Download Duality theory of locally precompact groups

Duality theory of
locally precompact groups
Gábor Lukács
[email protected]
University of Manitoba
Winnipeg, Manitoba, Canada
Financial support of the Killam Trust is gratefully acknowledged
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12
Ethical issues
This research (including this presentation) was prepared
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12
Ethical issues
This research (including this presentation) was prepared
without animal experiments;
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12
Ethical issues
This research (including this presentation) was prepared
without animal experiments;
without child labour;
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12
Ethical issues
This research (including this presentation) was prepared
without animal experiments;
without child labour;
without using Microsoft products.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.0/12
Pontryagin-van Kampen duality
Let L be a locally compact abelian (LCA) group.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12
Pontryagin-van Kampen duality
Let L be a locally compact abelian (LCA) group.
L̂ is the group of continuous characters χ : L → T,
equipped with the compact open topology.
(T = R/Z.)
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12
Pontryagin-van Kampen duality
Let L be a locally compact abelian (LCA) group.
L̂ = Hco (L, T). (T = R/Z.)
L̂ is a locally compact abelian group.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12
Pontryagin-van Kampen duality
Let L be a locally compact abelian (LCA) group.
L̂ = Hco (L, T). (T = R/Z.)
L̂ is LCA.
ˆ
The evaluation αL : L → L̂ is a topological
isomorphism.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12
Pontryagin-van Kampen duality
Let L be a locally compact abelian (LCA) group.
L̂ = Hco (L, T). (T = R/Z.)
L̂ is LCA.
ˆ
αL : L → L̂ is a topological isomorphism.
[ ∼
M̂ ∼
= L̂/M ⊥ and L/M
= M ⊥ for every closed
subgroup M ≤ L. (M ⊥ = {χ ∈ L̂ | χ(M ) = 0}.)
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12
Pontryagin-van Kampen duality
Let L be a locally compact abelian (LCA) group.
L̂ = Hco (L, T). (T = R/Z.)
L̂ is LCA.
ˆ
αL : L → L̂ is a topological isomorphism.
[ ∼
M̂ ∼
= L̂/M ⊥ and L/M
= M ⊥ for every closed
subgroup M ≤ L. (M ⊥ = {χ ∈ L̂ | χ(M ) = 0}.)
L is discrete ⇐⇒ L̂ is compact.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.1/12
How to extend or generalize the
Pontryagin-van Kampen duality?
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.2/12
Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
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Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
N (G) = neighborhoods of identity in the group G.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12
Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
N (G) = neighborhoods of identity in the group G.
B ⊆ G is precompact if for every U ∈ N (G) there
is F ⊆ G finite such that B ⊆ F U .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12
Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
N (G) = neighborhoods of identity in the group G.
B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G
such that B ⊆ g1 U ∪ . . . ∪ gl U .
B
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12
Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
N (G) = neighborhoods of identity in the group G.
B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G
such that B ⊆ g1 U ∪ . . . ∪ gl U .
g1 U
B
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12
Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
N (G) = neighborhoods of identity in the group G.
B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G
such that B ⊆ g1 U ∪ . . . ∪ gl U .
g1 U
g2 U
B
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12
Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
N (G) = neighborhoods of identity in the group G.
B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G
such that B ⊆ g1 U ∪ . . . ∪ gl U .
g1 U
g2 U
···
···
···
B
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12
Precompact subset
Unless otherwise stated, all topological groups and
spaces are Tychonoff.
N (G) = neighborhoods of identity in the group G.
B ⊆ G is precompact if ∀U∈ N (G) ∃g1 , . . . , gl ∈ G
such that B ⊆ g1 U ∪ . . . ∪ gl U .
g1 U
g2 U
···
···
···
···
gl U
B
···
···
···
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.3/12
Completion
{gα } ⊆ G is a Cauchy net (two-sided uniformity) if
gα gβ−1 −→ e, and
gα−1 gβ −→ e.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12
Completion
{gα } ⊆ G is a Cauchy net (two-sided uniformity) if
gα gβ−1 −→ e, and
gα−1 gβ −→ e.
G is complete if every Cauchy net converges.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12
Completion
{gα } ⊆ G is a Cauchy net (two-sided uniformity) if
gα gβ−1 −→ e, and
gα−1 gβ −→ e.
G is complete if every Cauchy net converges.
(Raı̆kov, 1946) Every topological group G admits a
group completion Ḡ.
Ḡ is unique up to topological isomorphism.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12
Completion
{gα } ⊆ G is a Cauchy net (two-sided uniformity) if
gα gβ−1 −→ e, and
gα−1 gβ −→ e.
G is complete if every Cauchy net converges.
(Raı̆kov, 1946) Every topological group G admits a
group completion Ḡ.
Ḡ is unique up to topological isomorphism.
(Weil, 1937) G is precompact if and only if Ḡ is
compact.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.4/12
Comfort-Ross duality
A a discrete abelian group;
K := Â its (compact) dual.
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Comfort-Ross duality
A a discrete abelian group;
K := Â its (compact) dual.
For H ≤ K, τH is the initial group topology with
respect to
ιH : A −→ TH
a 7−→ (χ(a))χ∈H .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12
Comfort-Ross duality
A a discrete abelian group;
K := Â its (compact) dual.
For H ≤ K, τH is the initial group topology with
respect to ιH : A −→ TH .
Comfort and Ross (1964):
(A, τH ) is precompact.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12
Comfort-Ross duality
A a discrete abelian group;
K := Â its (compact) dual.
For H ≤ K, τH is the initial group topology with
respect to ιH : A −→ TH .
Comfort and Ross (1964):
(A, τH ) is precompact.
\
If (A, τ ) is precompact, then τ = τH for H = (A,
τ ).
That is, every precompact group topology on A has
the form τH .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12
Comfort-Ross duality
A a discrete abelian group;
K := Â its (compact) dual.
For H ≤ K, τH is the initial group topology with
respect to ιH : A −→ TH .
Comfort and Ross (1964):
(A, τH ) is precompact.
\
If (A, τ ) is precompact, then τ = τH for H = (A,
τ ).
(A, τH ) is Hausdorff ⇐⇒ H is dense in K.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.5/12
Locally precompact groups
G is locally precompact if N (G) contains a
precompact set.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12
Locally precompact groups
G is locally precompact if N (G) contains a
precompact set.
(Weil, 1937) G is locally precompact if and only if
Ḡ is locally compact.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12
Locally precompact groups
G is locally precompact if N (G) contains a
precompact set.
(Weil, 1937) G is locally precompact if and only if
Ḡ is locally compact.
A locally precompact group G can be encoded as a
discrete group Gd , a locally compact group Ḡ, and a
dense injection iG : Gd → Ḡ.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12
Locally precompact groups
G is locally precompact if N (G) contains a
precompact set.
(Weil, 1937) G is locally precompact if and only if
Ḡ is locally compact.
G locally precompact,
G 7−→ (iG : Gd → Ḡ).
Let D be discrete, L be locally compact. Given a
dense injection i : D → L, G = i(L) is locally
precompact, and i = iG .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12
Locally precompact groups
G is locally precompact if N (G) contains a
precompact set.
(Weil, 1937) G is locally precompact if and only if
Ḡ is locally compact.
Equivalence of categories:
G locally precompact,
G 7−→ (iG : Gd → Ḡ).
D discrete, L is LC, i : D → L is a dense injection,
i(L) ←−[ (i : D → L).
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.6/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
ˆ: M −→ Mop is an equivalence of categories.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
ˆ: M −→ Mop is an equivalence of categories.
f ∈ ob M is injective ⇐⇒ fˆ is dense.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
ˆ: M −→ Mop is an equivalence of categories.
f ∈ ob M is injective ⇐⇒ fˆ is dense.
⇓
The full subcategory DI of dense injections in M is
self-dual.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
ˆ: M −→ Mop is an equivalence of categories.
f ∈ ob M is injective ⇐⇒ fˆ is dense.
DI (= dense injections in M) is self-dual.
Consider the following full subcategories of DI:
ob D = {i : D → L | D is discrete}.
ob K = {i : L → K | K compact}.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
ˆ: M −→ Mop is an equivalence of categories.
DI (= dense injections in M) is self-dual.
Consider the following full subcategories of DI:
ob D = {i : D → L | D is discrete}.
ob K = {i : L → K | K compact}.
ˆ: D −→ Kop is an equivalence of categories.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
ˆ: M −→ Mop is an equivalence of categories.
DI (= dense injections in M) is self-dual.
Consider the following full subcategories of DI:
ob D = {i : D → L | D is discrete}.
ob K = {i : L → K | K compact}.
ˆ: D −→ Kop is an equivalence of categories.
D is equivalent to the category LPA.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of morphisms
Set M = LCA·→· , the category of morphisms in LCA.
ˆ: M −→ Mop is an equivalence of categories.
DI (= dense injections in M) is self-dual.
Consider the following full subcategories of DI:
ob D = {i : D → L | D is discrete}.
ob K = {i : L → K | K compact}.
ˆ: D −→ Kop is an equivalence of categories.
D is equivalent to the category LPA.
K is the category of group compactifications of LCA
groups.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.7/12
Duality of LPA groups
Let G be an LPA group.
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Duality of LPA groups
Let G be an LPA group.
b̄ → G
cd is its dual;
ibG : G
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
Let G be an LPA group.
b̄ → G
cd is its dual;
ibG : G
b̄ , τ ) of the group
cd is the completion ((G)
G
d G
b̄ equipped with the precompact topology τ .
(G)
d
G
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
Let G be an LPA group.
b̄ → G
cd is its dual;
ibG : G
b̄ , τ ) of the group
cd is the completion ((G)
G
d G
b̄ equipped with the precompact topology τ .
(G)
d
G
Let i : L → K be a compactification (L is LCA).
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
Let G be an LPA group.
b̄ → G
cd is its dual;
ibG : G
b̄ , τ ) of the group
cd is the completion ((G)
G
d G
b̄ equipped with the precompact topology τ .
(G)
d
G
Let i : L → K be a compactification (L is LCA).
G = î(K̂) is its dual (î : K̂ → L̂) ;
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
Let G be an LPA group.
b̄ → G
cd is its dual;
ibG : G
b̄ , τ ) of the group
cd is the completion ((G)
G
d G
b̄ equipped with the precompact topology τ .
(G)
d
G
Let i : L → K be a compactification (L is LCA).
G = î(K̂) is its dual;
G carries the subgroup topology induced by L̂.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
(GL, 2006) The functors
b̄ → G
cd )
G 7−→ (ibG : G
î(K̂) ←−[ (i : L → K)
form a duality between LPA and K.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
(GL, 2006) The functors
b̄ → G
cd )
G 7−→ (ibG : G
î(K̂) ←−[ (i : L → K)
form a duality between LPA and K.
Special cases:
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
(GL, 2006) The functors
b̄ → G
cd )
G 7−→ (ibG : G
î(K̂) ←−[ (i : L → K)
form a duality between LPA and K.
Special cases:
G = L: Its dual is L̂ → bL̂ (Bohr compactification),
so one obtains the Pontryagin duality.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Duality of LPA groups
(GL, 2006) The functors
b̄ → G
cd )
G 7−→ (ibG : G
î(K̂) ←−[ (i : L → K)
form a duality between LPA and K.
Special cases:
G = L: Its dual is L̂ → bL̂ (Bohr compactification),
so one obtains the Pontryagin duality.
b̄ is discrete. This
G precompact: Ḡ is compact, so G
is precisely the Comfort-Ross duality.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.8/12
Dual properties
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Gδ -topology in LCA
Gδ -topology:
∞
T
Un is a Gδ -neighborhood of x0 if
n=1
each Un is a neighborhood of x0 .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Gδ -topology in LCA
Gδ -topology: N (δG) = {
∞
T
Un | Un ∈ N (G)}.
n=1
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Gδ -topology in LCA
Gδ -topology: N (δG) = {
∞
T
Un | Un ∈ N (G)}.
n=1
Let L be an LCA group.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Gδ -topology in LCA
Gδ -topology: N (δG) = {
∞
T
Un | Un ∈ N (G)}.
n=1
Let L be an LCA group.
Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Gδ -topology in LCA
Gδ -topology: N (δG) = {
∞
T
Un | Un ∈ N (G)}.
n=1
Let L be an LCA group.
Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}.
{Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Gδ -topology in LCA
Gδ -topology: N (δG) = {
∞
T
Un | Un ∈ N (G)}.
n=1
Let L be an LCA group.
Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}.
{Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL.
a ∈ L is in the Gδ -closure of X ⊆ L ⇔ for each
Σ ∈ Σ(L̂), there is x ∈ X such that x − a ∈ Σ⊥ .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Gδ -topology in LCA
Gδ -topology: N (δG) = {
∞
T
Un | Un ∈ N (G)}.
n=1
Let L be an LCA group.
Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}.
{Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL.
a ∈ L is in the Gδ -closure of X ⊆ L ⇔ for each
Σ ∈ Σ(L̂), there is x ∈ X such that x|Σ = a|Σ .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Gδ -topology in LCA
Gδ -topology: N (δG) = {
∞
T
Un | Un ∈ N (G)}.
n=1
Let L be an LCA group.
Σ(L̂) = {Σ ≤ L̂ | Σ is open and σ-compact}.
{Σ⊥ | Σ ∈ Σ(L̂)} is a base at 0 for δL.
clδL X = {a ∈ L | ∀Σ ∈ Σ(L̂)∃x ∈ X, x|Σ = a|Σ }.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.10/12
Local pseudocompactness
G is locally pseudocompact if N (G) contains a
pseudocompact set.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12
Local pseudocompactness
G is locally pseudocompact if N (G) contains a
pseudocompact set.
(Comfort and Trigos-Arrieta, 1995) The following
are equivalent:
G is locally pseudocompact;
β Ḡ = βG (Stone-Čech-compactification);
υ Ḡ = υG (Hewitt-realcompactification);
Gδ -dense in Ḡ and Ḡ is locally compact.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12
Local pseudocompactness
G is locally pseudocompact if N (G) contains a
pseudocompact set.
(Comfort and Trigos-Arrieta, 1995) The following
are equivalent:
G is locally pseudocompact;
Gδ -dense in Ḡ and Ḡ is locally compact.
Let G be an LPA group.
b̄
clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G)∃g
∈ G, g|Σ = a|Σ }.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12
Local pseudocompactness
(Comfort and Trigos-Arrieta, 1995) The following
are equivalent:
G is locally pseudocompact;
Gδ -dense in Ḡ and Ḡ is locally compact.
Let G be an LPA group.
b̄
clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G)∃g
∈ G, g|Σ = a|Σ }.
⇓
b̄ a is τ -continuous}.
clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G),
G|Σ
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12
Local pseudocompactness
(Comfort and Trigos-Arrieta, 1995) The following
are equivalent:
G is locally pseudocompact;
Gδ -dense in Ḡ and Ḡ is locally compact.
Let G be an LPA group.
b̄ a is τ -continuous}.
clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G),
G|Σ
⇓
G is locally pseudocompact ⇔ ∀Σ, τG|Σ = τḠ|Σ .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12
Local pseudocompactness
Let G be an LPA group.
G is locally pseudocompact ⇔ ∀Σ, τG|Σ = τḠ|Σ .
⇓
(GL, 2006) Let G be a locally precompact abelian
group. The following are equivalent:
G is locally pseudocompact;
τG|Σ is the Bohr-topology on Σ for every
b̄
Σ ∈ Σ(G).
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12
Local pseudocompactness
Let G be an LPA group.
G is locally pseudocompact ⇔ ∀Σ, τG|Σ = τḠ|Σ .
(GL, 2006) Let G be a locally precompact abelian
group. The following are equivalent:
G is locally pseudocompact;
τG|Σ is the Bohr-topology on Σ for every
b̄
Σ ∈ Σ(G).
Special case:
G precompact (Hernández and Macario, 2003).
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.11/12
Realcompactness
Index of precompactness ip(G) = smallest cardinal τ
such that for every U ∈ N (G), there is S ⊆ G such that
|S| ≤ τ and G = SU .
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12
Realcompactness
ip(G) = sup inf{|S| | G = SU }.
U ∈N (G)
(GL, 2006) Let G be a locally precompact group.
Then G is realcompact if and only if
ip(G) is not Ulam-measurable, and
G is Gδ -closed in Ḡ.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12
Realcompactness
ip(G) = sup inf{|S| | G = SU }.
U ∈N (G)
(GL, 2006) Let G be a locally precompact group.
Then G is realcompact if and only if
ip(G) is not Ulam-measurable, and
G is Gδ -closed in Ḡ.
κ is Ulam-measurable if there is a σ-additive
measure µ : P (κ) → {0, 1} such that
µ(κ) = 1;
µ({λ}) = 0 for every λ ∈ κ.
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12
Realcompactness
ip(G) = sup inf{|S| | G = SU }.
U ∈N (G)
(GL, 2006) Let G be a locally precompact group.
Then G is realcompact if and only if
ip(G) is not Ulam-measurable, and
G is Gδ -closed in Ḡ.
Let G be an LPA group.
b̄ a is τ -continuous}.
clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G),
G|Σ
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12
Realcompactness
(GL, 2006) Let G be a locally precompact group.
Then G is realcompact if and only if
ip(G) is not Ulam-measurable, and
G is Gδ -closed in Ḡ.
Let G be an LPA group.
b̄ a is τ -continuous}.
clδḠ G = {a ∈ Ḡ | ∀Σ ∈ Σ(G),
G|Σ
⇓
G = clδḠ G ⇔ if a ∈ Ḡ is τG|Σ -continuous for every
b̄ then a ∈ G.
Σ ∈ Σ(G),
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12
Realcompactness
(GL, 2006) Let G be a locally precompact group.
Then G is realcompact if and only if
ip(G) is not Ulam-measurable, and
G is Gδ -closed in Ḡ.
Let G be an LPA group.
G = clδḠ G ⇔ if a ∈ Ḡ is τG|Σ -continuous for every
b̄ then a is τ -continuous.
Σ ∈ Σ(G),
G
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12
Realcompactness
(GL, 2006) Let G be a locally precompact abelian
group. Then G is realcompact if and only if
ip(G) is not Ulam-measurable, and
b̄ such that ψ is τ -continuevery character ψ of G
|Σ
G|Σ
b̄ is τ -continuous.
ous on every Σ ∈ Σ(G)
G
⇑
Let G be an LPA group.
G = clδḠ G ⇔ if a ∈ Ḡ is τG|Σ -continuous for every
b̄ then a is τ -continuous.
Σ ∈ Σ(G),
G
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12
Realcompactness
(GL, 2006) Let G be a locally precompact abelian
group. Then G is realcompact if and only if
ip(G) is not Ulam-measurable, and
b̄ such that ψ is τ -continuevery character ψ of G
|Σ
G|Σ
b̄ is τ -continuous.
ous on every Σ ∈ Σ(G)
G
Special cases:
G LCA (Comfort, Hernández, Trigos-Arrieta, 1996);
G precompact (Hernández and Macario, 2003).
Category Theory Octoberfest, October 21–22, 2006, University of Ottawa, Ontario, Canada – p.12/12