AND W*-TENSOR PRODUCTS
CORNELIU CONSTANTINESCU
For every topological spaceX let C(X) denote the C*-algebra of bounded con- tinuous scalar valued functions onX. For hyperstonian spacesS, T, we identify C(S) ¯⊗ C(T) ( ¯⊗ denotes the W*-tensor product) as the inductive limit of an in- ductive system (C(A))A∈Λ, where Λ is a downward directed set of subsets ofS×T (Theorem 4).
AMS 2010 Subject Classification: 46L06.
Key words: hyperstonian spaces, W*-tensor products.
We use the notation and terminology of [1]. For W*-tensor products of W*-algebras we use [2] and for inductive limits of C*-algebras we use [3].
Let S, T be hyperstonian spaces, M and N the sets of positive normal measures on S and T, respectively, Λ the set of sets K of pairwise disjoint compact sets of S×T such that
X
K∈K
(µ⊗ν)(K) =µ(S)ν(T) for all (µ, ν)∈ M × N. For everyK∈Λ put
K˘ := [
K∈K
K.
If (µ, ν)∈ M × N and K∈Λ such that
(Suppµ)×(Suppν)⊂(S×T)\K˘
(where Supp denotes the support of the corresponding measure), then µ(S)ν(T) = X
K∈K
(µ⊗ν)(K) = 0 so µ⊗ν = 0 and ˘Kis dense inS×T.
REV. ROUMAINE MATH. PURES APPL.,56(2011),4, 269–273
For K,L ∈ Λ we put K L if for every L ∈ L there is a K∈K with L⊂K; in this case ˘L⊂K.˘ is an order relation on Λ. LetK,L∈Λ. Put
M:={K∩L| (K, L)∈K×L}. The sets of Mare pairwise disjoint and
X
(K,L)∈K×L
(µ⊗ν)(K∩L) = X
K∈K
X
L∈L
(µ⊗ν)(K∩L) = X
K∈K
(µ⊗ν)(K) =µ(S)ν(T) for (µ, ν) ∈ M × N, so M∈Λ. Since KM,LM, Λ is upward directed.
For K,L∈Λ withKLput
ϕL,K :C(˘K)→ C(˘L), f 7→f|L,˘
where f|L˘ denotes the restriction of f to ˘L. ϕL,K is an injective C*-homo- morphism and for K,L,M∈Λ withKLM we have
ϕM,L◦ϕL,K =ϕM,K,
so (C(˘K), ϕL,K) is an inductive system of C*-algebras. We denote by E its inductive limit and by
ϕK :C(˘K)→E
the associated injective C*-homomorphisms for every K∈Λ.
LetR be a hyperstonian space such that C(S) ¯⊗ C(T) =C(R), where ¯⊗denotes the W*-tensor product,
i:C(S×T) =C(S)⊗ C(T)→ C(S) ¯⊗ C(T) =C(R)
the inclusion map, andϕ:R→S×T the surjective continuous map such that i(f ⊗g) = (f⊗g)◦ϕ
for all (f, g)∈ C(S)× C(T) ([1], Proposition 4.1.2.15).
We considerC(S) and C(T) as the canonical von Neumann algebras (of multiplication operators) on the Hilbert spaces |
µ∈M
L2(µ) and |
ν∈N
L2(ν), respectively, where | denotes the Hilbert sum. ThenC(R) is a von Neumann algebra on the Hilbert space
H:=
|
µ∈M
L2(µ)
⊗
|
ν∈N
L2(ν)
=
= |
(µ,ν)∈M×N
L2(µ)⊗L2(ν)
= |
(µ,ν)∈M×N
L2 µ⊗ν .
Let (µ, ν)∈ M × N. Then µ⊗ν ∈
z }| {¨ C(S)⊗
z }| {¨ C(T),
where z }| {¨
C(S) andz }| {¨
C(T) denote the preduals ofC(S) andC(T), respectively. With the usual identification of
z }| {¨ C(S)⊗
z }| {¨
C(T) with a subset of the predual of C(S) ¯⊗ C(T) =C(R),
µ⊗ν is identified with a positive normal measure on R, which we denote by µ⊗¯ν. Forf ∈ C(S×T) =C(S)⊗ C(T),f ◦ϕ∈ C(R) and
Z
fd(µ⊗ν) = Z
(f◦ϕ) d(µ⊗¯ ν).
Lemma1. For every compact setK ofS×T and(µ, ν)∈ M×N we have (µ⊗ν)(K) = (µ⊗¯ ν)(−1ϕ (K)).
Proof. Let U be the downward directed set of clopen (i.e., closed and open) sets of S×T containing K. Then
K = \
U∈U
U, −1ϕ (K) = \
U∈U
−1ϕ (U).
By the above, (µ⊗ν)(K) = inf
U∈U(µ⊗ν)(U) = inf
U∈U(µ⊗¯ν)(−1ϕ (U)) = (µ⊗¯ν)(−1ϕ (K)).
Lemma 2. The set U := [
(µ,ν)∈M×N
−1ϕ ((Suppµ)×(Suppν))
is an open dense set of R.
Proof. LetV be a clopen set ofRwithU∩V =∅andeRV the characteristic function ofV relative toR, i.e., equal to 1 onV and equal to 0 onR\V. For (µ, ν)∈ M × N,ξ1, ξ2∈L2(µ), andη1, η2 ∈L2(ν), we have
eRV(ξ1⊗η1)|ξ2⊗η2
= Z
eRV
(ξ1⊗η1)(ξ2⊗η2)
◦ϕ
dµ⊗¯ ν= 0, where h· | · i denotes the scalar product. It follows that
eRVξ|η
= 0 for all ξ, η ∈L2(µ)L2(ν) (where denotes the algebraic tensor product), and so for allξ, η∈L2(µ)⊗L2(ν) =L2(µ⊗ν) and for allξ, η ∈H. ThuseRV = 0 and U is dense inR.
Proposition 3. ForK∈Λ, Kˆ := [
K∈K
interior of −1ϕ (K) is an open dense set of R.
Proof. For (µ, ν)∈ M × N, by Lemma 1, (µ⊗¯ν)(ˆK) = X
K∈K
(µ⊗¯ν) (interior of −1ϕ (K)) =
= X
K∈K
(µ⊗¯ν)(−1ϕ (K)) = X
K∈K
(µ⊗ν)(K) =µ(S)ν(T).
By Lemma 2, ˆKis dense inR.
For every K∈Λ and x ∈ C(ˆK), denote by ˜x the continuous extension of x toR (Proposition 3) and put
ψK :C(˘K)→ C(R), x7→
^ z }| { (x◦ϕ)|K.ˆ
ψK is an injective C*-homomorphism and forK,L∈Λ withKLwe have ψL◦ϕL,K =ψK.
Thus there is an injective C*-homomorphism ψ:E→ C(R) with ψ◦ϕK =ψK
for every K∈Λ.
LetU be a clopen set ofR. ϕ(U) is a compact set ofS×T. LetAbe a maximal set of pairwise disjoint clopen sets of (S×T)\ϕ(U) and put
K:=A∪ {ϕ(U)}.
Then K∈Λ,eS×Tϕ(U)∈ C(˘K),
eS×Tϕ(U)◦ϕ=eRU, so,
eRU =ψKeS×Tϕ(U)=ψϕKeS×Tϕ(U)∈ψ(E).
Since the linear combinations of functions of the form eRU is dense inC(R) we have ψ(E) =C(R), i.e., ψis surjective. We have proved
Theorem4. E is a W*-algebra andψ:E → C(S) ¯⊗ C(T) is an isomor- phism of W*-algebras.
In particular, ifA and B are sets then
l∞(A) ¯⊗l∞(B)≈l∞(A×B).
REFERENCES [1] Corneliu Constantinescu,C*-algebras. Elsevier, 2001.
[2] Masamichi Takesaki,Theory of Operator Algebra, I. Springer, 2002.
[3] N.E. Wegge-Olsen,K-theory and C*-algebras. Oxford University Press, 1993.
Received 25 March 2010 Bodenacherstr. 53
CH 8121 Benglen constant@math.ethz.ch
