Glimm ideals of the minimal tensor product of
C∗ -algebras
Dave McConnell
22nd January 2013
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
Outline
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm ideals of the minimal tensor product of C∗ -algebras
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Primitive ideals
Let A be a C∗ -algebra and Prim(A) the set of kernels of irreducible
representations of A.
I
For S ⊆ Prim(A), define k(S) = ∩{P : P ∈ S}, the kernel of
S.
I
For I an ideal of A, define hull(I ) = {P ∈ Prim(A) : P ⊇ I }
I
Hull-kernel topology on Prim(A) : S ⊆ Prim(A) has closure
S̄ = hull(k(S)), with S homeomorphic to Prim(A/k(S)),
I
Any I / A, we have I = k(hull(I )).
I
Open sets are thus
U(I ) = {P ∈ Prim(A) : P 6⊇ I }
for I an ideal of A, and U(I ) homeomorphic to Prim(I ).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Properties of the hull-kernel topology
I
Prim A is a T0 space,
I
Not Hausdorff in general,
I
For P ∈ Prim(A), {P} = hull(P). So {P} closed iff P
maximal (iff A/P simple).
I
Prim(A) is locally compact, and compact if A is unital,
I
A commutative ⇒ Prim(A) = P(A) = χ(A), hence is a
locally compact Hausdorff space.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Factor representations
I
Call a representation π : A → B(H) factorial if π(A)00 is a
factor,
I
Clearly every irreducible representation of A is factorial,
I
Denote by Fac(A) the set of kernels of factorial
representations of A, then Prim(A) ⊆ Fac(A),
I
Fac(A) is a topological space in the hull-kernel topology,
I
Prim(A) is dense in Fac(A), with equality when A separable.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
The complete regularisation of a topological space
A space X is said to be completely regular (Tychonoff) if it is
Hausdorff, and given a closed F ⊆ X and a point y ∈ X \F , then
there is f ∈ C (X ) with f (y ) = 1 and f (F ) = {0}.
Theorem
Given any topological space X there is a completely regular space
ρX and a continuous surjection ρ : X → ρX such that f 7→ f ◦ ρ is
a ∗-isomorhpism of C b (ρX ) onto C b (X ).
I
Define an equivalence relation ≈ on X : x1 ≈ x2 iff
f (x1 ) = f (x2 ) for all f ∈ C b (X ),
I
Let ρX = X / ≈ and ρ : X → ρX be the quotient map,
I
Topology τcr induced by the functions in C b (X ),
I
If X is completely regular then ρX = X .
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Universal property: the Tychonoff functor
I
I
Suppose X and Y are topological spaces and φ : X → Y a
continuous map,
φ defines a continuous map φρ : ρX → ρY such that
φρ (ρX (x)) = ρY (φ(x)), that is,
X
ρX
ρX
I
φ
/Y
φρ
ρY
/ ρY
commutes.
X 7→ ρX defines a covariant functor from the category of
topological spaces and continuous maps to the subcategory of
completely regular spaces.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm(A)
Let A be a C∗ -algebra.
I As a topological space, Glimm(A) is usually defined as
ρ(Prim(A)),
I Equivalently Glimm(A) = ρ(Fac(A)) (Kaniuth),
I For P ∈ Prim(A), I ∈ Fac(A) such that
ρ(P) = ρ(I ) = p ∈ Glimm(A) we define the corresponding
Glimm ideal Gp of A as
\
Gp =
{P ∈ Prim(A) : ρ(P) = p} = k([P])
\
=
{I ∈ Fac(A) : ρ(I ) = p} = k([I ])
I
It follows that the ≈-equivalence classes satisfy
[P] = hull(Gρ(P) ) for P ∈ Prim(A), and [I ] = hullf (Gρ(I ) ) for
I ∈ Fac(A).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm(A ⊗ B)
I
Dauns and Hofmann: can represent A as the section algebra
of a C∗ -bundle over Glimm(A), with fibres
{A/Gp : p ∈ Glimm(A)},
I
ZM(A) ≡ C b (Glimm(A)) ≡ C (βGlimm(A)).
Question: for the minimal tensor product of C∗ -algebras A and B,
can we determine Glimm(A ⊗ B) in terms of Glimm(A) and
Glimm(B)
1. as a topological space,
2. as a collection of ideals of A ⊗ B.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
Outline
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm ideals of the minimal tensor product of C∗ -algebras
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
The minimal tensor product
Let A and B be C∗ -algebras.
I
We denote by A B the algebraic tensor product of A and B,
and by A ⊗ B their minimal C∗ -tensor product,
I
For ∗-homomorphisms π : A → C and σ : B → D, let
π σ : A B → C D be the ∗-homomorphism defined via
(π σ)(a ⊗ b) = π(a) ⊗ σ(b)
on elementary tensors a ⊗ b, and π ⊗ σ : A ⊗ B → C ⊗ D the
∗-homomorphism extending π ⊗ σ,
I
Recall that if π and σ are injective (resp. surjective) then so is
π ⊗ σ.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Ideals of A ⊗ B, Property (F)
For any C∗ -algebra C let Id0 (C ) be the set of all proper
norm-closed 2-sided ideals of C
I If (I , J) ∈ Id0 (A) × Id0 (B) and qI : A → A/I , qJ : B → B/J
the quotient maps then qI qJ : A B → (A/I ) (B/J) has
ker(qI qJ ) = I B + A J,
I
I
I
which has closure I ⊗ B + A ⊗ J in A ⊗ B by injectivity.
Alternatively, we may consider the kernel of
qI ⊗ qJ : A ⊗ B → (A/I ) ⊗ (B/J),
Clearly ker(qI ⊗ qJ ) ⊇ I ⊗ B + A ⊗ J, but this inclusion may
be strict,
If ker(qI ⊗ qJ ) = I ⊗ B + A ⊗ J for all (I , J) ∈ Id0 (A) × Id0 (B)
we say that A ⊗ B satisfies Tomiyama’s property (F).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Embedding Id0 (A) × Id0 (B) in Id0 (A ⊗ B)
We define a map Φ : Id0 (A) × Id0 (B) → Id0 (A ⊗ B) via
Φ(I , J) = ker(qI ⊗ qJ ).
I
I
The restriction of Φ to Prim(A) × Prim(B) is a
homeomorphism onto its image, which is a dense subset of
Prim(A ⊗ B) (Wulfsohn),
The restriction of Φ to Fac(A) × Fac(B) is a homeomorphism
onto a dense subset of Fac(A ⊗ B) (Lazar).
Theorem
(Lazar, 2010) Let Y be a T1 -space and
f : Prim(A) × Prim(B) → Y a continuous map. Then there is a
unique continuous map f¯ : Prim(A ⊗ B) → Y extending f ◦ Φ−1 .
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
Outline
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm ideals of the minimal tensor product of C∗ -algebras
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Prim(A), Fac(A) and Glimm(A)
I
I
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Complete regularisation map
ρA×B : Prim(A) × Prim(B) → ρ(Prim(A) × Prim(B)) is
continuous,
Lazar: there is a unique continuous map
ρA×B : Prim(A ⊗ B) → ρ(Prim(A) × Prim(B)) such that
ρA×B ◦ Φ = ρA×B
I
Universal property: ρA×B factors through Glimm(A ⊗ B) to
give a continuous (surjective) map
ψ : Glimm(A ⊗ B) → ρ(Prim(A) × Prim(B)) with
ψ ◦ ρα = ρA×B ,
where ρα : Prim(A ⊗ B) → Glimm(A ⊗ B) is the complete
regularisation map.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Proof that ψ is a homeomorphism
ψ is a homeomorphism ⇔ the ∗-homomorphism
ψ ∗ : C b (ρ(Prim(A) × Prim(B))) → C b (Glimm(A ⊗ B)),
ψ ∗ (g ) = g ◦ ψ
is surjective. Take f ∈ C b (Glimm(A ⊗ B)), then
f ◦ ρα ◦ Φ ∈ C b (Prim(A) × Prim(B)).
∃ g ∈ C b (ρ(Prim(A) × Prim(B))) with g ◦ ρA×B = f ◦ ρα ◦ Φ. It
follows that
f ◦ ρα = g ◦ ρA×B = g ◦ ψ ◦ ρα ,
so that f = g ◦ ψ = ψ ∗ (g ).Hence ψ ∗ is surjective.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
We now have a homeomorphism
ψ : Glimm(A ⊗ B) → ρ(Prim(A) × Prim(B)) determined by
(ψ ◦ ρα ◦ Φ)(P, Q) = ρA×B (P, Q)
for all (P, Q) ∈ Prim(A) × Prim(B).
I
Question: Is ρ(Prim(A) × Prim(B)) (homeomorphic to) the
product space Glimm(A) × Glimm(B), i.e. is
ρ(Prim(A) × Prim(B)) = ρ(Prim(A)) × ρ(Prim(B))?
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
The complete regularisation of a product space
Let X and Y be topological spaces.
I
Relationship between ρ(X × Y ) and ρ(X ) × ρ(Y ) studied by
Morita, Ishii.
I
It is easily seen that there is a bijection
ρ(X × Y ) → ρ(X ) × ρ(Y ) sending
ρX ×Y (x, y ) 7→ (ρX (x), ρY (y )),
that is, (x1 , y1 ) ≈ (x2 , y2 ) iff x1 ≈ x2 and y1 ≈ y2 .
I
f ∈ C b (X ), g ∈ C b (Y ) then (x, y ) 7→ f (x)g (y ) is continuous
on X × Y .
I
This shows that the topology on ρ(X ) × ρ(Y ) is weaker.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Complete regularisation of Prim(A) × Prim(B)
In the case of two C∗ -algebras A and B we may thus consider
ρ(Prim(A) × Prim(B)) = (Glimm(A) × Glimm(B), τcr )
where τcr is the topology induced by C b (Prim(A) × Prim(B)),and
ρ(Prim(A)) × ρ(Prim(B)) = (Glimm(A) × Glimm(B), τp ),
where τp is the product topology, and τp ≤ τcr .
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Products of primitive ideal spaces
I
I
Characterisation due to Ishii of spaces X for which
ρ(X × Y ) = ρ(X ) × ρ(Y ) for any space Y ,
We establish sufficient conditions on A for which τcr = τp on
Glimm(A) × Glimm(B) for any C∗ -algebra B:
1. Prim(A) is compact (e.g. A unital),
2. A is σ-unital and Glimm(A) is locally compact (iff Glimm(A)
first countable),
3. The complete regularisation map ρA is open (e.g. Prim(A)
Hausdorff).
I
On the other hand, local compactness of Glimm(A) is always
a necessary condition.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm(A ⊗ B) in terms of Glimm(A) × Glimm(B)
Theorem
There is a homeomorphism
ψ : Glimm(A ⊗ B) → (Glimm(A) × Glimm(B), τcr ) determined by
(ψ ◦ ρα ◦ Φ)(P, Q) = (ρA (P), ρB (Q))
for all (P, Q) ∈ Prim(A) × Prim(B). Hence ψ defines a
continuous bijection onto (Glimm(A) × Glimm(B), τp ).
I
Characterises Glimm(A ⊗ B) completely as a topological
space (despite not knowing Prim(A ⊗ B)).
I
Not sufficient to determine the Glimm ideals of (A ⊗ B).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
A continuous map Fac(A ⊗ B) → Fac(A) × Fac(B)
For I ∈ Id0 (A ⊗ B) let
I A = {a ∈ A : a ⊗ B ⊆ I }
IB
= {b ∈ B : A ⊗ b ⊆ I }
Define a map Ψ : Id0 (A ⊗ B) → Id0 (A) × Id0 (B) via
Ψ(I ) = (I A , I B )
For all (I , J) ∈ Id0 (A) × Id0 (B) we have
1. Ψ ◦ Φ(I , J) = (I , J) ,
2. Ψ(I ⊗ B + A ⊗ J) = (I , J).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Properties of Ψ
I
Ψ maps Fac(A ⊗ B) onto Fac(A) × Fac(B) (Guichardet),
I
Not known in general whether Ψ maps
Prim(A ⊗ B) → Prim(A) × Prim(B),
I
Hence we will base the definition of Glimm(A) on the
complete regularisation of Fac(A),
I
For all (I , J) ∈ Id0 (A) × Id0 (B) we have
Ψ−1 (hullf (I ) × hullf (J)) = hullf (I ⊗ B + A ⊗ J),
where hullf denotes the hull of factorial ideals,
I
It follows that Ψ is continuous
Fac(A ⊗ B) → Fac(A) × Fac(B).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Recall that for I ∈ Fac(A), the Glimm ideal corresponding to [I ]
(or to ρA (I )) is given by
\
GI = k([I ]) = {J ∈ Fac(A) : J ≈ I },
and the equivalence class satisfies
[I ] = hullf (GI ) = {J ∈ Fac(A) : J ⊇ GI }.
For M ∈ Fac(A ⊗ B), we have (Φ ◦ Ψ)(M) ⊇ M. Hence
(Φ ◦ Ψ)(M) ∈ {M}, so that (Φ ◦ Ψ)(M) ≈ M.
Proposition
Let (I , J) ∈ Fac(A) × Fac(B) and M ∈ Fac(A ⊗ B). Then
M ≈ Φ(I , J) ⇔ Ψ(M) ≈ (I , J). It follows that
M ⊇ GΦ(I ,J) ⇔ M A ⊇ GI and M B ⊇ GJ
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Let ∆ : Id0 (A) × Id0 (B) → Id0 (A ⊗ B) be given by
∆(I , J) = I ⊗ B + A ⊗ J
Theorem
For all (I , J) ∈ Fac(A) × Fac(B), the Glimm ideal of A ⊗ B
corresponding to [Φ(I , J)] is given by
GΦ(I ,J) = ∆(GI , GJ ),
where GI and GJ are the Glimm ideals corresponding to [I ] and [J]
respectively. Hence, regarding the Glimm spaces as sets of ideals,
the homeomorphism ψ satisfies
1. ψ = Ψ,
2. ψ −1 = ∆.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Proof
I
Recall that Ψ−1 (hullf (GI ) × hullf (GJ )) = hullf (∆(I , J)),
I
By the proposition,
M ∈ [Φ(I , J)] = hullf (GΦ(I ,J) ) ⇔ Ψ(M) ∈ hullf (GI ) × hullf (GJ )
⇔ M ∈ hullf (∆(GI , GJ ))
I
It follows that [Φ(I , J)] = hullf (∆(GI , GJ )), and so
GΦ(I ,J) := k ([Φ(I , J)]) = k (hullf (∆(GI , GJ )))
= ∆(GI , GJ )
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm ideals of A ⊗ B
I
The space of Glimm ideals of A ⊗ B thus consists of
{G ⊗ B + A ⊗ H : (G , H) ∈ Glimm(A) × Glimm(B)},
with topology homeomorphic to
(Glimm(A) × Glimm(B), τcr ) in the natural way.
I
Extends a Theorem of Kaniuth by eliminating the assumption
of (F).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
Outline
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Glimm ideals of the minimal tensor product of C∗ -algebras
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Sectional representation
Consider the fibred space A = {A/Gp : p ∈ Glimm(A)}:
I
I
Each element a ∈ A defines a section â : Glimm(A) → A
such that â(p) = a + Gp ,
Dauns-Hofmann: there is a topology on A such that
1. the restriction of this topology to each fibre A/Gp is the norm
topology, and
2. the collection {â : a ∈ A} consists of all continuous sections
Glimm(A) → A,
3. norm functions p 7→ kâ(p)k upper semicontinuous on
Glimm(A) for all a ∈ A.
I
p 7→ kâ(p)k continuous for all a ∈ A iff ρA is an open map
(Lee). In this case we say that A defines a continuous
C∗ -bundle over Glimm(A).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Sectional representation of A ⊗ B
As a consequence of the previous two Theorems, we can represent
A ⊗ B as the section algebra of an upper semicontinuous bundle
s.t.:
I
The base space is given by (Glimm(A) × Glimm(B), τcr ),
I
Fibre algebras given by
A⊗B
: (p, q) ∈ Glimm(A) × Glimm(B)
∆(Gp , Gq )
I
If in fact ∆(Gp , Gq ) = Φ(Gp , Gq ) for all
(p, q) ∈ Glimm(A) × Glimm(B), then the fibre algebras are
given by (A/Gp ) ⊗ (B/Gq ).
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Continuous C∗ -bundles
Corollary
Suppose that A and B are C∗ -algebras such that Φ = ∆ on
Glimm(A) × Glimm(B). Then
ρα : Prim(A ⊗ B) → Glimm(A ⊗ B) is open if and only if
ρA : Prim(A) → Glimm(A) and ρB : Prim(B) → Glimm(B) are
open.
Hence the Dauns-Hofmann representation of A ⊗ B defines a
continuous C∗ -bundle over Glimm(A ⊗ B) if and only if A and B
define continuous C∗ -bundles over Glimm(A) and Glimm(B)
respectively.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
The centre of the multiplier algebra
Corollary
(to the Dauns-Hofmann Theorem) There is a ∗-isomorphism
µA : C b (Glimm(A)) → ZM(A) with the property that
µA (f )a + Gp = f (p)(a + Gp )
for f ∈ C b (Glimm(A)), a ∈ A, p ∈ Glimm(A).
It follows that for C∗ -algebras A and B, we may identify
ZM(A ⊗ B) ≡ C b (Glimm(A) × Glimm(B), τcr )
≡ C (β(Glimm(A) × Glimm(B), τcr )),
where β denotes the Stone-Čech compactification.
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Embedding ZM(A) ⊗ ZM(B) ⊆ ZM(A ⊗ B)
I
Applying the same Corollary to A and B, we have
ZM(A) ⊗ ZM(B) ≡ C b (Glimm(A)) ⊗ C b (Glimm(B))
≡ C (βGlimm(A)) ⊗ C (βGlimm(B))
≡ C (βGlimm(A) × βGlimm(B))
I
As a consequence, we see that
ZM(A) ⊗ ZM(B) = ZM(A ⊗ B) if and only if
β(Glimm(A) × Glimm(B), τcr ) = βGlimm(A) × βGlimm(B)
Glimm ideals of the minimal tensor product of C∗ -algebras
Prim(A), Fac(A) and Glimm(A)
Ideals of A ⊗ B
The Glimm space of A ⊗ B
The Dauns-Hofmann Theorem
Conditions for ZM(A) ⊗ ZM(B) = ZM(A ⊗ B)
I
Call a completely regular space X pseudocompact if every
f ∈ C (X ) is bounded,
I
Glicksberg’s Theorem: if X , Y both infinite then
β(X × Y ) = βX × βY if and only if X × Y is pseudocompact,
I
Note that βGlimm(A) × βGlimm(B) contains a copy of
Glimm(A) × Glimm(B) with the product topology,
Theorem
ZM(A) ⊗ ZM(B) = ZM(A ⊗ B) if and only if τcr = τp on
Glimm(A) × Glimm(B) and either
1. One of Glimm(A) or Glimm(B) is finite, or
2. Glimm(A) × Glimm(B) is pseudocompact.
Glimm ideals of the minimal tensor product of C∗ -algebras