@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 12, no. 2, 2011

pp. 175-185

Morse theory for C*-algebras: a geometric

interpretation of some noncommutative
manifolds

Vida Milani∗, Ali Asghar Rezaei and Seyed M. H. Mansourbeigi

Abstract

The approach we present is a modification of the Morse theory for uni-

tal C*-algebras. We provide tools for the geometric interpretation of

noncommutative CW complexes. Some examples are given to illustrate

these geometric information. The main object of this work is a classi-

fication of unital C*-algebras by noncommutative CW complexes and

the modified Morse functions on them.

2010 MSC: 06B30, 46L35, 46L85, 55P15, 55U10.

Keywords: C*-algebra, critical points, CW complexes, homotopy equiva-
lence, homotopy type, Morse function, Noncommutative CW

complex, poset, pseudo-homotopy type, *-representation, sim-

plicial complex.

1. Introduction

Morse theory is an approach in the study of smooth manifolds by the tools
from calculus. The classical Morse theory provides a connection between the
topological structure of a manifold M and the homotopy type of critical points
of a function f : M → R (the Morse functions).

On a smooth manifold M, a point a ∈ M is a critical point for a smooth
function f : M → R, if the induced map f∗ : Ta(M) → R is zero. The real
number f(a) is then called a critical value. The function f is a Morse function
if i) all the critical values are distinct and ii) its critical points are non degen-
erate, i.e. the Hessian matrix of second derivatives at the critical points has a

∗Corresponding author.



176 V. Milani, A. A. Rezaei and S. M. H. Mansourbeigi

non vanishing determinant. The number of negative eigenvalues of this Hessian
matrix is the index of f at the critical point. The classical Morse theory states:

Theorem ([14]): There exists a Morse function on any differentiable manifold
and any differentiable manifold has the homotopy type of a CW complex with
one λ-cell for each critical point of index λ.

So once we have information around the critical points of a Morse function
on M, we can reconstruct M by a sequence of surgeries.
A C*-algebraic approach which links operator theory and algebraic geometry, is
obtained via a suitable set of equivalence classes of extensions of commutative
C*-algebras. This provides a functor from locally compact spaces into abelian
groups([7], [12], [15]).

If J and B are two C*-algebras, an extension of B by J is a C*-algebra A
together with morphisms j : J → A and η : A → B such that the following
sequence is exact:

(1.1) 0 −−−−→ J
j

−−−−→ A
η

−−−−→ B

The aim of the extension problem is the characterization of those C*-algebras
A satisfying the above exact sequence. This has something to do with algebraic
topology techniques. In the construction of a CW complex, if Xk−1 is a suitable
subcomplex, Ik the unit ball and Sk−1 its boundary, then the various solutions
for the extension problem of C(Xk−1) by C0(I

k −Sk−1) correspond to different
ways of attaching Ik to Xk−1 along S

k−1, via an attaching map ϕk : S
k−1 →

Xk−1 which identify points x ∈ S
k−1 with their image ϕk(x) in the disjoint

union Xk−1 ∪ I
k.

After the construction of noncommutative geometry [1], there have been at-
tempts to formulate the classical tools of differential geometry and topology
in terms of C*-algebras (in some sense the dualization of the notions, [3], [4],
[11]). The dual concept of CW complexes , with some regards, is the notion
of noncommutative CW complexes ([7] and [15]). The approach of this paper
is the geometric study of these structures. So many works have been done on
the combinatorial structures of noncommutative simplicial complexes and their
decompositions, for example [2], [6], [9], [10]. Following these works, together
with some topological constructions , we show how a modification of the clas-
sical Morse theory to the level of C*-algebras will provide an innovative way
to explain the geometry of noncommutative CW complexes through the criti-
cal ideals of the modified Morse function. This leads up to some classification
theory.

This paper is prepared as follows. After introducing the notion of primitive
spectrum of a C*-algebra, we will proceed the topological structure in detail
and present some examples. Then we will study the noncommutative CW
complexes and interpret their geometry by introducing the modified Morse
function. All these provide tools for the modified Morse theory for C*-algebras.
The last section is devoted to the proof of the following theorem:



Morse theory for C*-algebras 177

Main Theorem: Every unital C*-algebra with an acceptable Morse function
on it is of pseudo-homotopy type as a noncommutative CW complex, having a
k-th decomposition cell for each critical chain of order k.

2. The Structure of the Primitive Spectrum

The technique we follow to link the geometry, topology and algebra is the
primitive spectrum point of view. In fact as we will see in our case it is a
promising candidate for the noncommutative analogue of a topological manifold
M. We review some preliminaries on the primitive spectrum. Details can be
found in [5], [11], [13].

Let A be a unital C*-algebra. The primitive spectrum of A is the space of
kernels of irreducible *-representations of A. It is denoted by Prim(A). The
topology on this space is given by the closure operation as follows:

For any subset U ⊆ Prim(A), the closure of U is defined by

(2.1) U := {I ∈ Prim(A) :
⋂

J∈U

J ⊂ I}

Obviously U ⊆ U. This operation defines a topology on Prim(A) (the hull-
kernel topology), making it into a T0-space ([8]).

Definition 2.1. A subset U ⊂ Prim(A) is called absorbing if it satisfies the
following condition:

(2.2) I ∈ U,I ⊆ J ⇒ J ∈ U.

Lemma 2.2. The closed subsets of Prim(A) are exactly its absorbing subsets.

Proof. It is clear from the definition of closed sets. �

In the special case, when M is a compact topological space, and A = C(M)
is the commutative unital C*-algebra of complex continuous functions on M,
a homeomorphism between M and prim(A) is obtained in the following way.
For each x ∈ M let

Ix := {f ∈ A : f(x) = 0};

Ix is a closed maximal ideal of A. It is in fact the kernel of the evaluation map

(ev)x :A −→ C

f 7−→ f(x).

Now

(2.3) I : M → Prim(A)

defined by I(x) := Ix is the desired homeomorphism.
Let A be an arbitrary unital C*-algebra. To each I ∈ Prim(A), there

corresponds an absorbing set

WI := {J ∈ Prim(A) : J ⊇ I},

and an open set
OI := {J ∈ Prim(A) : J ⊆ I},



178 V. Milani, A. A. Rezaei and S. M. H. Mansourbeigi

containing I.
Being a T0-space, Prim(A) can be made into a partially ordered set (poset)

by setting,

I < J ⇔ I ⊂ J for I,J ∈ Prim(A),

Remark 2.3. The following statements are equivalent:
i) I ⊆ J.
ii) OI ⊆ OJ.
iii) WI ⊇ WJ.

The topology of Prim(A) can be given equivalently by means of this partial
order,

I < J ⇔ J ∈ {I},

where {I} is the closure of the one point set {I}.
Since A is unital, Prim(A) is compact([8]). Let Prim(A) =

⋃n
i=0 OIi be a

finite open covering.
In general, let us suppose we have a topological space M together with an

open covering U = {Ui} which is also a topology for M. An equivalence relation
on M is set by declaring that any two points x,y ∈ M are equivalent if every
open set Ui containing either x or y contains the other too. In this way the
quotient space of M is made into a finite lattice.

In the same way an equivalence relation on Prim(A) is given by

I ∼ J ⇔ (J ∈ OI ⇔ I ∈ OJ).

This is of course the trivial relation I ∼ J ⇔ I = J.
In each OIi choose one Ii with respect to the above equivalence relation.

Since I ⊆ J implies OI ⊆ OJ, OIis can be chosen so that Ii 6= Ij for i 6= j. Let
I0,I1, ...,In be chosen in this way so that Prim(A) is made into a finite lattice
for which the points are the equivalence classes of [I0], ..., [In]. For simplicity
we show each class [Ii] by its only representative Ii. Let

Ji0,...,ik := Ii0 ∩ ... ∩ Iik,

where 1 ≤ i0, ..., ik ≤ n,1 ≤ k ≤ n.
Set

Wi0,...,ik := {J ∈ Prim(A) : J ⊇ Ji0,...,ik}.

This is a closed subset of Prim(A).
In what follows we see that the above construction makes it possible to obtain

a cell complex decomposition for Prim(C(X)) when X has a CW complex
structure. In fact the closed sets Wi0,...,ik corresponding to Ji0,...,ik play the
role of chains in the construction.

Remark 2.4. If Ji0,...,ik = 0 for some 1 ≤ i0, ..., ik ≤ n, 1 ≤ k ≤ n, then
Wi0,...,ik = Prim(A). Also for each pair of indices (i0, ..., it) ,σ(i0, ..., it+m),

Wi0,...,it ⊆ Wσ(i0,...,it+m)

where σ is a permutation on t + m elements and 1 ≤ i0, ..., it+m ≤ n.



Morse theory for C*-algebras 179

Remark 2.5. A sequence

X0 ⊂ X1 ⊂ ... ⊂ Xn = X

is an n-dimensional CW complex structure for a compact topological space X,
where X0 is a finite discrete space consisting of 0-cells, and for k = 1, ...,n each
k-skeleton Xk is obtained by attaching λk number of k-disks to Xk−1 via the
attaching maps

ϕk :
⋃

λk

Sk−1 → Xk−1.

In other words

(2.4) Xk =
Xk−1

⋃

(∪λkI
k)

x ∼ ϕk(x)
:= Xk−1

⋃

ϕk

(∪λkI
k)

where Ik := [0,1]k and Sk−1 := ∂Ik. The quotient map is denoted by

ρ : Xk−1
⋃

(∪λkI
k) → Xk.

For more details see [12].

Now let

X0 ⊂ X1 ⊂ ... ⊂ Xn = X

be an n-dimensional CW complex structure for the compact space X. A cell
complex structure is induced on Prim(C(X)) by the following procedure:

Let Ak = C(Xk), k = 0,1, ...,n. Set A = C(X) = C(Xn) = An. Let
I : X → Prim(C(X)) be the homeomorphism of relation (4). For each k-cell
Ck in the k-skeleton Xk, let

ICk =
⋂

x∈Ck

Ix = {f ∈ A : f(x) = 0;x ∈ Ck},

for 0 ≤ k ≤ n . By considering the restriction of functions on X to Xk, ICk
will be an ideal in Ak.

Definition 2.6. In the above notations, ICk is called a k-ideal in A (or Ak)
and the restriction of its corresponding closed set Wi0,...,ik in Prim(Ak), i.e.

Wi0,...,ik = {J ∈ Prim(Ak) : J ⊇ ICk }

is called a k-chain.

In the following two examples we identify the k-ideals and the k-chains for
the CW complex structures of the closed interval [0,1] and the 2-torus S1 ×S1.

Example 2.7. Let X0 = {0,1} and X1 = [0,1] be the zero and the one skeleton
for a CW complex structure of [0,1]. Then we have A0 = C(X0) ≃ C ⊕ C and
A = A1 = C(X1). The 0-ideals I0 and I1 and their corresponding 0-chains W0
and W1 are as follow:

I0 = {f ∈ A0 : f(0) = 0} ≃ C,I1 = {f ∈ A0 : f(1) = 0} ≃ C,

W0 = {J ∈ Prim(A0) : J ⊇ I0} ≃ {0},W1 = {J ∈ Prim(A0) : J ⊇ I1} ≃ {1}.



180 V. Milani, A. A. Rezaei and S. M. H. Mansourbeigi

Corresponding to the 1-chain C1 = [0,1], the only 1-ideal is

I =
⋂

x∈C1

Ix = {f ∈ A : f(x) = 0;x ∈ [0,1]} = 0,

with the corresponding 1-chain

WI = {J ∈ Prim(A) : J ⊇ I} = Prim(A) ≃ [0,1].

Example 2.8. Let

X0 = {0},X1 = {α,β},X2 = T
2 = S1 × S1

be the skeletons of a CW complex structure for the 2-torus T 2, where α,β
are homeomorphic images of S1 (closed curves with the origin 0). Let A0 =
C(X0) = C , A1 = C(X1) and A = A2 = C(T

2). The 0-ideal and its corre-
sponding 0-chain are as follow:

I0 = {f ∈ A0 : f(0) = 0},

W0 = {J ∈ Prim(A0) : J ⊇ I0} ≃ Prim(A0) = {0}.

Also the 1-ideals I1,I2 and 1-chains WI1,WI2 are

I1 = {f ∈ A1 : f(α) = 0} = ∩x∈αIx ≃ C,

I2 = {f ∈ A1 : f(β) = 0} = ∩x∈βIx ≃ C,

WI1 = {J ∈ Prim(A1) : J ⊇ I1} ≃ {α},

WI2 = {J ∈ Prim(A1) : J ⊇ I2} ≃ {β}.

Finally the only 2-ideal and its corresponding 2-chain are

I = {f ∈ A : f(T 2) = 0} ≃ 0,

WI = {J ∈ Prim(A) : J ⊇ I} ≃ T
2.

3. The Noncommutative CW Complexes

In this section we see how the construction of the primitive spectrum of the
previous section helps us to study the noncommutative CW complexes.
For a continuous map φ : X → Y between compact topological spaces X and
Y , the C*-morphism induced on their associated C*-algebras is denoted by
C(φ) : C(Y ) → C(X) which is defined by C(φ)(g) := g ◦ φ for g ∈ C(Y ).

Definition 3.1. Let A1, A2 and C be C*-algebras. A pull back for C via
morphisms α1 : A1 → C and α2 : A2 → C is the C*-subalgebra of A1 ⊕ A2
denoted by PB(C,α1,α2) defined by

PB(C,α1,α2) := {a1 ⊕ a2 ∈ A1 ⊕ A2 : α1(a1) = α2(a2)}.

For any C*-algebra A, let

SnA := C(Sn → A),InA := C([0,1]n → A),In0 A := C0((0,1)
n → A),

where Sn is the n-dimensional unit sphere.
We review the definition of noncommutative CW complexes from [7], [15].



Morse theory for C*-algebras 181

Definition 3.2. A 0-dimensional noncommutative CW complex is any finite
dimensional C*-algebra A0. Recursively an n-dimensional noncommutative
CW complex is any C*-algebra appearing in the following diagram

(3.1)

0 −−−−→ In0 Fn −−−−→ An
π

−−−−→ An−1 −−−−→ 0
∥

∥

∥





y

fn





y

ϕn

0 −−−−→ In0 Fn −−−−→ I
nFn

δ
−−−−→ Sn−1Fn −−−−→ 0

Where the rows are extensions, An−1 an (n − 1)-dimensional noncommutative
CW complex, Fn some finite (linear) dimensional C*-algebra of dimension λn, δ
the boundary restriction map, ϕn an arbitrary morphism (called the connecting
morphism), for which

(3.2) An = PB(S
n−1Fn,δ,ϕn) := {(α,β) ∈ I

nFn ⊕ An−1 : δ(α) = ϕn(β)},

and fn and π are respectively projections onto the first and second coordinates.
With these notations {A0, ...,An} is called the noncommutative CW complex

decomposition of dimension n for A = An.
For each k = 0,1, ...,n, Ak is called the k-th decomposition cell.

Proposition 3.3. Let X be an n-dimensional CW complex containing cells of
each dimension k = 0, ...,n. Then there exists a noncommutative CW complex
decomposition of dimension n for A = C(X).

Conversely if {A0, ...,An} be a noncommutative CW complex decomposition
for the C*-algebra A such that Ais (i = 0, ..,n) are unital, Then there exists
an n-dimensional CW complex structure on Prim(A).

Proof. Let
X0 ⊂ X1 ⊂ ... ⊂ Xn = X

be a CW complex structure for X where Xk(for each k ≤ n) is the k-skeleton
defined in relation (2.4). Let Ak = C(Xk) and i :

⋃

λk
Sk−1 →

⋃

λk
Ik ,

ϕk :
⋃

λk
Sk−1 → Xk−1 be the injection and attaching maps respectively, and

C(i) and C(ϕk) be their induced maps. Let

PB := PB(C(
⋃

λk

Sk−1),C(ϕk),C(i)),

and define

θ :C(Xk) −→ PB

f 7−→ (f ◦ ρ)1 ⊕ (f ◦ ρ)2,

Where (f ◦ ρ)1 and (f ◦ ρ)2 are the restrictions of (f ◦ ρ) to
⋃

λk
Ik and Xk−1

respectively.
θ is well defined since C(ϕk)((f ◦ ρ)1) = C(i)((f ◦ ρ)2). Also C(ϕk)(h) =

C(i)(g)for (h,g) ∈ PB, and so if f ∈ C(Xk) be defined by

f(y) =

{

g(y) y ∈
⋃

λk
Ik

h(y) y ∈ Xk−1



182 V. Milani, A. A. Rezaei and S. M. H. Mansourbeigi

then θ(f) = (h,g). Now the noncommutative CW complex decomposition of
dimension n for A = C(X) is given by {A0, ...,An}.

Conversely let An be as in (3.2), and

ϕ
∗
n : S

n−1 → Prim(An−1)

be the attaching map induced by the connecting morphism

ϕn : An−1 → S
n−1Fn

of diagram (3.1). Then using the notation in relation (2.4),

Prim(An) = Prim(An−1)
⋃

ϕ∗
n

In.

We note that ϕ∗n = C(ϕn). Furthermore for k ≤ n, ϕ
∗
k(S

k−1) is a closed
subset of Prim(Ak−1). It is of the form

ϕ∗k(S
k−1) = {J ∈ Prim(Ak−1) : Ik−1 ⊂ J}

for some ideal Ik−1 in Ak−1. In fact Ik−1 =
⋂

J∈ϕ∗
k
(Sk−1) J. So Prim(A)

has an n-dimensional CW-structure with Xk = Prim(Ak) as its k-skeleton for
k = 0, ...,n. �

Example 3.4. Following the notations of diagram (3.1), a 1-dimensional non-
commutative CW complex decomposition for A = C([0,1]) = C(I) is given by

A0 = C ⊕ C,A1 = C([0,1]).

Let F1 = C, then

I10F1 = C0((0,1)),I
1F1 = C([0,1]),S

0F1 = C ⊕ C

and ϕ1 = id. Also

C(I) = PB(S0F1,δ,ϕ1) = {f⊕(λ⊕µ) ∈ C([0,1])⊕(C⊕C) : f(0) = λ,f(1) = µ}

together with the maps

π :A1 −→ A0

f ⊕ (λ ⊕ µ) 7−→ λ ⊕ µ,

f1 :A1 −→ I
1F1 = A1

f ⊕ (λ ⊕ µ) 7−→ f,

and

δ :I1F1 = A1 −→ S
0
F1 = C ⊕ C

f 7−→ f(0) ⊕ f(1).



Morse theory for C*-algebras 183

4. Modified Morse Theory on C*-Algebras

In this section, following the study of the Morse theory for the cell complexes
in [2], [6], [9], [10], with some modification, we define the Morse function for
the C*-algebras and state and prove the modified Morse theory for the non-
commutative CW complexes. This is a classification theory in the category of
C*-algebras and noncommutative CW complexes.

Definition 4.1. If A, B are two C*-algebras, two morphisms α,β : A → B
are homotopic, written α ∼ β,if there exists a family {Ht}t∈[0,1] of morphisms
Ht : A → B such that for each a ∈ A the map t 7→ Ht(a) is a norm continuous
path in B with H0 = α and H1 = β.The C*-algebras A and B are said to have
the same homotopy type, if there exists morphisms ϕ : A → B and ψ : B → A
such that ϕ ◦ ψ ∼ idB and ψ ◦ ϕ ∼ idA. In this case the morphisms ϕ and ψ
are called homotopy equivalence.

Definition 4.2. Let A and B be unital C*-algebras. We say A is of pseudo-
homotopy type as B if C(Prim(A)) and B have the same homotopy type.

Remark 4.3. In the case of unital commutative C*-algebras, by the GNS
construction [11], C(Prim(A)) = A, . So the notions of pseudo-homotopy type
and the same homotopy type are equivalent.

For a unital C*-algebra A let

Σ = {Wi1,...,ik}1≤i1,...,ik≤n,1≤k≤n

be the set of all k-chains (k = 1, ...,n) in Prim(A), and

Γ = {Ii1,...,ik}1≤i1,...,ik≤n,1≤k≤n

be the set of all k-ideals corresponding to the k-chains of Σ for k = 1, ...,n.

Lemma 4.4. Γ is an absorbing set.

Proof. This follows from the fact that for each Ii1,...,ik ∈ Γ, J ∈ Prim(A) the
relation Ii1,...,ik ⊂ J is equivalent to J = Ii1,...,it for some t ≤ k, meaning
J ∈ Γ. �

Definition 4.5. Let f : Σ → R be a function. The k-chain Wk = Wi1,...,ik is
called a critical chain of order k for f, if for each (k+1)-chain Wk+1 containing
Wk and for each (k − 1)-chain Wk−1 contained in Wk, we have

f(Wk−1) ≤ f(Wk) ≤ f(Wk+1).

The corresponding ideal Ik to Wk is called the critical ideal of order k.

Definition 4.6. Let f has a critical chain of order k. We say f is an acceptable
Morse function, if it has a critical chain of order i, for all i ≤ k.

Definition 4.7. A function f : Σ → R is called a modified Morse function on
the C*-algebra A, if for each k-chain Wk in Σ, there is at most one (k+1)-chain
Wk+1 containing Wk and at most one (k-1)-chain Wk−1 contained in Wk, such
that

f(Wk+1) ≤ f(Wk) ≤ f(Wk−1).



184 V. Milani, A. A. Rezaei and S. M. H. Mansourbeigi

Here we state the discrete Morse theory of Forman from [9], and state and
prove our modification of it.

Theorem (Discrete Morse Theory): Suppose ∆ is a simplicial complex
with a discrete Morse function. Then ∆ is homotopy equivalent to a CW
complex with one cell of dimension p for each critical p-simplex.

Lemma 4.8. If f is an acceptable modified Morse function on A, then Prim(A)
is homotopy equivalent to a CW complex with exactly one cell of dimension p
for each critical chain of order p.

Proof. In the discrete Morse theory it suffices to substitute Γ for the simplicial
complex ∆. Since Γ is absorbing, it satisfies the properties of the simplicial
complex ∆ in the discrete Morse theorem. It follows that Prim(A) is homotopy
equivalent to a CW complex with exactly one cell of dimension p for each critical
chain of order p. �

Now we state our main theorem which provides a condition for a unital C*-
algebra to admit a noncommutative CW-complex decomposition. This is what
we call the geometric condition.

Theorem 4.9. Every unital C*-algebra A with an acceptable modified Morse
function f on it, is of pseudo-homotopy type as a noncommutative CW complex
having a k-th decomposition cell for each critical chain of order k.

Proof. If A is a unital C*-algebra, then the acceptable modified Morse function
on A is in fact a function on the simplicial complex of all k-ideals in Prim(A)
(a function on Γ). From lemma 4.8 we conclude that Prim(A) is homotopy
equivalent to a finite dimensional CW complex Ω. From the proposition 3.3
there is a noncommutative CW-complex decomposition for C(Ω) making it into
a noncommutative CW complex. Now C(Prim(A))and C(Ω) have the same
homotopy type, which means A is of psuedo-homotopy type of the noncommu-
tative CW complex C(Ω). Furthermore since f is acceptable, from the proof of
proposition 3.3 it follows that if there exists a critical k-chain for f, then there
exists C*-algebras Ai for each i ≤ k so that {A0, ...,Ak} is a noncommutative
CW complex decomposition for C(Prim(A)) yielding a noncommutative CW
complex decomposition for C(Ω) . �



Morse theory for C*-algebras 185

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(Received February 2011 – Accepted July 2011)

V. Milani (v-milani@cc.sbu.ac.ir, vmilani3@math.gatech.edu)
Dept. of Math., Faculty of Math. Sci., Shahid Beheshti University, Tehran,
Iran.
School of Mathematics, Georgia Institute of Technology, Atlanta GA, USA.

A. A. Rezaei (A Rezaei@sbu.ac.ir)
Dept. of Math., Faculty of Math. Sci., Shahid Beheshti University, Tehran,
Iran.

S. M. H. Mansourbeigi (s.mansourbeigi@ieee.org)
Dept. of Electrical Engineering, Polytechnic University, NY, USA.


	Morse theory for C*-algebras: a geometric[2pt] interpretation of some noncommutative[2pt] manifolds. By V. Milani, A. A. Rezaei and S. M. H. Mansourbeigi