2x∗1x1 are generated by x∗1x2 (with (x∗1x2)
∗(x∗1x2)
invertible) and x∗1x1. As a remark, we note that x∗1x1 and x∗2x2 commute.
We also note that
x1x
∗
1 + x2x
∗
2 = 1 + c
and hence x1x
∗
1 and x2x
∗
2 commute. Indeed
x1x
∗
1 =
(√
ct1α+ t2γ
)(√
ct2α
∗ + t1γ
)
= cαα∗ +
√
ct21αγ +
√
ct22γα
∗ + γ2
= c− cq−2γ2 +
√
ct21q
−1γα+
√
ct22q
−1α∗γ + γ2,
while
x2x
∗
2 =
(
t2α
∗ − q−1
√
ct1γ
)(
t1α− q−1
√
ct2γ
)
= α∗α− q−1
√
ct22α
∗γ − q−1
√
ct21γα+ q−2cγ2
= 1− γ2 − q−1
√
ct22α
∗γ − q−1
√
ct21γα+ q−2cγ2.
Before proceeding further, we recall some operator-theoretic properties often used implicitly
in the following analysis, including that range(T ) = ker(T ∗)⊥ and ker(T ∗) = ker(T ∗T ) for general
bounded linear operators T on a Hilbert space H easily derived from 〈T ∗(v), w〉 = 〈v, T (w)〉 and
〈(T ∗T )(v), v〉 = 〈T (v), T (v)〉 for all v, w ∈ H respectively. In C∗-algebra theory, a projection
refers to a self-adjoint idempotent. For an operator T in the C∗-algebra B(H) of all bounded
linear operators on H, we recall that T is a projection if and only if T is geometrically the
orthogonal projection from H onto a closed subspace of H.
We will need some basic knowledge of Fredholm operators, i.e., operators T ∈ B(H) with its
quotient class [T ] an invertible element of the Calkin algebra B(H)/K(H), which can be found
in [4, 6]. Any such operator has closed finite-codimensional range and finite-dimensional kernel,
and the intersection of R\{0} and the spectrum Sp(T ) of any positive Fredholm operator T
is a compact subset of (0,∞). Also we note that the set of all Fredholm operators is closed
under taking adjoint and composition of operators. For any positive Fredholm operator T we
will denote by T−1/2 the positive operator f(T ) defined by functional calculus, where f is the
nonnegative continuous function on {0} t K such that f(•) = •−1/2 on K := Sp(T )\{0} and
f(0) = 0.
Below we recall a folklore result with a proof.
Lemma 1. For any Fredholm operator T on a Hilbert space H,
T̃ := T (T ∗T )−1/2
is a partial isometry sending the closed subspace (ker(T ))⊥ ≡ range(T ∗) isometrically onto the
closed subspace range(T ) while annihilating ker(T ).
Nonstandard Quantum Complex Projective Line 5
Proof. Note that since T ∗T is a positive Fredholm operator, the set K is a compact subset of
(0,∞).
By the spectral theory of self-adjoint operators,
T̃ ∗T̃ ≡ (T ∗T )−1/2T ∗T (T ∗T )−1/2 ≡ f(T ∗T )(T ∗T )f(T ∗T ) = χK(T ∗T )
for the characteristic function χK on Sp(T ∗T ), and hence T̃ ∗T̃ is the orthogonal projection
from H onto range(T ∗T ). Thus T̃ annihilates
ker
(
T̃
)
≡ ker
(
T̃ ∗T̃
)
≡
(
range
(
T̃ ∗T̃
))⊥
= (range(T ∗T ))⊥ ≡ ker(T ∗T ) ≡ ker(T )
and is metric preserving on
range(T ∗T ) ≡ (ker(T ∗T ))⊥ ≡ (ker(T ))⊥ ≡ range(T ∗)
due to〈
T̃ (v), T̃ (w)
〉
=
〈(
T̃ ∗T̃
)
(v), w
〉
= 〈v, w〉 for all v, w ∈ range(T ∗T ),
i.e., T̃ is a partial isometry sending range(T ∗T ) isometrically onto range(T̃ ) while annihilating
ker(T ).
It remains to show that range(T̃ ) = range(T ). In fact, since 1
f |K is a well-defined continuous
function on K, the restriction of (T ∗T )−1/2 ≡ f(T ∗T ) to range(T ∗T ) is an invertible linear
operator on range(T ∗T ) and hence
range
(
(T ∗T )−1/2
)
= range(T ∗T ) ≡ (ker(T ))⊥.
Thus we get
range(T̃ ) ≡ range
(
T (T ∗T )−1/2
)
= T ((ker(T ))⊥) = range(T ). �
At each fixed t1 ∈ T, by applying Lemma 1 to the Fredholm operator values of the norm
continuous T-families x1 and x2, we get two partial isometries
x̃1 := x1(x
∗
1x1)
−1/2
and
x̃2 := x2(x
∗
2x2)
−1/2 = x2(x
∗
2x2)
−1/2
where
(x∗2x2)
−1/2 = (x∗2x2)
−1/2 ≡
(
(x∗2x2)
−1)1/2 ≡√(x∗2x2)
−1
is a well-defined invertible operator on `2(Z≥) since ker(x2) = 0 and hence the spectrum of the
positive Fredholm operator x∗2x2 is a compact subset of (0,∞), implying the invertibility of x∗2x2
and making the functional calculus (x∗2x2)
−1/2 ≡
(
(x∗2x2)
−1)1/2 meaningful.
Theorem 1. The C∗-algebra C
(
CP 1
q,c
)
coincides with C∗({x̃∗1x̃2, x∗1x1}), i.e. the C∗-algebra
generated by two T-families x∗1x1 ≥ 0 and x̃∗1x̃2 of operators on `2(Z≥) such that at each fixed
t1∈T, x̃∗1x̃2 is an isometry of index −2 (with a zero kernel and a range of codimension 2),
where x̃∗1 and x̃2 = x2(x
∗
2x2)
−1/2 are isometries of index −1 while x̃1 = x1(x
∗
1x1)
−1/2 and x̃∗2 are
surjective partial isometries of index 1.
6 N. Ciccoli and A.J.-L. Sheu
Proof. By Lemma 1, the surjective Fredholm operator x1 with kernel of dimension 1 yields
a surjective partial isometry x̃1 = x1(x
∗
1x1)
−1/2 of index 1 and the injective Fredholm x2 with
cokernel of dimension 1 yields an isometry x̃2 = x2(x
∗
2x2)
−1/2 of index −1.
Now both x̃2 and the adjoint x̃∗1 of the surjective partial isometry x̃1 are isometries of index−1,
and hence x̃∗1x̃2 is an isometry of index −2 with (x̃∗1x̃2)
∗(x̃∗1x̃2) = 1.
From the definition of x̃i, we get
x̃∗1x̃2 = (x∗1x1)
−1/2x∗1x2(x
∗
2x2)
−1/2 ∈ C∗({x∗1x2, x∗1x1, x∗2x2}) = C
(
CP 1
q,c
)
.
Since (x∗1x1)
1/2(x∗1x1)
−1/2 is the orthogonal projection onto range(x∗1x1) ≡ range(x∗1), by spectral
theory:
x∗1x2 = (x∗1x1)
1/2(x∗1x1)
−1/2x∗1x2 = (x∗1x1)
1/2x̃∗1x̃2(x
∗
2x2)
1/2 ∈ C∗({x̃∗1x̃2, x∗1x1, x∗2x2}).
So we get
C
(
CP 1
q,c
)
≡ C∗({x∗1x2, x∗1x1, x∗2x2}) = C∗({x̃∗1x̃2, x∗1x1, x∗2x2}).
Furthermore the generator x∗2x2 is redundant since
x∗2x2 = 1 + cq−2 − q−2x∗1x1 =
(
1 + cq−2
)
(x̃∗1x̃2)
∗(x̃∗1x̃2)− q−2x∗1x1
can be generated by x̃∗1x̃2 and x∗1x1. Thus
C
(
CP 1
q,c
)
≡ C∗({x̃∗1x̃2, x∗1x1, x∗2x2}) = C∗({x̃∗1x̃2, x∗1x1}). �
We remark that for each i ∈ {1, 2},
x̃∗i x̃i = (x∗ixi)
−1/2x∗ixi(x
∗
ixi)
−1/2 = χSp(x∗i xi)\{0}(x
∗
ixi) ∈ C∗({x∗ixi})
and hence belongs to C
(
CP 1
q,c
)
≡ C∗({x̃∗1x̃2, x∗1x1, x∗2x2}), since any C∗-algebra is closed under
functional calculus by continuous functions vanishing at 0.
3 Invariant subspace decomposition
In this section, fixing an arbitrary value of the parameter t1 ≡ t2 ∈ T, we study and treat
any T-family of operators on `2(Z≥), including any element of C
(
CP 1
q,c
)
, as an operator in
B
(
`2(Z≥)
)
.
As such the index-1 surjective partial isometry x̃1 has ker(x̃1) = ker(x1) = Cv1 for some unit
vector v1, and
p1 := 1− x̃∗1x̃1 ∈ C∗({x̃∗1x̃2, x̃∗1x̃1}) ⊂ C
(
CP 1
q,c
)
is the rank-1 orthogonal projection onto Cv1. Note that (x̃∗1x̃1)(v1) = 0 is equivalent to x̃1(v1) =
0 (or equivalently v1 ⊥ range(x̃∗1) = range(x̃∗1x̃1)).
Note that
p2 := x̃∗1x̃1 − (x̃∗2x̃1)
∗(x̃∗2x̃1) = x̃∗1x̃1 − (x̃∗1x̃2)(x̃
∗
2x̃1) = x̃∗1(1− x̃2x̃∗2)x̃1
is also a rank-1 projection onto Cv2 for some unit vector v2. In fact p2 clearly annihilates ker(x̃1)
and can be viewed as the conjugation of the rank-1 projection 1− x̃2x̃∗2 (onto the kernel of x̃∗2)
by the unitary operator x̃1|(ker(x̃1))⊥ , from (ker(x̃1))
⊥ onto `2(Z≥). In an explicit description,
Nonstandard Quantum Complex Projective Line 7
v2 can be taken as the inverse image under x̃1|(ker(x̃1))⊥ , of any unit vector in the 1-dimensional
range of 1− x̃2x̃∗2, and, in particular, v2 ∈ (ker(x̃1))
⊥ ≡ range(x̃∗1x̃1). The inequalities
0 ≤ p2 = x̃∗1x̃1 − (x̃∗1x̃2)(x̃
∗
2x̃1) ≤ x̃∗1x̃1
relate the three projections p2, (x̃∗1x̃2)(x̃
∗
2x̃1), and x̃∗1x̃1 in C∗({x̃∗1x̃2, x̃∗1x̃1}), and clarify their
geometric relation: p2 and (x̃∗1x̃2)(x̃
∗
2x̃1) are projections onto two mutually orthogonal subspaces
which add up to the range of the projection x̃∗1x̃1, i.e.,
range(p2)⊕⊥ range((x̃∗1x̃2)(x̃
∗
2x̃1)) = range(x̃∗1x̃1),
indicating, in particular, v2 ∈ range(p2) ⊂ range(x̃∗1x̃1).
Now v2 ⊥ v1 since v2 is in the range of the self-adjoint operator x̃∗1x̃1 and hence is perpen-
dicular to ker(x̃∗1x̃1) = Cv1. Furthermore,
vi+2n := (x̃∗1x̃2)
n(vi)
with n ≥ 0 and i ∈ {1, 2} are orthonormal vectors, since
v1 ⊥ range(x̃∗1) ⊃ range(x̃∗1x̃2) 3 (x̃∗1x̃2)(v1)
and
v2 ⊥ range((x̃∗1x̃2)(x̃
∗
2x̃1)) = range(x̃∗1x̃2) with v1 ⊥ v2 .
Thus V := Span{v1, v2} ⊥ range(x̃∗1x̃2) or more precisely, by combining with the fact that the
index-(−2) isometry x̃∗1x̃2 has range(x̃∗1x̃2) of codimension 2,
H = V ⊕⊥ range(x̃∗1x̃2) as Hilbert space orthogonal direct sum.
Hence, since x̃∗1x̃2 is an isometry, we inductively get:
range(x̃∗1x̃2)
k−1 = (x̃∗1x̃2)
k−1(V)⊕⊥ range(x̃∗1x̃2)
k
for all k ≥ 1.
Clearly, for each i ∈ {1, 2}, the operator x̃∗1x̃2 restricted to the closed linear span Hi ⊂
`2(Z≥) of {vi+2n : n ≥ 0} is a unilateral shift S, while the orthogonal projection onto Cvi is
pi ∈ C∗({x̃∗1x̃2, x̃∗1x̃1}).
Since x̃∗1x̃2 is a unilateral shift simultaneously on both H1 and H2, it generates a C∗-algebra
C∗({x̃∗1x̃2}|H1⊕H2) containing two “synchronized” copies of the ideal of compact operators, i.e.,
C∗({x̃∗1x̃2}|H1⊕H2) ⊃
{
T ⊕ T : T ∈ K
(
`2(Z≥)
)} ∼= K(`2(Z≥)
)
,
where H1 and H2 are identified with the same Hilbert space `2(Z≥) in a canonical way, i.e.,
identifying vi+2n for i ∈ {1, 2} with the canonical orthonormal basis vector en ∈ `2(Z≥).
However our goal is to show that C
(
CP 1
q,c
)
or for now C∗({x̃∗1x̃2, x̃∗1x̃1}|H1⊕H2) contains the
direct sum K(H1) ⊕ K(H2) of all “non-synchronized” pairs of compact operators. This can be
achieved by noticing that for any k,m ∈ Z≥,
ε
(1)
k,m := (x̃∗1x̃2)
kp1((x̃
∗
1x̃2)
∗)m|H1⊕H2 ∈ C∗({x̃∗1x̃2, x̃∗1x̃1}|H1⊕H2)
is a typical matrix unit in K(H1) ⊕ 0 sending v1+2m to v1+2k while eliminating all other vi+2n
with i + 2n 6= 1 + 2m, and we get K(H1) ⊕ 0 as the closure of the linear span of ε
(1)
k,m with
k,m ∈ Z≥. Similarly the elements
ε
(2)
k,m := (x̃∗1x̃2)
kp2((x̃
∗
1x̃2)
∗)m|H1⊕H2 ∈ C∗({x̃∗1x̃2, x̃∗1x̃1}|H1⊕H2)
8 N. Ciccoli and A.J.-L. Sheu
with k,m ∈ Z≥ linearly span a dense subspace of 0⊕K(H2). Thus
K(H1)⊕K(H2) = (K(H1)⊕ 0) + (0⊕K(H2)) ⊂ C∗({x̃∗1x̃2, x̃∗1x̃1}|H1⊕H2).
Next we want to show that each vk with k ≥ 1 is an eigenvector of x∗1x1 or equivalently of
x∗2x2 = 1 + cq−2 − q−2x∗1x1, and hence each Hi is invariant under x∗1x1 and x∗2x2.
Proposition 1. The isometry x̃∗1x̃2 intertwines the positive operators x∗1x1 and (1 + c)− x∗2x2,
i.e.,
(x∗1x1)(x̃
∗
1x̃2) = (x̃∗1x̃2)(1 + c− x∗2x2).
Proof. A direct computation shows
(x∗1x1)(x̃
∗
1x̃2) = x∗1x1(x
∗
1x1)
−1/2x∗1x̃2 = (x∗1x1)
−1/2x∗1x1x
∗
1x̃2
= (x∗1x1)
−1/2x∗1(1 + c− x2x∗2)x̃2 = x̃∗1(1 + c− x2x∗2)x̃2
= (1 + c)x̃∗1x̃2 − x̃∗1x2x∗2x̃2 = (1 + c)x̃∗1x̃2 − x̃∗1x2x∗2x2(x∗2x2)−1/2
= (1 + c)x̃∗1x̃2 − x̃∗1x2(x∗2x2)−1/2x∗2x2 = (1 + c)x̃∗1x̃2 − (x̃∗1x̃2)(x
∗
2x2)
= (x̃∗1x̃2)(1 + c− x∗2x2). �
Proposition 2. The isometry x̃∗1x̃2 intertwines the (possibly degenerate) eigenspaces Eλ(x∗2x2)
and E1+c−λ(x∗1x1), where Eλ(T ) := ker(λ−T ) for linear operators T and λ ∈ C. More precisely,
(x̃∗1x̃2)(Eλ(x∗2x2)) ⊂ E1+c−λ(x∗1x1),
and
(x̃∗1x̃2)
−1(E1+c−λ(x∗1x1)) ⊂ (Eλ(x∗2x2)),
where (x̃∗1x̃2)
−1(E1+c−λ(x∗1x1)) is the inverse image of E1+c−λ(x∗1x1) under (the non-surjective)
x̃∗1x̃2.
Proof. The commutation relation
(x∗1x1)(x̃
∗
1x̃2) = (x̃∗1x̃2)((1 + c)− x∗2x2)
implies that if v ∈ Eλ(x∗2x2) then (x̃∗1x̃2)(v) ∈ E1+c−λ(x∗1x1), because
(x∗1x1)((x̃
∗
1x̃2)(v)) = (x̃∗1x̃2)((1 + c)− x∗2x2)(v)
= (x̃∗1x̃2)((1 + c)− λ)v = ((1 + c)− λ)((x̃∗1x̃2)(v)).
On the other hand, if (x̃∗1x̃2)(v) ∈ E1+c−λ(x∗1x1), then
((1 + c)− λ)((x̃∗1x̃2)(v)) = (x∗1x1)((x̃
∗
1x̃2)(v)) = (x̃∗1x̃2)((1 + c)− x∗2x2)(v)
= (1 + c)(x̃∗1x̃2)(v)− (x̃∗1x̃2)((x
∗
2x2)(v)),
and hence (x̃∗1x̃2)(λv) = (x̃∗1x̃2)((x
∗
2x2)(v)). Since x̃∗1x̃2 is injective, we get λv = (x∗2x2)(v), i.e.,
v ∈ Eλ(x∗2x2). �
Corollary 1. If λ is an eigenvalue of x∗2x2, then 1 + c− λ is an eigenvalue of x∗1x1.
Proof. If Eλ(x∗2x2) 6= 0 then E1+c−λ(x∗1x1) ⊃ (x̃∗1x̃2)(Eλ(x∗2x2)) 6= 0 since x̃∗1x̃2 is injective. �
Nonstandard Quantum Complex Projective Line 9
The equality x∗1x1 + q2x∗2x2 = q2 + c implies
Eλ(x∗2x2) = Eq2+c−q2λ(x∗1x1)
for any λ ∈ R, or equivalently
Eλ(x∗1x1) = Eq−2(q2+c−λ)(x
∗
2x2).
Proposition 3. The orthonormal vectors vk, k ∈ N, are eigenvectors of x∗1x1 (and of x∗2x2 ≡
1 + q−2c − q−2x∗1x1), and hence each of H1 and H2 is invariant under all of the generators
x̃∗1x̃2, x∗1x1, and x∗2x2 of C
(
CP 1
q,c
)
. More explicitly, (x∗1x1)(vk) = ckvk for all k ∈ N, where ck is
defined recursively by
ck+2 = c− q−2c+ q−2ck for k ∈ N, with c2 = 1 + c and c1 = 0,
which can be rewritten as
c2n = q−2(n−1) + c and c2n+1 =
(
1− q−2n
)
c
for all n ∈ Z≥.
Proof. We prove (x∗1x1)(vk) = ckvk and the formula ck+2 = c− q−2c+ q−2ck inductively on k.
First v1 ∈ (range(x̃∗1))
⊥ = ker(x̃1), so
(x∗1x1)(v1) =
(
(x∗1x1)
1/2x∗1x1(x
∗
1x1)
−1/2)(v1) =
(
(x∗1x1)
1/2x∗1x̃1
)
(v1) = 0.
Next since v2 ∈ range(x̃∗1(1− x̃2x̃∗2)x̃1), so v2 = x̃∗1(w) for some unit vector
w ∈ range(1− x̃2x̃∗2) = ker(x̃2x̃
∗
2) = ker(x̃∗2) = ker(x∗2)
and hence
(x∗1x1)(v2) = (x∗1x1)(x̃
∗
1(w)) = (x∗1x1)(x
∗
1x1)
−1/2x∗1(w)
= (x∗1x1)
−1/2(x∗1x1)x
∗
1(w) = (x∗1x1)
−1/2x∗1(1 + c− x2x∗2)(w)
= x̃∗1((1 + c)w − 0) = (1 + c)x̃∗1(w) = (1 + c)v2.
Now assume that (x∗1x1)(vk) = ckvk, i.e., vk ∈ Eck(x∗1x1), for k ∈ N. Then
vk+2 ≡ (x̃∗1x̃2)(vk) ∈ (x̃∗1x̃2)(Eck(x∗1x1)) ≡ (x̃∗1x̃2)(Eq−2(q2+c−ck)(x
∗
2x2))
⊂ E1+c−q−2(q2+c−ck)(x
∗
1x1) = Ec−q−2c+q−2ck(x∗1x1),
and hence (x∗1x1)(vk+2) = ck+2vk+2 for ck+2 := c− q−2c+ q−2ck.
The recursive formula ck+2 = c−q−2c+q−2ck rewritten as ck+2−c = q−2(ck−c) immediately
leads to ci+2n − c = q−2n(ci − c) and hence
ci+2n = q−2n(ci − c) + c
for any i ∈ {1, 2} and n ∈ N. More explicitly, we have c2n = q−2(n−1)+c and c2n+1 =
(
1−q−2n
)
c
for all n ∈ N. �
Corollary 2. The element x∗1x2 = (x∗1x1)
1/2x̃∗1x̃2(x
∗
2x2)
1/2 is a weighted shift on H1 and H2
with respect to the orthonormal bases {v2n−1}n≥1 and {v2n}n≥1 respectively. More precisely,
(x∗1x2)(vk) =
√
ck+2
√
1 + q−2c− q−2ckvk+2
for the constants ck specified in the above proposition.
10 N. Ciccoli and A.J.-L. Sheu
Proof. This is a simple consequence of (x∗1x1)(vk) = ckvk and
(x∗2x2)(vk) ≡
(
1 + q−2c− q−2x∗1x1
)
(vk) =
(
1 + q−2c− q−2ck
)
vk. �
With each of H1 and H2 invariant under the self-adjoint operators x∗ixi, it is clear that the
orthogonal complement H0 := (H1 ⊕ H2)
⊥ in `2(Z≥) is also invariant under each x∗ixi. On
the other hand, since we know the orthonormal vectors v1, v2 ∈ (range(x̃∗1x̃2))
⊥ for the index-
(−2) isometry x̃∗1x̃2, we get a Wold-von Neumann decomposition (Theorem 3.5.17 of [6]) for the
isometry x̃∗1x̃2 as
x̃∗1x̃2 = x̃∗1x̃2|H0 ⊕ x̃∗1x̃2|H1 ⊕ x̃∗1x̃2|H2
with
H0 ≡
(
Span
({
(x̃∗1x̃2)
k(vi) : i ∈ {1, 2} and k ∈ Z≥
}))⊥
.
Here (x̃∗1x̃2)|H0 is a unitary operator on H0 (if H0 6= 0) since x̃∗1x̃2|H0 is an index-0 isometry in
view of x̃∗1x̃2|Hi being an index-(−1) isometry for each i ∈ {1, 2}.
It is not clear whether H0 is actually trivial or not, so we remark that any discussion involving
H0 below is only needed and valid when H0 6= 0.
We already know that (x̃∗1x̃2)|Hi is a unilateral shift for each i ∈ {1, 2}. So with respect to
the decomposition
`2(Z≥) = H0 ⊕H1 ⊕H2
into orthogonal subspaces, the generators x̃∗1x̃2, x
∗
1x1, x
∗
2x2 and hence all elements of C
(
CP 1
q,c
)
can be viewed as block diagonal operators. Then it is easy to see that Propositions 1 and 2 hold
for the restrictions of x̃∗1x̃2, x
∗
1x1, x
∗
2x2 to each Hi.
Lemma 2. The spectrum Sp(x∗1x1|H0) of x∗1x1|H0 is invariant under the function
f1 : s 7→ c− q−2c+ q−2s ≡ c+ q−2(s− c)
and its inverse function. Similarly, the spectrum Sp(x∗2x2|H0) of x∗2x2|H0 is invariant under the
function
f2 : s 7→ 1− q−2 + q−2s ≡ 1 + q−2(s− 1)
and its inverse function.
Proof. By Proposition 1,
(x∗1x1|H0)(x̃∗1x̃2|H0) = (x̃∗1x̃2|H0)(1 + c− x∗2x2|H0)
with x̃∗1x̃2|H0 unitary, we get x∗1x1|H0 and 1 + c− x∗2x2|H0 unitarily equivalent and hence
Sp(x∗1x1|H0) = Sp(1 + c− x∗2x2|H0) = 1 + c− Sp(x∗2x2|H0).
On the other hand, from x∗1x1 + q2x∗2x2 = q2 + c, we have
Sp(x∗2x2|H0) = q−2
(
q2 + c− Sp(x∗1x1|H0)
)
= 1 + q−2c− q−2 Sp(x∗1x1|H0).
Hence
Sp(x∗1x1|H0) = 1 + c−
(
1 + q−2c− q−2 Sp(x∗1x1|H0)
)
= c− q−2c+ q−2 Sp(x∗1x1|H0),
Nonstandard Quantum Complex Projective Line 11
which shows that under the invertible function f1 : s ∈ R 7→ c − q−2c + q−2s ∈ R, the set
Sp(x∗1x1|H0) ⊂ R equals itself and hence the inverse function (f1)
−1 maps Sp(x∗1x1|H0) onto
itself too.
Since the invertible function g : s ∈ R 7→ 1 + c− s ∈ R maps Sp(x∗2x2|H0) onto Sp(x∗1x1|H0),
the conjugate g−1 ◦ f1 ◦ g and its inverse function map Sp(x∗2x2|H0) onto itself, where(
g−1 ◦ f1 ◦ g
)
(s) = 1 + c− f1(1 + c− s)
= 1 + c−
(
c− q−2c+ q−2(1 + c− s)
)
= 1− q−2 + q−2s. �
Note that f1(s)− c = q−2(s− c) and f2(s)− 1 = q−2(s− 1) for all s ∈ R with q > 1. So the
only bounded backward f1-orbit is the constant f1-orbit {c}, and similarly the only bounded
backward f2-orbit is the constant f2-orbit {1}, where by a backward fi-orbit, we mean the set
{(fi)−n(s) : n ∈ N} for a point s ∈ R. On the other hand, any forward fi-orbit converges to the
constant fi-orbit, i.e., lim
n→∞
(f1)
n(s) = c and lim
n→∞
(f2)
n(s) = 1 for any s. Since each spectrum
Sp(x∗ixi|H0) is a compact and hence bounded set that is invariant under backward iterations
of fi, it can contain only the constant fi-orbit. We have, therefore
Corollary 3. Sp(x∗1x1|H0) = {c} and Sp(x∗2x2|H0) = {1}, i.e., x∗1x1|H0 = c id and x∗2x2|H0 = id.
Proposition 4. There is a unital C∗-algebra isomorphism from C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 to the
pullback T ⊕C(T)T of two copies of T σ→ C(T), sending x̃∗1x̃2|H1⊕H2 to S⊕S. This isomorphism
provides an exact sequence
0→ K(H1)⊕K(H2)→ C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2
∼= T ⊕C(T) T
σ→ C(T)→ 0
of C∗-algebras with σ(x̃∗1x̃2|H1⊕H2) = idT, σ(x∗1x1|H1⊕H2) = c, and σ(x∗2x2|H1⊕H2) = 1.
Proof. We note that the eigenvalues ck of x∗1x1|H1⊕H2 satisfying ck+2 = c−q−2c+q−2ck = f1(ck)
form two forward f1-orbits and hence lim
k→∞
ck = c. This limit is also clear from the explicit
formulae of c2n and c2n+1 given in Proposition 3. Similarly, one can verify that the eigenvalues c′k
of x∗2x2|H1⊕H2 form two forward f2-orbit and hence lim
k→∞
c′k = 1.
So x∗1x1|H1⊕H2 ≡ c ⊕ c mod K(H1) ⊕ K(H2) and x∗2x2|H1⊕H2 ≡ 1 ⊕ 1 mod K(H1) ⊕ K(H2),
while x̃∗1x̃2|H1⊕H2 = SH1 ⊕ SH2 for copies SHi of the unilateral shift.
It has been shown earlier that K(H1)⊕K(H2) ⊂ C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 , so it is not hard to
see that C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 is the pullback of two copies of T σ→ C(T). In fact, SH1⊕SH2 ≡
x̃∗1x̃2|H1⊕H2 generates {T ⊕T : T ∈ T } as a C∗-subalgebra of C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 and hence
x̃∗1x̃2|H1⊕H2 ∈ {T ⊕ T : T ∈ T }+ (K(H1)⊕K(H2)) ⊂ C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 .
On the other hand,
x∗1x1|H1⊕H2 ∈ (c⊕ c) + (K(H1)⊕K(H2)) ⊂ {T ⊕ T : T ∈ T }+ (K(H1)⊕K(H2))
and hence
C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 ⊂ {T ⊕ T : T ∈ T }+ (K(H1)⊕K(H2)).
So we get
C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 = {T ⊕ T : T ∈ T }+ (K(H1)⊕K(H2)) = T ⊕C(T) T ,
where the second equality is due to that any S ⊕ T ∈ T ⊕ T with σ(S) = σ(T ) can be written
as
S ⊕ T = (T ⊕ T ) + ((S − T )⊕ 0) ∈ T ⊕ T + (K(H1)⊕K(H2)).
12 N. Ciccoli and A.J.-L. Sheu
Replacing T ⊕C(T) T in the canonical exact sequence
0→ K(H1)⊕K(H2)→ T ⊕C(T) T
σ→ C(T)→ 0
by the isomorphic C∗-algebra C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 , we get the stated exact sequence with
σ(x̃∗1x̃2|H1⊕H2) = idT, σ(x∗1x1|H1⊕H2) = c, and σ(x∗2x2|H1⊕H2) = 1. �
Theorem 2. The restriction map
T ∈ C∗({x̃∗1x̃2, x∗1x1}) 7→ T |H1⊕H2 ∈ C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2
is a C∗-algebra isomorphism, and hence C∗({x̃∗1x̃2, x∗1x1}) is isomorphic to the pullback T ⊕C(T)T
of two copies of T σ→ C(T) with x̃∗1x̃2 corresponding to S ⊕ S.
Proof. Clearly we only need to consider the case with H0 6= 0.
Since x̃∗1x̃2|H0 is unitary, as shown in the above discussion of Wold–von Neumann decompo-
sition, and C(T) is the universal C∗-algebra generated by a single unitary generator, there is a
unique C∗-algebra homomorphism
h : C(T)→ C∗({x̃∗1x̃2|H0})
sending idT to x̃∗1x̃2|H0 while fixing all scalars in C ⊂ C(T).
Clearly with x∗1x1|H0 = 1 and x∗2x2|H0 = c,
h ◦ σ : C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2 → C∗({x̃∗1x̃2|H0}) = C∗({x̃∗1x̃2, x∗1x1})|H0
is a well-defined C∗-algebra homomorphism sending x̃∗1x̃2|H1⊕H2 to x̃∗1x̃2|H0 and x∗ixi|H1⊕H2 to
x∗ixi|H0 for i ∈ {1, 2}. Hence the restriction map
T ∈ C∗({x̃∗1x̃2, x∗1x1}) 7→ T |H1⊕H2 ∈ C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2
gives a well-defined isomorphism. �
In Theorem 2, we treat elements of C∗({x̃∗1x̃2, x∗1x1}) as operators instead of families of
operators by fixing implicitly the value of T-parameter at any t1 ∈ T, i.e., the statement of
Theorem 2 is a pointwise result at any t1 ∈ T. It is clear that collectively the restriction map
C
(
CP 1
q,c
)
→ C
(
CP 1
q,c
)∣∣
H̃1⊕H̃2
is still a C∗-algebra isomorphism where elements of C
(
CP 1
q,c
)
are T-families of operators on
`2(Z≥) and H̃1 ⊕ H̃2 represents a T-family of Hilbert subspaces H1 ⊕H2 of `2(Z≥) constructed
pointwise for each t1 ∈ T as described above.
4 Superfluous circle parameter
In this section, we show that the T-parameter is superfluous for the C∗-algebra C
(
CP 1
q,c
)∣∣
H̃1⊕H̃2
consisting of T-families of operators on H1 ⊕ H2, and hence C
(
CP 1
q,c
)
|H̃1⊕H̃2
∼= C
(
CP 1
q,c
)
is
isomorphic to C∗({x̃∗1x̃2, x∗1x1})|H1⊕H2
∼= T ⊕C(T) T (for any t1 ∈ T fixed) as obtained in
Proposition 4.
Recall that by a simple change of orthonormal basis ek tkek of `2(Z≥) for any fixed t ∈ T,
the weighted shift operator α becomes α̃ = tα with respect to the new orthonormal basis, while
the self-adjoint operator γ remains the same operator γ̃ = γ.
Nonstandard Quantum Complex Projective Line 13
Note that the earlier concrete description of x∗1x2, x
∗
1x1, and x∗2x2 as families of operators
parametrized by t1 ∈ T (with t2 = t1) viewed as a representation of C
(
CP 1
q,c
)
≡ C∗({x∗1x2, x∗1x1,
x∗2x2}) can be first “consolidated” by a change of orthonormal basis converting α to α̃ := t21α
and γ to γ̃ = γ, so that we can rewrite the description as
x∗1x1 = c+ (1− c)γ2 +
√
ct1
2
α∗γ +
√
ct21γα
= c+ (1− c)γ̃2 +
√
cα̃∗γ̃ +
√
cγ̃α̃,
x∗2x2 = 1 + q−2(c− 1)γ2 − q−2
√
ct21γα− q−2
√
ct1
2
α∗γ
= 1 + q−2(c− 1)γ̃2 − q−2
√
cγ̃α̃− q−2
√
cα̃∗γ̃,
x∗1x2 =
√
ct1
2
(α∗)2 − cq−1α∗γ + γα∗ − q−1
√
ct21γ
2
= t21
(√
ct1
4
(α∗)2 − cq−1t12α∗γ + t1
2
γα∗ − q−1
√
cγ2
)
= t21
(√
c(α̃∗)2 − cq−1α̃∗γ̃ + γ̃α̃∗ − q−1
√
cγ̃2
)
,
where it is understood that α̃, γ̃ with respect to suitable orthonormal basis of `2(Z≥) are the
same familiar matrix operators α, γ, and hence we can simply replace α̃, γ̃ by α, γ in the above
formulas for x∗1x2, x
∗
1x1, and x∗2x2.
So we have
x∗1x1 = c+ (1− c)γ2 +
√
cα∗γ +
√
cγα,
x∗2x2 = 1 + q−2(c− 1)γ2 − q−2
√
cγα− q−2
√
cα∗γ,
x∗1x2 = t21
(√
c(α∗)2 − cq−1α∗γ + γα∗ − q−1
√
cγ2
)
,
where only x∗1x2 still involves t1 = t2 as a factor. From Proposition 3 and Corollary 2, there is
an orthonormal basis {v2k, v2k−1 : k ∈ N} of H1⊕H2 consisting of eigenvectors of x∗1x1 and x∗2x2
and with respect to which x∗1x2 is a double weighted shift. So after the change of orthonormal
basis v2k
(
t21
)k
v2k and v2k−1
(
t21
)k
v2k−1, the factor t21 in the formula of x∗1x2 can be dropped
while the formulas of x∗1x1 and x∗2x2 remain the same, i.e., we have C∗({x∗1x2, x∗1x1})|H1⊕H2 for
any fixed t1 ∈ T unitarily equivalent to C∗({x∗1x2, x∗1x1})|H1⊕H2 for t1 := 1. So we conclude that
the parameter t1 ∈ T is “edundant” in the sense that representations of the generators x∗1x2,
x∗1x1, and x∗2x2 of C
(
CP 1
q,c
)
as operators on `2(Z≥) by the above formulas for different t1’s in T
are unitarily equivalent representations.
So we can now say that C
(
CP 1
q,c
) ∼= C∗({x∗1x2, x∗1x1}) where C∗({x∗1x2, x∗1x1}) is considered
as in the previous section for the operators x∗1x2, x
∗
1x1 without specifying any value of the t1-
parameter. Thus we conclude that C
(
CP 1
q,c
)
is isomorphic to the pullback T ⊕C(T) T of two
copies of T σ→ C(T) by Proposition 4, and hence is isomorphic to the algebra C
(
S2µc
)
of Podleś
quantum 2-sphere by the result of [8].
We now summarize our conclusion in the following theorem, where the operators X1 :=√
cα + γ and X2 := −q−1
√
cγ + α∗ on `2(Z≥) are respectively the values of the T-families x1
and x2 at t1 = 1 = t2.
Theorem 3. The C∗-algebra C
(
CP 1
q,c
) ∼= C∗({X∗1X2, X
∗
1X1}) for the linear operators X1 :=√
cα+ γ and X2 := −q−1
√
cγ + α∗ on `2(Z≥), and is isomorphic to the pullback
T ⊕C(T) T ≡
{
(T, S) ∈ T ⊕ T : σ(T ) = σ(S)
}
of two copies of the standard Toeplitz C∗-algebra T along the symbol map T σ→ C(T).
We remark that the above change of orthonormal basis v2k
(
t21
)k
v2k and v2k−1
(
t21
)k
v2k−1
is “compatible” and hence works well with the elements x∗1x2, x
∗
1x1, and x∗2x2 of C
(
CP 1
q,c
)
, but is
not suitable for manipulating more fundamental elements like x1 and x2 in C
(
S3q
)
≡ C(SUq(2)).
14 N. Ciccoli and A.J.-L. Sheu
Acknowledgements
N. Ciccoli was partially supported by INDAM-GNSAGA and Fondo Ricerca di Base 2017
“Geometria della quantizzazione”. A.J.-L. Sheu was partially supported by University of Pe-
rugia – Visiting Researcher Program, the grant H2020-MSCA-RISE-2015-691246-QUANTUM
DYNAMICS, and the Polish government grant 3542/H2020/2016/2.
References
[1] Bonechi F., Ciccoli N., Qiu J., Tarlini M., Quantization of Poisson manifolds from the integrability of the
modular function, Comm. Math. Phys. 331 (2014), 851–885, arXiv:1306.4175.
[2] Bonechi F., Ciccoli N., Staffolani N., Tarlini M., On the integration of Poisson homogeneous spaces, J. Geom.
Phys. 58 (2008), 1519–1529, arXiv:0711.0361.
[3] Dijkhuizen M.S., Noumi M., A family of quantum projective spaces and related q-hypergeometric orthogonal
polynomials, Trans. Amer. Math. Soc. 350 (1998), 3269–3296, arXiv:q-alg/9605017.
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demic Press, New York – London, 1972.
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(Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 56–66.
[6] Murphy G.J., C∗-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990.
[7] Podleś P., Quantum spheres, Lett. Math. Phys. 14 (1987), 193–202.
[8] Sheu A.J.-L., Quantization of the Poisson SU(2) and its Poisson homogeneous space – the 2-sphere, Comm.
Math. Phys. 135 (1991), 217–232.
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arXiv:math.OA/9802083.
[10] Sheu A.J.-L., Covariant Poisson structures on complex projective spaces, Comm. Anal. Geom. 10 (2002),
61–78, arXiv:math.SG/9802082.
https://doi.org/10.1007/s00220-014-2050-9
https://arxiv.org/abs/1306.4175
https://doi.org/10.1016/j.geomphys.2008.07.001
https://doi.org/10.1016/j.geomphys.2008.07.001
https://arxiv.org/abs/0711.0361
https://doi.org/10.1090/S0002-9947-98-01971-0
https://arxiv.org/abs/q-alg/9605017
https://doi.org/10.1007/BFb0101178
https://doi.org/10.1007/BF00416848
https://doi.org/10.1007/BF02098041
https://doi.org/10.1007/BF02098041
https://doi.org/10.1090/conm/228/03296
https://arxiv.org/abs/math.OA/9802083
https://doi.org/10.4310/CAG.2002.v10.n1.a4
https://arxiv.org/abs/math.SG/9802082
1 Introduction
2 Nonstandard quantum CPq,c1
3 Invariant subspace decomposition
4 Superfluous circle parameter
References
|
| id | nasplib_isofts_kiev_ua-123456789-210775 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T14:21:04Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ciccoli, Nicola Sheu, Albert Jeu-Liang 2025-12-17T14:36:14Z 2020 Nonstandard Quantum Complex Projective Line. Nicola Ciccoli and Albert Jeu-Liang Sheu. SIGMA 16 (2020), 073, 14 pages 1815-0659 2020 Mathematics Subject Classification: 58B32; 46L85 arXiv:2002.03439 https://nasplib.isofts.kiev.ua/handle/123456789/210775 https://doi.org/10.3842/SIGMA.2020.073 In our attempt to explore how the quantum nonstandard complex projective spaces ℂⁿq,c studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to establish the known identification of (ℂ¹q,c) with the pull-back of two copies of the Toeplitz *-algebra along the symbol map in a more direct way via an operator theoretic analysis, which also provides some interesting non-obvious details, such as a prominent generator of (ℂ¹q,c) being a concrete weighted double shift. N. Ciccoli was partially supported by INDAM-GNSAGA and Fondo Ricerca di Base 2017 "Geometria della quantizzazione". A.J.-L. Sheu was partially supported by the University of Perugia Visiting Researcher Program, the grant H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS, and the Polish government grant 3542/H2020/2016/2. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Nonstandard Quantum Complex Projective Line Article published earlier |
| spellingShingle | Nonstandard Quantum Complex Projective Line Ciccoli, Nicola Sheu, Albert Jeu-Liang |
| title | Nonstandard Quantum Complex Projective Line |
| title_full | Nonstandard Quantum Complex Projective Line |
| title_fullStr | Nonstandard Quantum Complex Projective Line |
| title_full_unstemmed | Nonstandard Quantum Complex Projective Line |
| title_short | Nonstandard Quantum Complex Projective Line |
| title_sort | nonstandard quantum complex projective line |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210775 |
| work_keys_str_mv | AT ciccolinicola nonstandardquantumcomplexprojectiveline AT sheualbertjeuliang nonstandardquantumcomplexprojectiveline |