The Glimm space of the minimal tensor product
of C∗ -algebras
David McConnell
ICMS, Edinburgh, 19th April 2013
Primitive ideals and the hull-kernel topology
Let A be a C∗ -algebra and Prim(A) the set of kernels of irreducible
representations of A.
I Hull-kernel topology on Prim(A) has open sets
U(I ) = {P ∈ Prim(A) : P 6⊇ I }
for I an ideal of A,
I
I
I
I
The map P 7→ P ∩ I is a homeomorphism U(I ) → Prim(I ),
T
I = {P ∈ Prim(A) : P ⊇ I },
T
In particular {P : P ∈ Prim(A)} = {0}.
Prim(A) is a locally compact T0 space, but non-Hausdorff in
general.
The Glimm space of the minimal tensor product of C∗ -algebras
Representation over Prim(A)
Let A be a commutative C∗ -algebra.
I
Irreducible representations of A are the characters on A,
I
Hull-kernel topology is precisely the weak-∗ topology,
I
Gelfand-Naimark: A = C0 (Prim(A)).
Non-commutative generalisation:
I
I
Try to represent A as the section algebra of a bundle over Prim(A),
with fibres {A/P : P ∈ Prim(A)},
`
Identify a ∈ A with the section â : Prim(A) → A/P, where
â(P) = a + P,
I
J.M.G Fell: A is ∗-isomorphic to the section aglebra of a bundle over
Prim(A) iff Prim(A) hull-kernel Hausdorff.
I
J. Dauns and K.H. Hofmann obtained a sectional representation
theorem valid for any C∗ -algebra A over the complete regularisation
of Prim(A).
The Glimm space of the minimal tensor product of C∗ -algebras
Glimm(A)
I
Define an equivalence relation ≈ on Prim(A) via P ≈ Q if and only
if f (P) = f (Q) for all f ∈ C b (Prim(A)),
I
As a set let Glimm(A) = Prim(A)/ ≈, and
ρA : Prim(A) → Glimm(A) be the quotient map,
I
For f ∈ C b (Prim(A)) define f ρ : Glimm(A) → C via
f ρ ([P]) = f (P),
I
Topology τcr on Glimm(A) induced by {f ρ : f ∈ C b (Prim(A)},
I
Glimm(A) is completely regular (Tychonoff), and the map f 7→ f ρ
is a ∗-isomorphism of C b (Prim(A)) onto C b (Glimm(A)),
I
To each ≈-equivalence
class p = [P] define the closed two-sided
T
ideal Gp = [P], the Glimm ideal of A corresponding to p.
I
Note that if Prim(A) is Hausdorff then Prim(A) = Glimm(A).
The Glimm space of the minimal tensor product of C∗ -algebras
The Dauns-Hofmann Theorem
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)).
I
For each p ∈ Glimm(A) we have Gp = Mp · A, where
Mp = {f ∈ C b (Glimm(A)) : f (p) = 0}.
Let X be a locally compact Hausdorff space,
I
G.G. Kasparov: A can be represented as the section algebra of an
upper-semicontinuous C∗ -bundle over X if and only if there is a
∗-homomorphism µ : C0 (X ) → ZM(A) such that µ(C0 (X )) · A = A,
I
Equivalently can take µ0 : C0 (X ) → C b (Glimm(A)) with this
property,
I
This occurs if and only if there is a continuous map
φ : Prim(A) → X , or equivalently, a continuous map
φρ : Glimm(A) → X .
The Glimm space of the minimal tensor product of C∗ -algebras
Glimm(A ⊗ B)
For C∗ -algebras A and B, let A ⊗ B be their minimal C∗ -tensor product.
In order to address these questions for A ⊗ B, need to 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.
Related questions studied in:
I
E. Kaniuth, Minimal primal ideal spaces and norms of inner
derivations of tensor products of C∗ -algebras, Math. Proc.
Cambridge Philos. Soc., 119 (1996), no.2, 297-308.
I
R.J. Archbold, Continuous bundles of C∗ -algebras and tensor
products, Quart. J. Math. Oxford, Ser. (2), 50 (1999), no. 198,
131-146.
I
A.J. Lazar, The space of ideals in the minimal tensor product of
C∗ -algebras, Math. Proc. Cambridge Philos. Soc., 148 (2010), no.
2, 243-252.
The Glimm space of the minimal tensor product of C∗ -algebras
The minimal tensor product
Let A and B be C∗ -algebras.
I
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)
for elementary tensors a ⊗ b ∈ 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.
The Glimm space of the minimal tensor product of C∗ -algebras
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,
which has closure I ⊗ B + A ⊗ J in A ⊗ B by injectivity.
I
Alternatively, we may consider the kernel of
qI ⊗ qJ : A ⊗ B → (A/I ) ⊗ (B/J),
I
Clearly ker(qI ⊗ qJ ) ⊇ I ⊗ B + A ⊗ J, but this inclusion may be strict,
I
I
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).
S. Wassermann: B(H) ⊗ B(H) does not satisfy (F).
The Glimm space of the minimal tensor product of C∗ -algebras
Maps between Id0 (A) × Id0 (B) and Id0 (A ⊗ B)
We define maps Φ, ∆0 : Id0 (A) × Id0 (B) → Id0 (A ⊗ B) via
Φ(I , J)
0
∆ (I , J)
=
ker(qI ⊗ qJ )
=
I ⊗B +A⊗J
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),
I
For I / A ⊗ B, define ideals of A and B via
I A = {a ∈ A : a ⊗ B ⊆ I }, I B = {b ∈ B : A ⊗ b ⊆ I },
I
Setting Ψ(I ) = (I A , I B ) gives a map
Ψ : Id 0 (A ⊗ B) → Id 0 (A) × Id 0 (B),
I
For (I , J) ∈ Id 0 (A) × Id 0 (B),
(Ψ ◦ Φ)(I , J) = (Ψ ◦ ∆0 )(I , J) = (I , J)
The Glimm space of the minimal tensor product of C∗ -algebras
The complete regularisation of Prim(A) × Prim(B)
I
Consider the relation ≈ on Prim(A) × Prim(B),
I
Easily seen that (P1 , Q1 ) ≈ (P2 , Q2 ) if and only if P1 ≈ P2 and
Q1 ≈ Q2 ,
I
Hence as a set, the complete regularisation of Prim(A) × Prim(B)
agrees with Glimm(A) × Glimm(B),
I
Topology τcr induced by C b (Prim(A) × Prim(B)) is stronger than
the product topology τp .
The Glimm space of the minimal tensor product of C∗ -algebras
Characterisation of Glimm(A ⊗ B)
Theorem
Let A and B be C∗ -algebras and ρA , ρB and ρα the complete
regularisation maps of Prim(A), Prim(B) and Prim(A ⊗ B) respectively.
Denote by ∆ the restriction of ∆0 to Glimm(A) × Glimm(B). Then
(i) ∆ is a homeomorphism of (Glimm(A) × Glimm(B), τcr ) onto
Glimm(A ⊗ B),
(ii) ∆ is an open bijection of (Glimm(A) × Glimm(B), τp ) onto
Glimm(A ⊗ B),
(iii) For all (P, Q) ∈ Prim(A) × Prim(B) we have
(∆ ◦ (ρA × ρB )) (P, Q) = (ρα ◦ Φ)(P, Q),
(iv) ∆
−1
= Ψ.
Extends a result of E. Kaniuth by eliminating the assumption of (F).
The Glimm space of the minimal tensor product of C∗ -algebras
τcr and τp
Theorem
Let A be a C∗ -algebra satisfying one of the following conditions:
(i) Prim(A) is compact (e.g. A unital),
(ii) A is σ-unital and Glimm(A) is locally compact,
(iii) The complete regularisation map ρA is open (e.g. Prim(A)
Hausdorff, A quasi-standard).
Then τcr = τp on Glimm(A) × Glimm(B) for any C∗ -algebra B, so that
Glimm(A ⊗ B) can be identified with (Glimm(A) × Glimm(B), τp ).
On the other hand, there is a separable, liminal C∗ -algebra A such that
τcr 6= τp on Glimm(A) × Glimm(A). Hence Glimm(A ⊗ A) is not
homeomorphic to this space with the product topology.
The Glimm space of the minimal tensor product of C∗ -algebras
Sectional representation
Consider the fibred space A = {A/Gp : p ∈ Glimm(A)}:
I
Each element a ∈ A defines a section â : Glimm(A) → A such that
â(p) = a + Gp ,
I
kak = sup{kâ(p)k : p ∈ Glimm(A)},
I
norm functions p →
7 kâ(p)k upper semicontinuous on Glimm(A) for
all a ∈ A.
Dauns-Hofmann: there is a topology on A such that
I
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,
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). Equivalently, A is a continuous C0 (Glimm(A))-algebra.
The Glimm space of the minimal tensor product of C∗ -algebras
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 ).
I
This assumption is strictly weaker than property (F).
The Glimm space of the minimal tensor product of C∗ -algebras
Continuous C∗ -bundles
Corollary
Suppose that A and B are C∗ -algebras such that Φ = ∆ on
Glimm(A) × Glimm(B), and denote by ρA , ρB and ρα the complete
regularisation maps of Prim(A), Prim(B) and Prim(A ⊗ B) respectively.
Then the following are equivalent:
(i) The Dauns-Hofmann representations of A and B define continuous
C∗ -bundles over Glimm(A) and Glimm(B) respectively,
(ii) ρA and ρB are open maps,
(iii) ρα is an open map,
(iv) The Dauns-Hofmann representation of A ⊗ B defines a continuous
C∗ -bundle over Glimm(A ⊗ B).
The Glimm space of the minimal tensor product of C∗ -algebras
Embedding ZM(A) ⊗ ZM(B) in ZM(A ⊗ B)
I
We may regard M(A) ⊗ M(B) ⊆ M(A ⊗ B), with action on A ⊗ B
determined via
(x ⊗ y )(a ⊗ b) = (xa) ⊗ (yb), (a ⊗ b)(x ⊗ y ) = (ax) ⊗ (by )
for elementary tensors a ⊗ b ∈ A ⊗ B and x ⊗ y ∈ M(A) ⊗ M(B),
I
Akemann, Pedersen and Tomiyama: If A is σ-unital and non-unital,
and B infinite dimensional, then this inclusion is strict.
We will describe the centres of both algebras in terms of Glimm(A) and
Glimm(B), and give necessary and sufficient conditions for equality.
I
R. Haydon and S. Wassermann: for any C∗ -algebras C and D,
Z (C ⊗ D) = Z (C ) ⊗ Z (D),
I
In particular, we have Z (M(A) ⊗ M(B)) = ZM(A) ⊗ ZM(B),
I
Moreover, easily seen that ZM(A) ⊗ ZM(B) ⊆ ZM(A ⊗ B).
The Glimm space of the minimal tensor product of C∗ -algebras
The structure space of the centre
Corollary to Dauns-Hofmann Theorem:
ZM(A)
=
C b (Prim(A))
=
C b (Glimm(A))
=
C (βGlimm(A))
It follows that
ZM(A ⊗ B)
=
C (βGlimm(A ⊗ B))
=
C (β (Glimm(A) × Glimm(B), τcr ))
While
ZM(A) ⊗ ZM(B)
=
C (βGlimm(A)) ⊗ C (βGlimm(B))
=
C (βGlimm(A) × βGlimm(B))
The Glimm space of the minimal tensor product of C∗ -algebras
Conditions for ZM(A) ⊗ ZM(B) = ZM(A ⊗ B)
Theorem
ZM(A) ⊗ ZM(B) = ZM(A ⊗ B) if and only if either
1. One of Glimm(A) or Glimm(B) is finite, or
2. τcr = τp on Glimm(A) × Glimm(B), and Glimm(A) × Glimm(B) is
pseudocompact.
(Recall that a completely regular space X is said to be pseudocompact if
every f ∈ C (X ) is bounded.)
The embedding ZM(A) ⊗ ZM(B) ,→ ZM(A ⊗ B) is dual to the
continuous surjection
ι : β (Glimm(A) × Glimm(B), τcr ) → βGlimm(A) × βGlimm(B),
extending the identity ι on Glimm(A) × Glimm(B). Hence equality holds
if and only ι is a homeomorphism. Note that τcr = τp is a necessary
condition.
The Glimm space of the minimal tensor product of C∗ -algebras
Stone-Čech compactification of a product space
I
Suppose that Glimm(B) is finite, hence discrete and compact,
I
Then ρB : Prim(B) → Glimm(B) is an open map, so that τcr = τp ,
I
Glimm(A) × Glimm(B) consists of n disjoint copies of Glimm(A),
I
Hence
β (Glimm(A) × Glimm(B))
= βGlimm(A) × Glimm(B)
= βGlimm(A) × βGlimm(B).
In the infinite case, we apply Glicksberg’s Theorem: for completely
regular spaces X and Y , the canonical map β(X × Y ) → βX × βY
is a homeomorphism if and only if X × Y is pseudocompact.
The Glimm space of the minimal tensor product of C∗ -algebras
A non-trivial example ZM(A) ⊗ ZM(B) = ZM(A ⊗ B)
Let X be a non-compact locally compact Hausdorff space, and H a
separable infinite dimensional Hilbert space.
I For x ∈ βX , Y := βX \{x} is locally compact, non-compact and
pseudocompact,
I Set A = C0 (Y , B(H)) (with pointwise operations and supremum
norm), then Glimm(A) = Y ,
I Define B as the C∗ -algebra of sequences (Tn ) ⊂ B(H), such that
Tn → T∞ ∈ K (H),
I Glimm(B) = N ∪ {∞}, and ρB : Prim(B) → Glimm(B) is open
(Archbold and Somerset),
I Since B is σ-unital and non-unital, M(A) ⊗ M(B) ( M(A ⊗ B)
(Akemann, Pedersen and Tomiyama),
I Glimm(A) × Glimm(B) is the product of a pseudocompact space
and a compact space, hence is pseudocompact, and τcr = τp ,
I It follows that ZM(A) ⊗ ZM(B) = ZM(A ⊗ B).
The Glimm space of the minimal tensor product of C∗ -algebras
Example cont’d.
I
A ⊗ B has a quotient isomorphic to B(H) ⊗ B(H), hence does not
have property (F),
I
On the other hand, Φ = ∆ on Glimm(A) × Glimm(B) (Archbold),
and τcr = τp ,
I
Hence A ⊗ B is ∗-isomorphic to the section algebra of a bundle over
Y × N ∪ {∞} (with the product topology),with fibre algebras for
y ∈ Y given by
(A ⊗ B)(y ,n)
(A ⊗ B)(y ,∞)
I
= B(H) ⊗ B(H), n ∈ N
= B(H) ⊗ K (H)
Since A and B define continuous C∗ -bundles and ∆ = Φ, so does
A ⊗ B.
The Glimm space of the minimal tensor product of C∗ -algebras