Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The triangle of operators, topologies, bornologies

? % Ngai-Ching Wong

Department of Applied Mathematics National Sun Yat-sen University Taiwan

¬È 
. Tàó. http:\\www.math.nsysu.edu.tw\ ∼wong

In memory of my teacher Yau-Chuen Wong (1935.10.2–1994.11.7)

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared.

the topological method, the bornological method.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis:

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared.

the bornological method.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis: the topological method,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared.

The approaches in the study of locally convex spaces of

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of 1 Grothendieck (via Banach space operators),

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 3 Hogbe-Nlend (via convex bounded sets) are compared.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s Abstract

In this talk, we shall discuss two common techniques in functional analysis: the topological method, the bornological method.

In terms of Pietsch’s operator ideals, we establish the equivalence of the notions of operators, topologies and bornologies. The approaches in the study of locally convex spaces of 1 Grothendieck (via Banach space operators), 2 Randtke (via continuous seminorms) and 3 Hogbe-Nlend (via convex bounded sets) are compared.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s 1 Abstract 2 Preliminaries Uniformities Bornologies 3 General setting Mackey-Aren duality Grothendieck’s construction Randtke’s construction Hogbe-Nlend’s construction The diamond The triangle 4 Applications to operator algebras 5 Applications to LCS’s Schwartz spaces and co-Schwartz spaces Nuclear spaces and co-nuclear spaces

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies But, ... . Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge. Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property.

In particular, every Cauchy sequence is convergent.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge. Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property.

But, ... .

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property.

Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property.

d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... . Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property.

Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... . Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property.

but not converge.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... . Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies while it being Cauchy is a metric property.

Note that the convergence of a sequence (or net) is a topological property,

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... . Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies while it being Cauchy is a metric property.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... . Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge. Note that the convergence of a sequence (or net) is a topological property,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Motivation – completeness

Consider R in its usual metric topology. In particular, every Cauchy sequence is convergent. But, ... . Consider a new metric

−1 −1 d1(x, y) = | tan x − tan y|

on R. d1 defines the same usual topology on R. Now, the sequence xn = n, n = 1, 2,...,

is d1-Cauchy, but not converge. Note that the convergence of a sequence (or net) is a topological property, while it being Cauchy is a metric property.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform, with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N. The uniform topology on X :

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform, with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N. The uniform topology on X :

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform, with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N. The uniform topology on X :

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform, with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

The uniform topology on X :

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform, with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N. The uniform topology on X :

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform,

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N.

The uniform topology on X : UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform,

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N.

The uniform topology on X : UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

Metric spaces are uniform,

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N.

The uniform topology on X : UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N.

The uniform topology on X : UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Continuity in this uniform space is the uniform continuity.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N.

The uniform topology on X : UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform, with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

X is a uniform space when a filter R is given on X × X s.t. Every vicinity N ∈ R contains all (x, x). N ∈ R =⇒ N−1 ∈ R. ∀N ∈ R, ∃M ∈ R s.t. M2 ∈ N.

The uniform topology on X : UN (x) = {y ∈ X :(x, y) ∈ N} forms a local base at x.

A net {xλ} is Cauchy if ∀ vicinity N, ∃λ s.t. α, β > λ =⇒ (xα, xβ) ∈ N. A uniform space X is complete if every Cauchy net converges. Metric spaces are uniform, with a (filter) base for R of Nk = {(x, y) ∈ X × X : d(x, y) < 1/k}. Continuity in this uniform space is the uniform continuity.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies B is bounded if ∀ 0-ngh U, ∃λ > 0 s.t. B ∈ λU. We have seen ‘topology’ ↔ ‘uniformity’. We will see ‘topology’ ↔ ‘bornology’ (= bounded set system) below.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

Every 0-ngh U gives a vicinity NU = {(x, y): x − y ∈ U}.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies We have seen ‘topology’ ↔ ‘uniformity’. We will see ‘topology’ ↔ ‘bornology’ (= bounded set system) below.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

Every 0-ngh U gives a vicinity NU = {(x, y): x − y ∈ U}. B is bounded if ∀ 0-ngh U, ∃λ > 0 s.t. B ∈ λU.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies We will see ‘topology’ ↔ ‘bornology’ (= bounded set system) below.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

Every 0-ngh U gives a vicinity NU = {(x, y): x − y ∈ U}. B is bounded if ∀ 0-ngh U, ∃λ > 0 s.t. B ∈ λU. We have seen ‘topology’ ↔ ‘uniformity’.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

Every 0-ngh U gives a vicinity NU = {(x, y): x − y ∈ U}. B is bounded if ∀ 0-ngh U, ∃λ > 0 s.t. B ∈ λU. We have seen ‘topology’ ↔ ‘uniformity’. We will see ‘topology’ ↔ ‘bornology’ (= bounded set system) below.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A = A1 ∪ A2 ∪ · · · ∪ Ak

such that Ai × Ai ⊆ N. So, uniformly bounded = totally bounded. We need a notion for the usual boundedness.

A is (uniformly) ’bounded’ in X if ∀ vicinity N, can write

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

Let X be a uniform space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies So, uniformly bounded = totally bounded. We need a notion for the usual boundedness.

A = A1 ∪ A2 ∪ · · · ∪ Ak

such that Ai × Ai ⊆ N.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

Let X be a uniform space. A is (uniformly) ’bounded’ in X if ∀ vicinity N, can write

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies So, uniformly bounded = totally bounded. We need a notion for the usual boundedness.

Ai × Ai ⊆ N.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

Let X be a uniform space. A is (uniformly) ’bounded’ in X if ∀ vicinity N, can write

A = A1 ∪ A2 ∪ · · · ∪ Ak

such that

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies So, uniformly bounded = totally bounded. We need a notion for the usual boundedness.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

Let X be a uniform space. A is (uniformly) ’bounded’ in X if ∀ vicinity N, can write

A = A1 ∪ A2 ∪ · · · ∪ Ak

such that Ai × Ai ⊆ N.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies We need a notion for the usual boundedness.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

Let X be a uniform space. A is (uniformly) ’bounded’ in X if ∀ vicinity N, can write

A = A1 ∪ A2 ∪ · · · ∪ Ak

such that Ai × Ai ⊆ N. So, uniformly bounded = totally bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s

Let X be a uniform space. A is (uniformly) ’bounded’ in X if ∀ vicinity N, can write

A = A1 ∪ A2 ∪ · · · ∪ Ak

such that Ai × Ai ⊆ N. So, uniformly bounded = totally bounded. We need a notion for the usual boundedness.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

In the LCS setting, we discuss convex vector bornologies, i.e., 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions:

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X ,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

In the LCS setting, we discuss convex vector bornologies, i.e., 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

called bounded sets, satisfying some natural conditions:

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X ,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In the LCS setting, we discuss convex vector bornologies, i.e., 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions:

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In the LCS setting, we discuss convex vector bornologies, i.e., 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions: 1 Every point (singleton set) is bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In the LCS setting, we discuss convex vector bornologies, i.e., 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

3 Finite unions of bounded sets are bounded.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions: 1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

In the LCS setting, we discuss convex vector bornologies, i.e.,

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions: 1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions: 1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

In the LCS setting, we discuss convex vector bornologies, i.e.,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions: 1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

In the LCS setting, we discuss convex vector bornologies, i.e., 4 Sums of two bounded sets are bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s Bornologies

A bornology of a set X , like a topology, is a family of subsets of X , called bounded sets, satisfying some natural conditions: 1 Every point (singleton set) is bounded. 2 Subsets of bounded sets are bounded. 3 Finite unions of bounded sets are bounded.

In the LCS setting, we discuss convex vector bornologies, i.e., 4 Sums of two bounded sets are bounded. 5 Convex and balanced hulls and scalar multiples of bounded sets are bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A is ‘uniformly (i.e. totally) bounded’ in X if ∀0-ngh V , can write A = A1 ∪ A2 ∪ · · · ∪ Ak , s.t. Ai − Ai ⊆ V , i = 1, 2,..., k.

In other words, ∃ai ∈ Ai s.t.

Ai ⊆ ai + V , i = 1, 2,..., k, or

k A ⊆ ∪i=1(ai + V ).

A ⊆ λV .

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

A is ’topologically’ bounded in X if ∀0-ngh V , ∃λ > 0 s.t.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A = A1 ∪ A2 ∪ · · · ∪ Ak , s.t. Ai − Ai ⊆ V , i = 1, 2,..., k.

In other words, ∃ai ∈ Ai s.t.

Ai ⊆ ai + V , i = 1, 2,..., k, or

k A ⊆ ∪i=1(ai + V ).

A is ‘uniformly (i.e. totally) bounded’ in X if ∀0-ngh V , can write

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

A is ’topologically’ bounded in X if ∀0-ngh V , ∃λ > 0 s.t. A ⊆ λV .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A = A1 ∪ A2 ∪ · · · ∪ Ak , s.t. Ai − Ai ⊆ V , i = 1, 2,..., k.

In other words, ∃ai ∈ Ai s.t.

Ai ⊆ ai + V , i = 1, 2,..., k, or

k A ⊆ ∪i=1(ai + V ).

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

A is ’topologically’ bounded in X if ∀0-ngh V , ∃λ > 0 s.t. A ⊆ λV . A is ‘uniformly (i.e. totally) bounded’ in X if ∀0-ngh V , can write

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Ai − Ai ⊆ V , i = 1, 2,..., k.

In other words, ∃ai ∈ Ai s.t.

Ai ⊆ ai + V , i = 1, 2,..., k, or

k A ⊆ ∪i=1(ai + V ).

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

A is ’topologically’ bounded in X if ∀0-ngh V , ∃λ > 0 s.t. A ⊆ λV . A is ‘uniformly (i.e. totally) bounded’ in X if ∀0-ngh V , can write A = A1 ∪ A2 ∪ · · · ∪ Ak , s.t.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, ∃ai ∈ Ai s.t.

Ai ⊆ ai + V , i = 1, 2,..., k, or

k A ⊆ ∪i=1(ai + V ).

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

A is ’topologically’ bounded in X if ∀0-ngh V , ∃λ > 0 s.t. A ⊆ λV . A is ‘uniformly (i.e. totally) bounded’ in X if ∀0-ngh V , can write A = A1 ∪ A2 ∪ · · · ∪ Ak , s.t. Ai − Ai ⊆ V , i = 1, 2,..., k.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies k A ⊆ ∪i=1(ai + V ).

Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

A is ’topologically’ bounded in X if ∀0-ngh V , ∃λ > 0 s.t. A ⊆ λV . A is ‘uniformly (i.e. totally) bounded’ in X if ∀0-ngh V , can write A = A1 ∪ A2 ∪ · · · ∪ Ak , s.t. Ai − Ai ⊆ V , i = 1, 2,..., k.

In other words, ∃ai ∈ Ai s.t.

Ai ⊆ ai + V , i = 1, 2,..., k, or

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Uniformities General setting Bornologies Applications to operator algebras Applications to LCS’s In a LCS X ,...

A is ’topologically’ bounded in X if ∀0-ngh V , ∃λ > 0 s.t. A ⊆ λV . A is ‘uniformly (i.e. totally) bounded’ in X if ∀0-ngh V , can write A = A1 ∪ A2 ∪ · · · ∪ Ak , s.t. Ai − Ai ⊆ V , i = 1, 2,..., k.

In other words, ∃ai ∈ Ai s.t.

Ai ⊆ ai + V , i = 1, 2,..., k, or

k A ⊆ ∪i=1(ai + V ).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies the bornological property, via TUE , or −1 the topological property, via T UF , of T ,

However, there are a lot of examples indicating that these two machineries are equivalent.

State either

where UE (resp. UF ) is the closed unit ball of E (resp. F ).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Motivation – linear operators

How to describe a linear operator T : E −→ F between Banach spaces E and F ?

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies However, there are a lot of examples indicating that these two machineries are equivalent.

the bornological property, via TUE , or −1 the topological property, via T UF , of T ,

where UE (resp. UF ) is the closed unit ball of E (resp. F ).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Motivation – linear operators

How to describe a linear operator T : E −→ F between Banach spaces E and F ? State either

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies However, there are a lot of examples indicating that these two machineries are equivalent.

−1 the topological property, via T UF , of T ,

where UE (resp. UF ) is the closed unit ball of E (resp. F ).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Motivation – linear operators

How to describe a linear operator T : E −→ F between Banach spaces E and F ? State either

the bornological property, via TUE , or

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies However, there are a lot of examples indicating that these two machineries are equivalent.

where UE (resp. UF ) is the closed unit ball of E (resp. F ).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Motivation – linear operators

How to describe a linear operator T : E −→ F between Banach spaces E and F ? State either

the bornological property, via TUE , or −1 the topological property, via T UF , of T ,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies However, there are a lot of examples indicating that these two machineries are equivalent.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Motivation – linear operators

How to describe a linear operator T : E −→ F between Banach spaces E and F ? State either

the bornological property, via TUE , or −1 the topological property, via T UF , of T ,

where UE (resp. UF ) is the closed unit ball of E (resp. F ).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Motivation – linear operators

How to describe a linear operator T : E −→ F between Banach spaces E and F ? State either

the bornological property, via TUE , or −1 the topological property, via T UF , of T ,

where UE (resp. UF ) is the closed unit ball of E (resp. F ). However, there are a lot of examples indicating that these two machineries are equivalent.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

(i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

−1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F )

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

−1 (i.e., T UF is a 0-neighborhood of E in the norm topology);

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) ⇔ T is continuous

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

T is of finite rank

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

(i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology); T is of finite rank

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

−1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F )

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

−1 (i.e., T UF is a 0-neighborhood of E in the weak topology);

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) ⇔ T is weak-norm continuous

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

T is compact

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology); T is compact

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

T is compact (i.e., TUE is totally bounded in F )

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies −1 ◦ (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on norm compact subsets of E 0

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Examples

T is bounded (i.e., TUE is a bounded subset of F ) −1 ⇔ T is continuous (i.e., T UF is a 0-neighborhood of E in the norm topology);

T is of finite rank (i.e., TUE ⊆ conv{y1, y2,..., yn} for some y1, y2, ..., yn in F ) −1 ⇔ T is weak-norm continuous (i.e., T UF is a 0-neighborhood of E in the weak topology);

T is compact (i.e., TUE is totally bounded in F ) ⇔ T is continuous in the topology of uniform convergence on 0 −1 ◦ norm compact subsets of E (i.e., T UF ⊇ K , the polar of a norm compact subset K of the dual space E 0 of E).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies a neighborhood of zero and a bounded set. It is, however, no longer true in the context of locally convex spaces (LCS’s, shortly). Mackey-Arens’ Theorem indicates that topologies (families of neighborhoods) and bornologies (families of bounded sets) are in dual pair.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

This is because the unit ball of a normed space simultaneously serves as

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies a bounded set. It is, however, no longer true in the context of locally convex spaces (LCS’s, shortly). Mackey-Arens’ Theorem indicates that topologies (families of neighborhoods) and bornologies (families of bounded sets) are in dual pair.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

This is because the unit ball of a normed space simultaneously serves as a neighborhood of zero and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies It is, however, no longer true in the context of locally convex spaces (LCS’s, shortly). Mackey-Arens’ Theorem indicates that topologies (families of neighborhoods) and bornologies (families of bounded sets) are in dual pair.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

This is because the unit ball of a normed space simultaneously serves as a neighborhood of zero and a bounded set.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Mackey-Arens’ Theorem indicates that topologies (families of neighborhoods) and bornologies (families of bounded sets) are in dual pair.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

This is because the unit ball of a normed space simultaneously serves as a neighborhood of zero and a bounded set. It is, however, no longer true in the context of locally convex spaces (LCS’s, shortly).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

This is because the unit ball of a normed space simultaneously serves as a neighborhood of zero and a bounded set. It is, however, no longer true in the context of locally convex spaces (LCS’s, shortly). Mackey-Arens’ Theorem indicates that topologies (families of neighborhoods) and bornologies (families of bounded sets) are in dual pair.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies eλ −→ e in σ(E, F ) if he − eλ, f i −→ 0, ∀f ∈ F .

fλ −→ f in σ(F , E) if he, f − fλi −→ 0, ∀e ∈ E. σ(E, F ) is the weakest vector topology on E s.t. E 0 = F . σ(F , E) is the weakest vector topology on F s.t. F 0 = E. Q: Which topology is the greatest?

Theorem (Mackey-Aren) The Mackey topology τ(E, F ) is the greatest topology on E s.t. E 0 = F.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey-Aren Theorem

Let hE, F i be a dual pair of vector space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies fλ −→ f in σ(F , E) if he, f − fλi −→ 0, ∀e ∈ E. σ(E, F ) is the weakest vector topology on E s.t. E 0 = F . σ(F , E) is the weakest vector topology on F s.t. F 0 = E. Q: Which topology is the greatest?

Theorem (Mackey-Aren) The Mackey topology τ(E, F ) is the greatest topology on E s.t. E 0 = F.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey-Aren Theorem

Let hE, F i be a dual pair of vector space.

eλ −→ e in σ(E, F ) if he − eλ, f i −→ 0, ∀f ∈ F .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies σ(E, F ) is the weakest vector topology on E s.t. E 0 = F . σ(F , E) is the weakest vector topology on F s.t. F 0 = E. Q: Which topology is the greatest?

Theorem (Mackey-Aren) The Mackey topology τ(E, F ) is the greatest topology on E s.t. E 0 = F.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey-Aren Theorem

Let hE, F i be a dual pair of vector space.

eλ −→ e in σ(E, F ) if he − eλ, f i −→ 0, ∀f ∈ F .

fλ −→ f in σ(F , E) if he, f − fλi −→ 0, ∀e ∈ E.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies σ(F , E) is the weakest vector topology on F s.t. F 0 = E. Q: Which topology is the greatest?

Theorem (Mackey-Aren) The Mackey topology τ(E, F ) is the greatest topology on E s.t. E 0 = F.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey-Aren Theorem

Let hE, F i be a dual pair of vector space.

eλ −→ e in σ(E, F ) if he − eλ, f i −→ 0, ∀f ∈ F .

fλ −→ f in σ(F , E) if he, f − fλi −→ 0, ∀e ∈ E. σ(E, F ) is the weakest vector topology on E s.t. E 0 = F .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Q: Which topology is the greatest?

Theorem (Mackey-Aren) The Mackey topology τ(E, F ) is the greatest topology on E s.t. E 0 = F.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey-Aren Theorem

Let hE, F i be a dual pair of vector space.

eλ −→ e in σ(E, F ) if he − eλ, f i −→ 0, ∀f ∈ F .

fλ −→ f in σ(F , E) if he, f − fλi −→ 0, ∀e ∈ E. σ(E, F ) is the weakest vector topology on E s.t. E 0 = F . σ(F , E) is the weakest vector topology on F s.t. F 0 = E.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Theorem (Mackey-Aren) The Mackey topology τ(E, F ) is the greatest topology on E s.t. E 0 = F.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey-Aren Theorem

Let hE, F i be a dual pair of vector space.

eλ −→ e in σ(E, F ) if he − eλ, f i −→ 0, ∀f ∈ F .

fλ −→ f in σ(F , E) if he, f − fλi −→ 0, ∀e ∈ E. σ(E, F ) is the weakest vector topology on E s.t. E 0 = F . σ(F , E) is the weakest vector topology on F s.t. F 0 = E. Q: Which topology is the greatest?

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey-Aren Theorem

Let hE, F i be a dual pair of vector space.

eλ −→ e in σ(E, F ) if he − eλ, f i −→ 0, ∀f ∈ F .

fλ −→ f in σ(F , E) if he, f − fλi −→ 0, ∀e ∈ E. σ(E, F ) is the weakest vector topology on E s.t. E 0 = F . σ(F , E) is the weakest vector topology on F s.t. F 0 = E. Q: Which topology is the greatest?

Theorem (Mackey-Aren) The Mackey topology τ(E, F ) is the greatest topology on E s.t. E 0 = F.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies For A ⊂ E, set

A◦ = {y ∈ F : | ha, yi | ≤ 1, ∀a ∈ A}.

For B ⊂ F , set

B◦ = {x ∈ E : | hx, bi | ≤ 1, ∀b ∈ B}.

T a locally convex topology of E =⇒ T◦ a convex vector bornology of E 0. M a convex vector bornology of E =⇒ M◦ a locally convex topology of E 0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

Let hE, F i be a dual pair.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies For B ⊂ F , set

B◦ = {x ∈ E : | hx, bi | ≤ 1, ∀b ∈ B}.

T a locally convex topology of E =⇒ T◦ a convex vector bornology of E 0. M a convex vector bornology of E =⇒ M◦ a locally convex topology of E 0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

Let hE, F i be a dual pair. For A ⊂ E, set

A◦ = {y ∈ F : | ha, yi | ≤ 1, ∀a ∈ A}.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T a locally convex topology of E =⇒ T◦ a convex vector bornology of E 0. M a convex vector bornology of E =⇒ M◦ a locally convex topology of E 0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

Let hE, F i be a dual pair. For A ⊂ E, set

A◦ = {y ∈ F : | ha, yi | ≤ 1, ∀a ∈ A}.

For B ⊂ F , set

B◦ = {x ∈ E : | hx, bi | ≤ 1, ∀b ∈ B}.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies M a convex vector bornology of E =⇒ M◦ a locally convex topology of E 0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

Let hE, F i be a dual pair. For A ⊂ E, set

A◦ = {y ∈ F : | ha, yi | ≤ 1, ∀a ∈ A}.

For B ⊂ F , set

B◦ = {x ∈ E : | hx, bi | ≤ 1, ∀b ∈ B}.

T a locally convex topology of E =⇒ T◦ a convex vector bornology of E 0.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

Let hE, F i be a dual pair. For A ⊂ E, set

A◦ = {y ∈ F : | ha, yi | ≤ 1, ∀a ∈ A}.

For B ⊂ F , set

B◦ = {x ∈ E : | hx, bi | ≤ 1, ∀b ∈ B}.

T a locally convex topology of E =⇒ T◦ a convex vector bornology of E 0. M a convex vector bornology of E =⇒ M◦ a locally convex topology of E 0.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

(Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

i.e., a barrel.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0, i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0, i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0, i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0, i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0, i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

V a 0-ngh of a LCS E ⇔ V ◦ equicontinuous in E 0. (Bipolar Theorem) A◦◦ = abs conv A. B σ(E, E 0) bounded in E =⇒ B◦ σ(E 0, E) closed, absolutely convex and absorbing in E 0, i.e., a barrel. E is a barrelled space if every barrel is a 0-ngh. E is barrelled ⇔ bounded sets in E 0 are exactly equicontinuous. Equicontinuous sets in E 0 are exactly σ(E 0, E) compact. B σ(E 0, E) compact =⇒ B◦ is a 0-ngh in E. B is bounded =⇒ B◦ is a barrel in E. This defines the strong β(E, E 0) topology on E. Usually, β(E, E 0) is stronger than the original topology. B = {f } =⇒ {f }◦ =∼ [−1, 1] × ker f , a subbasic σ(E, E 0) open set.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies M = finite dimensional bornology of F =⇒ M◦ = σ(E, F ). M = σ(E 0, E)-compact sets =⇒ M◦ = Mackey topology τ(E, E 0). 00 Thus, a Banach space E = E ⇔ the unit ball UE is weakly σ(E, E 0)-compact.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey topology

T = σ(E, F ) =⇒ T◦ = finite dimensional bornology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies M = σ(E 0, E)-compact sets =⇒ M◦ = Mackey topology τ(E, E 0). 00 Thus, a Banach space E = E ⇔ the unit ball UE is weakly σ(E, E 0)-compact.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey topology

T = σ(E, F ) =⇒ T◦ = finite dimensional bornology. M = finite dimensional bornology of F =⇒ M◦ = σ(E, F ).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 00 Thus, a Banach space E = E ⇔ the unit ball UE is weakly σ(E, E 0)-compact.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey topology

T = σ(E, F ) =⇒ T◦ = finite dimensional bornology. M = finite dimensional bornology of F =⇒ M◦ = σ(E, F ). M = σ(E 0, E)-compact sets =⇒ M◦ = Mackey topology τ(E, E 0).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Mackey topology

T = σ(E, F ) =⇒ T◦ = finite dimensional bornology. M = finite dimensional bornology of F =⇒ M◦ = σ(E, F ). M = σ(E 0, E)-compact sets =⇒ M◦ = Mackey topology τ(E, E 0). 00 Thus, a Banach space E = E ⇔ the unit ball UE is weakly σ(E, E 0)-compact.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies by the ideal of nuclear operators. Other examples are those of Schwartz LCS’s by the ideal of precompact operators, and infra–Schwartz LCS’s by the ideal of weakly compact operators, and their “co–spaces”.

A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Other examples are those of Schwartz LCS’s by the ideal of precompact operators, and infra–Schwartz LCS’s by the ideal of weakly compact operators, and their “co–spaces”.

by the ideal of nuclear operators.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them. A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies by the ideal of precompact operators, and infra–Schwartz LCS’s by the ideal of weakly compact operators, and their “co–spaces”.

Other examples are those of Schwartz LCS’s

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them. A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces by the ideal of nuclear operators.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies infra–Schwartz LCS’s by the ideal of weakly compact operators, and their “co–spaces”.

by the ideal of precompact operators, and

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them. A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces by the ideal of nuclear operators. Other examples are those of Schwartz LCS’s

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies by the ideal of weakly compact operators, and their “co–spaces”.

infra–Schwartz LCS’s

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them. A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces by the ideal of nuclear operators. Other examples are those of Schwartz LCS’s by the ideal of precompact operators, and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies their “co–spaces”.

by the ideal of weakly compact operators, and

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them. A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces by the ideal of nuclear operators. Other examples are those of Schwartz LCS’s by the ideal of precompact operators, and infra–Schwartz LCS’s

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies their “co–spaces”.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them. A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces by the ideal of nuclear operators. Other examples are those of Schwartz LCS’s by the ideal of precompact operators, and infra–Schwartz LCS’s by the ideal of weakly compact operators, and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Classifying LCS’s

It is a long tradition of classifying special classes of locally convex spaces by families of continuous operators among them. A famous example is, of course, Grothendieck’s identification of the class of nuclear locally convex spaces by the ideal of nuclear operators. Other examples are those of Schwartz LCS’s by the ideal of precompact operators, and infra–Schwartz LCS’s by the ideal of weakly compact operators, and their “co–spaces”.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type, or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

(resp. Schwartz, infra–Schwartz) if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type, or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

(resp. Schwartz, infra–Schwartz) ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear if ∀ continuous seminorm p on X ,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type, or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

(resp. Schwartz, infra–Schwartz) the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear if ∀ continuous seminorm p on X , ∃ q ≥ p such that

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type, or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

(resp. Schwartz, infra–Schwartz)

nuclear (resp. precompact, weakly compact). Here e denotes completion.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

(resp. Schwartz, infra–Schwartz)

(resp. precompact, weakly compact). Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear Here e denotes completion.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

infra–Schwartz)

weakly compact). Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear (resp. Schwartz, if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, Here e denotes completion.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear (resp. Schwartz, infra–Schwartz) if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear (resp. Schwartz, infra–Schwartz) if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion. Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear (resp. Schwartz, infra–Schwartz) if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion. Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear (resp. Schwartz, infra–Schwartz) if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion. Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type, or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space),

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Grothendieck’s operator theoretical approach

A LCS X is nuclear (resp. Schwartz, infra–Schwartz) if ∀ continuous seminorm p on X , ∃ q ≥ p such that the canonical map −1 −1 Qepq : Xeq = X /^q (0) −→ Xep = X /^p (0) is nuclear (resp. precompact, weakly compact). Here e denotes completion. Alternatively, the completion Xe of X is a topological projective limit lim Q X of Banach spaces of nuclear type ←− epq eq (resp. precompact type, weakly compact type). Call X a Grothendieck space of nuclear (resp. precompact, weakly compact) type, or shortly a Groth(N)–space (resp. Groth(Kp)–space, Groth(W)–space), where N (resp. Kp, W) is the ideal of nuclear (resp. precompact, weakly compact) operators between Banach spaces.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, the convex bornological vector space X equipped with the infracomplete bornology of X is the bornological inductive limit lim J X (A) of Banach spaces of −→ BA type A.

or shortly a co–Groth(A)–space, if for each infracomplete disk A in X there is an infracomplete disk B ⊇ A such that the canonical S S map JBA from X (A) = λ>0 λA into X (B) = λ>0 λB belongs to A(X (A), X (B)).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle co–Grothendieck spaces

A LCS X is a co–Grothendieck space of type A,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, the convex bornological vector space X equipped with the infracomplete bornology of X is the bornological inductive limit lim J X (A) of Banach spaces of −→ BA type A.

if for each infracomplete disk A in X there is an infracomplete disk B ⊇ A such that the canonical S S map JBA from X (A) = λ>0 λA into X (B) = λ>0 λB belongs to A(X (A), X (B)).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle co–Grothendieck spaces

A LCS X is a co–Grothendieck space of type A, or shortly a co–Groth(A)–space,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, the convex bornological vector space X equipped with the infracomplete bornology of X is the bornological inductive limit lim J X (A) of Banach spaces of −→ BA type A.

there is an infracomplete disk B ⊇ A such that the canonical S S map JBA from X (A) = λ>0 λA into X (B) = λ>0 λB belongs to A(X (A), X (B)).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle co–Grothendieck spaces

A LCS X is a co–Grothendieck space of type A, or shortly a co–Groth(A)–space, if for each infracomplete disk A in X

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, the convex bornological vector space X equipped with the infracomplete bornology of X is the bornological inductive limit lim J X (A) of Banach spaces of −→ BA type A.

such that the canonical S S map JBA from X (A) = λ>0 λA into X (B) = λ>0 λB belongs to A(X (A), X (B)).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle co–Grothendieck spaces

A LCS X is a co–Grothendieck space of type A, or shortly a co–Groth(A)–space, if for each infracomplete disk A in X there is an infracomplete disk B ⊇ A

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, the convex bornological vector space X equipped with the infracomplete bornology of X is the bornological inductive limit lim J X (A) of Banach spaces of −→ BA type A.

belongs to A(X (A), X (B)).

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle co–Grothendieck spaces

A LCS X is a co–Grothendieck space of type A, or shortly a co–Groth(A)–space, if for each infracomplete disk A in X there is an infracomplete disk B ⊇ A such that the canonical S S map JBA from X (A) = λ>0 λA into X (B) = λ>0 λB

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, the convex bornological vector space X equipped with the infracomplete bornology of X is the bornological inductive limit lim J X (A) of Banach spaces of −→ BA type A.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle co–Grothendieck spaces

A LCS X is a co–Grothendieck space of type A, or shortly a co–Groth(A)–space, if for each infracomplete disk A in X there is an infracomplete disk B ⊇ A such that the canonical S S map JBA from X (A) = λ>0 λA into X (B) = λ>0 λB belongs to A(X (A), X (B)).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle co–Grothendieck spaces

A LCS X is a co–Grothendieck space of type A, or shortly a co–Groth(A)–space, if for each infracomplete disk A in X there is an infracomplete disk B ⊇ A such that the canonical S S map JBA from X (A) = λ>0 λA into X (B) = λ>0 λB belongs to A(X (A), X (B)). In other words, the convex bornological vector space X equipped with the infracomplete bornology of X is the bornological inductive limit lim J X (A) of Banach spaces of −→ BA type A.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The topology on X defined by the family of all such seminorms is called the A–topology of X . A LCS X is said to be A–topological if the topology of X coincides with the A–topology.

if the canonical map Qep : X → Xep belongs to the injective hull Ainj of A.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Randtke’s topological approach

A continuous seminorm p on a LCS X is an A–continuous seminorm

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The topology on X defined by the family of all such seminorms is called the A–topology of X . A LCS X is said to be A–topological if the topology of X coincides with the A–topology.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Randtke’s topological approach

A continuous seminorm p on a LCS X is an A–continuous seminorm if the canonical map Qep : X → Xep belongs to the injective hull Ainj of A.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A LCS X is said to be A–topological if the topology of X coincides with the A–topology.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Randtke’s topological approach

A continuous seminorm p on a LCS X is an A–continuous seminorm if the canonical map Qep : X → Xep belongs to the injective hull Ainj of A. The topology on X defined by the family of all such seminorms is called the A–topology of X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Randtke’s topological approach

A continuous seminorm p on a LCS X is an A–continuous seminorm if the canonical map Qep : X → Xep belongs to the injective hull Ainj of A. The topology on X defined by the family of all such seminorms is called the A–topology of X . A LCS X is said to be A–topological if the topology of X coincides with the A–topology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The bornology on X defined by the family of all such bounded sets is called the A–bornology of X . A LCS X is said to be A–bornological if the bornology of X coincides with the A–bornology.

S if the canonical map JB from X (B) = λ>0 λB into X belongs to the bornologically surjective hull Absur of A.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Hogbe-Nlend’s bornological approach

An absolutely convex bounded set B in a LCS X is A–bounded

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The bornology on X defined by the family of all such bounded sets is called the A–bornology of X . A LCS X is said to be A–bornological if the bornology of X coincides with the A–bornology.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Hogbe-Nlend’s bornological approach

An absolutely convex bounded set B in a LCS X is S A–bounded if the canonical map JB from X (B) = λ>0 λB into X belongs to the bornologically surjective hull Absur of A.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A LCS X is said to be A–bornological if the bornology of X coincides with the A–bornology.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Hogbe-Nlend’s bornological approach

An absolutely convex bounded set B in a LCS X is S A–bounded if the canonical map JB from X (B) = λ>0 λB into X belongs to the bornologically surjective hull Absur of A. The bornology on X defined by the family of all such bounded sets is called the A–bornology of X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Hogbe-Nlend’s bornological approach

An absolutely convex bounded set B in a LCS X is S A–bounded if the canonical map JB from X (B) = λ>0 λB into X belongs to the bornologically surjective hull Absur of A. The bornology on X defined by the family of all such bounded sets is called the A–bornology of X . A LCS X is said to be A–bornological if the bornology of X coincides with the A–bornology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A locally convex space X is A–topological 0 (resp. A–bornological) if and only if its strong dual Xβ is A–bornological (resp. A–topological). One can discover the same is true for Grothendieck spaces and co–Grothendieck spaces by observing the duality of topology and bornology and the duality of projective limits and inductive limits. Thus these three different approaches of Grothendieck, Randtke and Hogbe-Nlend coincide.

and co-Grothendieck spaces are essentially A–bornological spaces.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Equivalence of three approaches

Grothendieck spaces are essentially A–topological spaces,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A locally convex space X is A–topological 0 (resp. A–bornological) if and only if its strong dual Xβ is A–bornological (resp. A–topological). One can discover the same is true for Grothendieck spaces and co–Grothendieck spaces by observing the duality of topology and bornology and the duality of projective limits and inductive limits. Thus these three different approaches of Grothendieck, Randtke and Hogbe-Nlend coincide.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Equivalence of three approaches

Grothendieck spaces are essentially A–topological spaces, and co-Grothendieck spaces are essentially A–bornological spaces.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies One can discover the same is true for Grothendieck spaces and co–Grothendieck spaces by observing the duality of topology and bornology and the duality of projective limits and inductive limits. Thus these three different approaches of Grothendieck, Randtke and Hogbe-Nlend coincide.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Equivalence of three approaches

Grothendieck spaces are essentially A–topological spaces, and co-Grothendieck spaces are essentially A–bornological spaces. A locally convex space X is A–topological 0 (resp. A–bornological) if and only if its strong dual Xβ is A–bornological (resp. A–topological).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Thus these three different approaches of Grothendieck, Randtke and Hogbe-Nlend coincide.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Equivalence of three approaches

Grothendieck spaces are essentially A–topological spaces, and co-Grothendieck spaces are essentially A–bornological spaces. A locally convex space X is A–topological 0 (resp. A–bornological) if and only if its strong dual Xβ is A–bornological (resp. A–topological). One can discover the same is true for Grothendieck spaces and co–Grothendieck spaces by observing the duality of topology and bornology and the duality of projective limits and inductive limits.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Equivalence of three approaches

Grothendieck spaces are essentially A–topological spaces, and co-Grothendieck spaces are essentially A–bornological spaces. A locally convex space X is A–topological 0 (resp. A–bornological) if and only if its strong dual Xβ is A–bornological (resp. A–topological). One can discover the same is true for Grothendieck spaces and co–Grothendieck spaces by observing the duality of topology and bornology and the duality of projective limits and inductive limits. Thus these three different approaches of Grothendieck, Randtke and Hogbe-Nlend coincide.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

Operators A x GG projective xx ; cG GG inductive x xx GG GG topologies T xx xx GG GGbornologies B xx xx GG GG xx xx GG GG xx x GG GG xx xx b b GG GG xx xx O,O O,O GG G x xx polar P◦ GG GG {x xx GG # Topologiesx P / Bornologies M : o : polar M◦ ÓÓ :: Ó :: Ó : ÓÓ : ÓÓ :: Grothendieck Ó :: Ó : ÓÓ Randtke : ÓÓ Hogbe-Nlend :: Ó :: Ó : ÓÓ : ÓÓ :: Ó :: Ó   }ÕÓÓ Spaces

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies OI1 A(X , Y ) is a nonzero vector subspace of L(X , Y ) for all X , Y in C; and OI2 RTS ∈ A(X0, Y0) whenever R ∈ L(Y , Y0), T ∈ A(X , Y ) and S ∈ L(X0, X ) for any X0, X , Y and Y0 in C.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Details of the “triangle”

(“Operators”) A family A = {A(X , Y ): X , Y ∈ C} of algebras of operators associated to each pair of spaces X and Y in C is called an operator ideal if

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies OI2 RTS ∈ A(X0, Y0) whenever R ∈ L(Y , Y0), T ∈ A(X , Y ) and S ∈ L(X0, X ) for any X0, X , Y and Y0 in C.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Details of the “triangle”

(“Operators”) A family A = {A(X , Y ): X , Y ∈ C} of algebras of operators associated to each pair of spaces X and Y in C is called an operator ideal if

OI1 A(X , Y ) is a nonzero vector subspace of L(X , Y ) for all X , Y in C; and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Details of the “triangle”

(“Operators”) A family A = {A(X , Y ): X , Y ∈ C} of algebras of operators associated to each pair of spaces X and Y in C is called an operator ideal if

OI1 A(X , Y ) is a nonzero vector subspace of L(X , Y ) for all X , Y in C; and OI2 RTS ∈ A(X0, Y0) whenever R ∈ L(Y , Y0), T ∈ A(X , Y ) and S ∈ L(X0, X ) for any X0, X , Y and Y0 in C.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies GB1 Mfin(Y ) ⊆ M(Y ) ⊆ Mvon(Y ) for all Y in C; and × M M GB2 L(X , Y ) ⊆ L (X , Y ) for all X and Y in C.

0 GT1 σ(X , X ) ⊆ P(X ) ⊆ Pori(X ) for all X in C; and GT2 L(X , Y ) ⊆ L(XP, YP) for all X and Y in C. (“Bornologies”) A family M = {M(Y ): Y ∈ C} of convex vector bornologies associated to each space Y in C is called a generating bornology if

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies”) A family P = {P(X ): X ∈ C} of locally convex topologies associated to each space X in C is called a generating topology if

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies GB1 Mfin(Y ) ⊆ M(Y ) ⊆ Mvon(Y ) for all Y in C; and × M M GB2 L(X , Y ) ⊆ L (X , Y ) for all X and Y in C.

GT2 L(X , Y ) ⊆ L(XP, YP) for all X and Y in C. (“Bornologies”) A family M = {M(Y ): Y ∈ C} of convex vector bornologies associated to each space Y in C is called a generating bornology if

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies”) A family P = {P(X ): X ∈ C} of locally convex topologies associated to each space X in C is called a generating topology if 0 GT1 σ(X , X ) ⊆ P(X ) ⊆ Pori(X ) for all X in C; and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies GB1 Mfin(Y ) ⊆ M(Y ) ⊆ Mvon(Y ) for all Y in C; and × M M GB2 L(X , Y ) ⊆ L (X , Y ) for all X and Y in C.

(“Bornologies”) A family M = {M(Y ): Y ∈ C} of convex vector bornologies associated to each space Y in C is called a generating bornology if

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies”) A family P = {P(X ): X ∈ C} of locally convex topologies associated to each space X in C is called a generating topology if 0 GT1 σ(X , X ) ⊆ P(X ) ⊆ Pori(X ) for all X in C; and GT2 L(X , Y ) ⊆ L(XP, YP) for all X and Y in C.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies GB1 Mfin(Y ) ⊆ M(Y ) ⊆ Mvon(Y ) for all Y in C; and × M M GB2 L(X , Y ) ⊆ L (X , Y ) for all X and Y in C.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies”) A family P = {P(X ): X ∈ C} of locally convex topologies associated to each space X in C is called a generating topology if 0 GT1 σ(X , X ) ⊆ P(X ) ⊆ Pori(X ) for all X in C; and GT2 L(X , Y ) ⊆ L(XP, YP) for all X and Y in C. (“Bornologies”) A family M = {M(Y ): Y ∈ C} of convex vector bornologies associated to each space Y in C is called a generating bornology if

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies × M M GB2 L(X , Y ) ⊆ L (X , Y ) for all X and Y in C.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies”) A family P = {P(X ): X ∈ C} of locally convex topologies associated to each space X in C is called a generating topology if 0 GT1 σ(X , X ) ⊆ P(X ) ⊆ Pori(X ) for all X in C; and GT2 L(X , Y ) ⊆ L(XP, YP) for all X and Y in C. (“Bornologies”) A family M = {M(Y ): Y ∈ C} of convex vector bornologies associated to each space Y in C is called a generating bornology if

GB1 Mfin(Y ) ⊆ M(Y ) ⊆ Mvon(Y ) for all Y in C; and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies”) A family P = {P(X ): X ∈ C} of locally convex topologies associated to each space X in C is called a generating topology if 0 GT1 σ(X , X ) ⊆ P(X ) ⊆ Pori(X ) for all X in C; and GT2 L(X , Y ) ⊆ L(XP, YP) for all X and Y in C. (“Bornologies”) A family M = {M(Y ): Y ∈ C} of convex vector bornologies associated to each space Y in C is called a generating bornology if

GB1 Mfin(Y ) ⊆ M(Y ) ⊆ Mvon(Y ) for all Y in C; and × M M GB2 L(X , Y ) ⊆ L (X , Y ) for all X and Y in C.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, a seminorm p of X0 is T(A)(X0)–continuous if and only if there is a T in A(X0, Y ) for some Y in C and a continuous seminorm q of Y such that

p(x) ≤ q(Tx), ∀x ∈ X0.

In this case, we call p an A–seminorm of X0.

(“Operators” → “Topologies”) For each X0 in C, the A–topology of X0, denoted by T(A)(X0), is the projective topology of X0 with respect to the family

{T ∈ A(X0, Y ): Y ∈ C}.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Transformations of “vertices”

Let A be an operator ideal, P a generating topology and M a generating bornology on C.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, a seminorm p of X0 is T(A)(X0)–continuous if and only if there is a T in A(X0, Y ) for some Y in C and a continuous seminorm q of Y such that

p(x) ≤ q(Tx), ∀x ∈ X0.

In this case, we call p an A–seminorm of X0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Transformations of “vertices”

Let A be an operator ideal, P a generating topology and M a generating bornology on C.

(“Operators” → “Topologies”) For each X0 in C, the A–topology of X0, denoted by T(A)(X0), is the projective topology of X0 with respect to the family

{T ∈ A(X0, Y ): Y ∈ C}.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In this case, we call p an A–seminorm of X0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Transformations of “vertices”

Let A be an operator ideal, P a generating topology and M a generating bornology on C.

(“Operators” → “Topologies”) For each X0 in C, the A–topology of X0, denoted by T(A)(X0), is the projective topology of X0 with respect to the family

{T ∈ A(X0, Y ): Y ∈ C}.

In other words, a seminorm p of X0 is T(A)(X0)–continuous if and only if there is a T in A(X0, Y ) for some Y in C and a continuous seminorm q of Y such that

p(x) ≤ q(Tx), ∀x ∈ X0.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle Transformations of “vertices”

Let A be an operator ideal, P a generating topology and M a generating bornology on C.

(“Operators” → “Topologies”) For each X0 in C, the A–topology of X0, denoted by T(A)(X0), is the projective topology of X0 with respect to the family

{T ∈ A(X0, Y ): Y ∈ C}.

In other words, a seminorm p of X0 is T(A)(X0)–continuous if and only if there is a T in A(X0, Y ) for some Y in C and a continuous seminorm q of Y such that

p(x) ≤ q(Tx), ∀x ∈ X0.

In this case, we call p an A–seminorm of X0.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In other words, a subset B of Y0 is B(A)(Y0)–bounded if and only if there is a T in A(X , Y0) for some X in C and a topologically bounded subset A of X such that

B ⊆ TA.

In this case, we call B an A–bounded subset of Y0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Operators” → “Bornologies”) For each Y0 in C, the A–bornology of Y0, denoted by B(A)(Y0), is the inductive bornology of Y0 with respect to the family

{T ∈ A(X , Y0): X ∈ C}.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies In this case, we call B an A–bounded subset of Y0.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Operators” → “Bornologies”) For each Y0 in C, the A–bornology of Y0, denoted by B(A)(Y0), is the inductive bornology of Y0 with respect to the family

{T ∈ A(X , Y0): X ∈ C}.

In other words, a subset B of Y0 is B(A)(Y0)–bounded if and only if there is a T in A(X , Y0) for some X in C and a topologically bounded subset A of X such that

B ⊆ TA.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Operators” → “Bornologies”) For each Y0 in C, the A–bornology of Y0, denoted by B(A)(Y0), is the inductive bornology of Y0 with respect to the family

{T ∈ A(X , Y0): X ∈ C}.

In other words, a subset B of Y0 is B(A)(Y0)–bounded if and only if there is a T in A(X , Y0) for some X in C and a topologically bounded subset A of X such that

B ⊆ TA.

In this case, we call B an A–bounded subset of Y0.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b b O (P)(X , Y ) = L (XP, Y ) be the vector space of those operators sending a P(X )–neighborhood of zero to a bounded set, respectively.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” → “Operators”) For X , Y in C, let

O(P)(X , Y ) = L(XP, Y ) be the vector space of all continuous operators from X into Y which is still continuous with respect to the P(X )–topology, and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” → “Operators”) For X , Y in C, let

O(P)(X , Y ) = L(XP, Y ) be the vector space of all continuous operators from X into Y which is still continuous with respect to the P(X )–topology, and b b O (P)(X , Y ) = L (XP, Y ) be the vector space of those operators sending a P(X )–neighborhood of zero to a bounded set, respectively.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Ob(M)(X , Y ) = Lb(X , Y M)

be the vector space of those operators sending a neighborhood of zero to an M(Y )–bounded set, respectively.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Bornologies” → “Operators”) For X , Y in C, let

O(M)(X , Y ) = L(X , Y ) ∩ L×(X , Y M) be the vector space of all continuous operators from X into Y which send bounded sets to M(Y )–bounded sets, and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Bornologies” → “Operators”) For X , Y in C, let

O(M)(X , Y ) = L(X , Y ) ∩ L×(X , Y M) be the vector space of all continuous operators from X into Y which send bounded sets to M(Y )–bounded sets, and

Ob(M)(X , Y ) = Lb(X , Y M) be the vector space of those operators sending a neighborhood of zero to an M(Y )–bounded set, respectively.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies a bounded subset A of Y is P◦(Y )–bounded if and only if its ◦ 0 polar A is a P(Yβ)–neighborhood of zero; and a neighborhood V of zero of X is a M◦(X )–neighborhood of ◦ 0 zero if and only if V is M(Xβ)–bounded.

(resp. M◦(X )–topology of X ) is defined to be the bornology (resp. topology) polar to P(X ) (resp. M(Y )). More precisely,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” ↔ “Bornologies”) For X , Y in C, the P◦(Y )–bornology of Y

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies a bounded subset A of Y is P◦(Y )–bounded if and only if its ◦ 0 polar A is a P(Yβ)–neighborhood of zero; and a neighborhood V of zero of X is a M◦(X )–neighborhood of ◦ 0 zero if and only if V is M(Xβ)–bounded.

(resp. M◦(X )–topology of X ) (resp. topology) polar to P(X ) (resp. M(Y )). More precisely,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” ↔ “Bornologies”) For X , Y in C, the P◦(Y )–bornology of Y is defined to be the bornology

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies a bounded subset A of Y is P◦(Y )–bounded if and only if its ◦ 0 polar A is a P(Yβ)–neighborhood of zero; and a neighborhood V of zero of X is a M◦(X )–neighborhood of ◦ 0 zero if and only if V is M(Xβ)–bounded.

(resp. M◦(X )–topology of X ) (resp. topology) (resp. M(Y )). More precisely,

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” ↔ “Bornologies”) For X , Y in C, the P◦(Y )–bornology of Y is defined to be the bornology polar to P(X )

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies (resp. M◦(X )–topology of X ) (resp. topology) (resp. M(Y )).

a bounded subset A of Y is P◦(Y )–bounded if and only if its ◦ 0 polar A is a P(Yβ)–neighborhood of zero; and a neighborhood V of zero of X is a M◦(X )–neighborhood of ◦ 0 zero if and only if V is M(Xβ)–bounded.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” ↔ “Bornologies”) For X , Y in C, the P◦(Y )–bornology of Y is defined to be the bornology polar to P(X ) More precisely,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies (resp. M◦(X )–topology of X ) (resp. topology) (resp. M(Y )).

a neighborhood V of zero of X is a M◦(X )–neighborhood of ◦ 0 zero if and only if V is M(Xβ)–bounded.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” ↔ “Bornologies”) For X , Y in C, the P◦(Y )–bornology of Y is defined to be the bornology polar to P(X ) More precisely, a bounded subset A of Y is P◦(Y )–bounded if and only if its ◦ 0 polar A is a P(Yβ)–neighborhood of zero; and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies a neighborhood V of zero of X is a M◦(X )–neighborhood of ◦ 0 zero if and only if V is M(Xβ)–bounded.

Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” ↔ “Bornologies”) For X , Y in C, the P◦(Y )–bornology of Y (resp. M◦(X )–topology of X ) is defined to be the bornology (resp. topology) polar to P(X ) (resp. M(Y )). More precisely, a bounded subset A of Y is P◦(Y )–bounded if and only if its ◦ 0 polar A is a P(Yβ)–neighborhood of zero; and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Mackey-Aren duality Outline Grothendieck’s construction Preliminaries Randtke’s construction General setting Hogbe-Nlend’s construction Applications to operator algebras The diamond Applications to LCS’s The triangle

(“Topologies” ↔ “Bornologies”) For X , Y in C, the P◦(Y )–bornology of Y (resp. M◦(X )–topology of X ) is defined to be the bornology (resp. topology) polar to P(X ) (resp. M(Y )). More precisely, a bounded subset A of Y is P◦(Y )–bounded if and only if its ◦ 0 polar A is a P(Yβ)–neighborhood of zero; and a neighborhood V of zero of X is a M◦(X )–neighborhood of ◦ 0 zero if and only if V is M(Xβ)–bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A locally convex topology P of H is called a generating topology if P consists of norm open sets in H such that all operators in M are P-to-P continuous on H, i.e., the pre-images of P-open sets being P-open. A convex vector bornology M of H is called a generating bornology if M consists of norm bounded subsets of H such that all operators in M are M-to-M bounded, i.e., sending M-bounded sets to M-bounded sets.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s The Hilbert spaces version of the triangle

Let M be a von Neumann algebra of bounded linear operators on a Hilbert space H, and A an arbitrary non-zero two-sided ideal of M.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A convex vector bornology M of H is called a generating bornology if M consists of norm bounded subsets of H such that all operators in M are M-to-M bounded, i.e., sending M-bounded sets to M-bounded sets.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s The Hilbert spaces version of the triangle

Let M be a von Neumann algebra of bounded linear operators on a Hilbert space H, and A an arbitrary non-zero two-sided ideal of M. A locally convex topology P of H is called a generating topology if P consists of norm open sets in H such that all operators in M are P-to-P continuous on H, i.e., the pre-images of P-open sets being P-open.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s The Hilbert spaces version of the triangle

Let M be a von Neumann algebra of bounded linear operators on a Hilbert space H, and A an arbitrary non-zero two-sided ideal of M. A locally convex topology P of H is called a generating topology if P consists of norm open sets in H such that all operators in M are P-to-P continuous on H, i.e., the pre-images of P-open sets being P-open. A convex vector bornology M of H is called a generating bornology if M consists of norm bounded subsets of H such that all operators in M are M-to-M bounded, i.e., sending M-bounded sets to M-bounded sets.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The A-bornology B(A) is the inductive bornology of H induced by operators in A, i.e., the smallest convex vector bornology b of H such that operators in A are norm-to-b bounded. The polar of a subset A in H is

A◦ = {x ∈ H : | ha, xi | ≤ 1, ∀a ∈ A}.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The A-topology T(A) is the projective topology of H induced by operators in A, i.e., the weakest locally convex topology t of H such that operators in A are t-to-norm continuous.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The polar of a subset A in H is

A◦ = {x ∈ H : | ha, xi | ≤ 1, ∀a ∈ A}.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The A-topology T(A) is the projective topology of H induced by operators in A, i.e., the weakest locally convex topology t of H such that operators in A are t-to-norm continuous. The A-bornology B(A) is the inductive bornology of H induced by operators in A, i.e., the smallest convex vector bornology b of H such that operators in A are norm-to-b bounded.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The A-topology T(A) is the projective topology of H induced by operators in A, i.e., the weakest locally convex topology t of H such that operators in A are t-to-norm continuous. The A-bornology B(A) is the inductive bornology of H induced by operators in A, i.e., the smallest convex vector bornology b of H such that operators in A are norm-to-b bounded. The polar of a subset A in H is

A◦ = {x ∈ H : | ha, xi | ≤ 1, ∀a ∈ A}.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The set O(M) = B(H, HM) of all norm-to-M bounded linear operators on H is a two-sided ideals of M.

The polar M◦ = {V ⊆ H : V ◦ is M-bounded} of a generating bornology M is a generating topology.

The A-bornology B(A) is a generating bornology.

2

The set O(P) = L(HP, H) of all P-to-norm continuous linear operators on H is a two-sided ideals of M.

3 The polar P◦ = {B ⊆ H : B◦ is a P-neighborhood of 0} of a generating topology P is a generating bornology.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of G. West (1995)

1 The A-topology T(A) is a generating topology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The set O(M) = B(H, HM) of all norm-to-M bounded linear operators on H is a two-sided ideals of M.

The polar M◦ = {V ⊆ H : V ◦ is M-bounded} of a generating bornology M is a generating topology.

2

The set O(P) = L(HP, H) of all P-to-norm continuous linear operators on H is a two-sided ideals of M.

3 The polar P◦ = {B ⊆ H : B◦ is a P-neighborhood of 0} of a generating topology P is a generating bornology.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of G. West (1995)

1 The A-topology T(A) is a generating topology. The A-bornology B(A) is a generating bornology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The polar M◦ = {V ⊆ H : V ◦ is M-bounded} of a generating bornology M is a generating topology.

The set O(M) = B(H, HM) of all norm-to-M bounded linear operators on H is a two-sided ideals of M.

3 The polar P◦ = {B ⊆ H : B◦ is a P-neighborhood of 0} of a generating topology P is a generating bornology.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of G. West (1995)

1 The A-topology T(A) is a generating topology. The A-bornology B(A) is a generating bornology.

2

The set O(P) = L(HP, H) of all P-to-norm continuous linear operators on H is a two-sided ideals of M.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The polar M◦ = {V ⊆ H : V ◦ is M-bounded} of a generating bornology M is a generating topology.

3 The polar P◦ = {B ⊆ H : B◦ is a P-neighborhood of 0} of a generating topology P is a generating bornology.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of G. West (1995)

1 The A-topology T(A) is a generating topology. The A-bornology B(A) is a generating bornology.

2

The set O(P) = L(HP, H) of all P-to-norm continuous linear operators on H is a two-sided ideals of M. The set O(M) = B(H, HM) of all norm-to-M bounded linear operators on H is a two-sided ideals of M.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The polar M◦ = {V ⊆ H : V ◦ is M-bounded} of a generating bornology M is a generating topology.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of G. West (1995)

1 The A-topology T(A) is a generating topology. The A-bornology B(A) is a generating bornology.

2

The set O(P) = L(HP, H) of all P-to-norm continuous linear operators on H is a two-sided ideals of M. The set O(M) = B(H, HM) of all norm-to-M bounded linear operators on H is a two-sided ideals of M.

3 The polar P◦ = {B ⊆ H : B◦ is a P-neighborhood of 0} of a generating topology P is a generating bornology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of G. West (1995)

1 The A-topology T(A) is a generating topology. The A-bornology B(A) is a generating bornology.

2

The set O(P) = L(HP, H) of all P-to-norm continuous linear operators on H is a two-sided ideals of M. The set O(M) = B(H, HM) of all norm-to-M bounded linear operators on H is a two-sided ideals of M.

3 The polar P◦ = {B ⊆ H : B◦ is a P-neighborhood of 0} of a generating topology P is a generating bornology. The polar M◦ = {V ⊆ H : V ◦ is M-bounded} of a generating bornology M is a generating topology.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies O(T(A)) = A, O(B(A)) = A. T(A)◦ = B(A), B(A)◦ = T(A). T(O(P)) = P, B(O(M)) = M.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The triangle of (ideal) operators, topologies and bornologies is commutative in a von Neumann algebra M:

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T(A)◦ = B(A), B(A)◦ = T(A). T(O(P)) = P, B(O(M)) = M.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The triangle of (ideal) operators, topologies and bornologies is commutative in a von Neumann algebra M: O(T(A)) = A, O(B(A)) = A.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies T(O(P)) = P, B(O(M)) = M.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The triangle of (ideal) operators, topologies and bornologies is commutative in a von Neumann algebra M: O(T(A)) = A, O(B(A)) = A. T(A)◦ = B(A), B(A)◦ = T(A).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

The triangle of (ideal) operators, topologies and bornologies is commutative in a von Neumann algebra M: O(T(A)) = A, O(B(A)) = A. T(A)◦ = B(A), B(A)◦ = T(A). T(O(P)) = P, B(O(M)) = M.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s

Operators A x GG projective xx ; cG GG inductive x xx GG GG topologies T xx xx GG GGbornologies B xx xx GG GG xx xx GG GG xx x GG GG xx xx b b GG GG xx xx O,O O,O GG G x xx polar P◦ GG GG {x xx GG # Topologiesx P / Bornologies M : o : polar M◦ ÓÓ :: Ó :: Ó : ÓÓ : ÓÓ :: Grothendieck Ó :: Ó : ÓÓ Randtke : ÓÓ Hogbe-Nlend :: Ó :: Ó : ÓÓ : ÓÓ :: Ó :: Ó   }ÕÓÓ Spaces

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies i.e., the ideal topology on A generated by weakly compact operators, is the finest locally convex topology on A which coincides with the strong∗ (i.e., the double strong) operator topology on bounded subsets of A. The completion of A under this topology is (A∗∗, τ(A∗∗, A∗)).

The W–topology of A,

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of Jarchow (1986)

Let H be a Hilbert space and A be a C ∗–subalgebra of B(H).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The completion of A under this topology is (A∗∗, τ(A∗∗, A∗)).

i.e., the ideal topology on A generated by weakly compact operators, is the finest locally convex topology on A which coincides with the strong∗ (i.e., the double strong) operator topology on bounded subsets of A.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of Jarchow (1986)

Let H be a Hilbert space and A be a C ∗–subalgebra of B(H). The W–topology of A,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The completion of A under this topology is (A∗∗, τ(A∗∗, A∗)).

is the finest locally convex topology on A which coincides with the strong∗ (i.e., the double strong) operator topology on bounded subsets of A.

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of Jarchow (1986)

Let H be a Hilbert space and A be a C ∗–subalgebra of B(H). The W–topology of A, i.e., the ideal topology on A generated by weakly compact operators,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies The completion of A under this topology is (A∗∗, τ(A∗∗, A∗)).

Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of Jarchow (1986)

Let H be a Hilbert space and A be a C ∗–subalgebra of B(H). The W–topology of A, i.e., the ideal topology on A generated by weakly compact operators, is the finest locally convex topology on A which coincides with the strong∗ (i.e., the double strong) operator topology on bounded subsets of A.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries General setting Applications to operator algebras Applications to LCS’s A theorem of Jarchow (1986)

Let H be a Hilbert space and A be a C ∗–subalgebra of B(H). The W–topology of A, i.e., the ideal topology on A generated by weakly compact operators, is the finest locally convex topology on A which coincides with the strong∗ (i.e., the double strong) operator topology on bounded subsets of A. The completion of A under this topology is (A∗∗, τ(A∗∗, A∗)).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp is precompact. Denote by Ppc (X ) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X . Kp = O(Ppc ) is the ideal of all precompact operators, and b b Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e., precompact–bounded) operators between LCS’s. A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact.

if there exists a (λn) in c0 and an equicontinuous 0 0 sequence {xn} in X such that 0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Schwartz spaces

A continuous seminorm p on a LCS X is said to be precompact

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies and b b Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e., precompact–bounded) operators between LCS’s. A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact.

p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp is precompact. Denote by Ppc (X ) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X . Kp = O(Ppc ) is the ideal of all precompact operators,

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Schwartz spaces

A continuous seminorm p on a LCS X is said to be precompact if there exists a (λn) in c0 and an equicontinuous 0 0 sequence {xn} in X such that 0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies and b b Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e., precompact–bounded) operators between LCS’s. A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact.

Denote by Ppc (X ) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X . Kp = O(Ppc ) is the ideal of all precompact operators,

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Schwartz spaces

A continuous seminorm p on a LCS X is said to be precompact if there exists a (λn) in c0 and an equicontinuous 0 0 sequence {xn} in X such that 0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X . p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp is precompact.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies and b b Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e., precompact–bounded) operators between LCS’s. A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact.

Kp = O(Ppc ) is the ideal of all precompact operators,

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Schwartz spaces

A continuous seminorm p on a LCS X is said to be precompact if there exists a (λn) in c0 and an equicontinuous 0 0 sequence {xn} in X such that 0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X . p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp is precompact. Denote by Ppc (X ) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact.

and b b Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e., precompact–bounded) operators between LCS’s.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Schwartz spaces

A continuous seminorm p on a LCS X is said to be precompact if there exists a (λn) in c0 and an equicontinuous 0 0 sequence {xn} in X such that 0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X . p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp is precompact. Denote by Ppc (X ) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X . Kp = O(Ppc ) is the ideal of all precompact operators,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Schwartz spaces

A continuous seminorm p on a LCS X is said to be precompact if there exists a (λn) in c0 and an equicontinuous 0 0 sequence {xn} in X such that 0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X . p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp is precompact. Denote by Ppc (X ) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X . Kp = O(Ppc ) is the ideal of all precompact operators, and b b Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e., precompact–bounded) operators between LCS’s.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Schwartz spaces

A continuous seminorm p on a LCS X is said to be precompact if there exists a (λn) in c0 and an equicontinuous 0 0 sequence {xn} in X such that 0 p(x) ≤ sup{|λnhx, xni| : n ≥ 1}, ∀x ∈ X . p is a precompact seminorm on a LCS X if and only if the canonical map Qp : X → Xp is precompact. Denote by Ppc (X ) the locally convex (Hausdorff) topology on X defined by all precompact seminorms on X . Kp = O(Ppc ) is the ideal of all precompact operators, and b b Kp = O (T(Kp)) is the ideal of all quasi–Schwartz (i.e., precompact–bounded) operators between LCS’s. A LCS X is said to be a Schwartz space if every continuous seminorm p on X is precompact. ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 9 X is a Kp–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Old results in new context

Let X be a LCS. The following are all equivalent. 1 X is a Schwartz space. 2 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ Kp(Xq, Xp). 3 Qp ∈ Kp(X , Xp) for every continuous seminorm p on X . 4 ∀ 0–neighborhood U in X , ∃ 0–neighborhood V ⊇ U such that the canonical map from X 0(U◦) into X 0(V ◦) is precompact. 5 L(X , N) = Kp(X , N) for every normed (or Banach) space N. b b 6 Kp(X , Y ) = L (X , Y ) for every LCS Y . b 7 Kp(X , Y ) = Kp(X , Y ) for every LCS Y . b 8 L(X , N) = Kp(X , N) for every normed (or Banach) space N. 9 X is a Kp–topological space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 Y is a co–Schwartz space. 2 ∀ bounded disk B in Y , ∃ bounded disk A ⊇ B such that the canonical map JAB ∈ Kp(Y (B), Y (A)).

3 JB ∈ Kp(Y (B), Y ) for each bounded disk B in Y .

4 L(N, Y ) = Kp(N, Y ) for every normed space N. b 5 L (X , Y ) = Kp(X , Y ) for every LCS X .

6 Y is a Kp–bornological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s co–Schwartz spaces

0 A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. The following are all equivalent.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 2 ∀ bounded disk B in Y , ∃ bounded disk A ⊇ B such that the canonical map JAB ∈ Kp(Y (B), Y (A)).

3 JB ∈ Kp(Y (B), Y ) for each bounded disk B in Y .

4 L(N, Y ) = Kp(N, Y ) for every normed space N. b 5 L (X , Y ) = Kp(X , Y ) for every LCS X .

6 Y is a Kp–bornological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s co–Schwartz spaces

0 A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. The following are all equivalent. 1 Y is a co–Schwartz space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 3 JB ∈ Kp(Y (B), Y ) for each bounded disk B in Y .

4 L(N, Y ) = Kp(N, Y ) for every normed space N. b 5 L (X , Y ) = Kp(X , Y ) for every LCS X .

6 Y is a Kp–bornological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s co–Schwartz spaces

0 A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. The following are all equivalent. 1 Y is a co–Schwartz space. 2 ∀ bounded disk B in Y , ∃ bounded disk A ⊇ B such that the canonical map JAB ∈ Kp(Y (B), Y (A)).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 4 L(N, Y ) = Kp(N, Y ) for every normed space N. b 5 L (X , Y ) = Kp(X , Y ) for every LCS X .

6 Y is a Kp–bornological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s co–Schwartz spaces

0 A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. The following are all equivalent. 1 Y is a co–Schwartz space. 2 ∀ bounded disk B in Y , ∃ bounded disk A ⊇ B such that the canonical map JAB ∈ Kp(Y (B), Y (A)).

3 JB ∈ Kp(Y (B), Y ) for each bounded disk B in Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b 5 L (X , Y ) = Kp(X , Y ) for every LCS X .

6 Y is a Kp–bornological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s co–Schwartz spaces

0 A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. The following are all equivalent. 1 Y is a co–Schwartz space. 2 ∀ bounded disk B in Y , ∃ bounded disk A ⊇ B such that the canonical map JAB ∈ Kp(Y (B), Y (A)).

3 JB ∈ Kp(Y (B), Y ) for each bounded disk B in Y .

4 L(N, Y ) = Kp(N, Y ) for every normed space N.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 6 Y is a Kp–bornological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s co–Schwartz spaces

0 A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. The following are all equivalent. 1 Y is a co–Schwartz space. 2 ∀ bounded disk B in Y , ∃ bounded disk A ⊇ B such that the canonical map JAB ∈ Kp(Y (B), Y (A)).

3 JB ∈ Kp(Y (B), Y ) for each bounded disk B in Y .

4 L(N, Y ) = Kp(N, Y ) for every normed space N. b 5 L (X , Y ) = Kp(X , Y ) for every LCS X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s co–Schwartz spaces

0 A LCS Y is said to be a co–Schwartz space if its strong dual Yβ is a Schwartz space. The following are all equivalent. 1 Y is a co–Schwartz space. 2 ∀ bounded disk B in Y , ∃ bounded disk A ⊇ B such that the canonical map JAB ∈ Kp(Y (B), Y (A)).

3 JB ∈ Kp(Y (B), Y ) for each bounded disk B in Y .

4 L(N, Y ) = Kp(N, Y ) for every normed space N. b 5 L (X , Y ) = Kp(X , Y ) for every LCS X .

6 Y is a Kp–bornological space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 1 X is a Schwartz space (resp. co–Schwartz space).

2 0 Xβ is a co–Schwartz space (resp. Schwartz space). 3 00 Xββ is a Schwartz space (resp. co–Schwartz space).

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Duality

Let X be an infrabarrelled LCS. The following are all equivalent.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 2 0 Xβ is a co–Schwartz space (resp. Schwartz space). 3 00 Xββ is a Schwartz space (resp. co–Schwartz space).

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Duality

Let X be an infrabarrelled LCS. The following are all equivalent.

1 X is a Schwartz space (resp. co–Schwartz space).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 3 00 Xββ is a Schwartz space (resp. co–Schwartz space).

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Duality

Let X be an infrabarrelled LCS. The following are all equivalent.

1 X is a Schwartz space (resp. co–Schwartz space).

2 0 Xβ is a co–Schwartz space (resp. Schwartz space).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Duality

Let X be an infrabarrelled LCS. The following are all equivalent.

1 X is a Schwartz space (resp. co–Schwartz space).

2 0 Xβ is a co–Schwartz space (resp. Schwartz space). 3 00 Xββ is a Schwartz space (resp. co–Schwartz space).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Let Pas(X ) be the locally convex (Hausdorff) topology on X generated by the family of all absolutely summing seminorms on X . A continuous operator T from a LCS X into a LCS Y is said to be absolutely summing if

T ∈ O(Pas)(X , Y ) = L(XPas , Y ).

if there exists a σ(X 0, X )–closed equicontinuous subset B of X 0 and a positive Radon measure µ on B such that Z p(x) ≤ |hx, x0i|dµ(x0), ∀x ∈ X . B

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Absolutely summing operators

A continuous seminorm p on a LCS X is called an absolutely summing seminorm

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies if

T ∈ O(Pas)(X , Y ) = L(XPas , Y ).

Let Pas(X ) be the locally convex (Hausdorff) topology on X generated by the family of all absolutely summing seminorms on X . A continuous operator T from a LCS X into a LCS Y is said to be absolutely summing

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Absolutely summing operators

A continuous seminorm p on a LCS X is called an absolutely summing seminorm if there exists a σ(X 0, X )–closed equicontinuous subset B of X 0 and a positive Radon measure µ on B such that Z p(x) ≤ |hx, x0i|dµ(x0), ∀x ∈ X . B

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies if

T ∈ O(Pas)(X , Y ) = L(XPas , Y ).

A continuous operator T from a LCS X into a LCS Y is said to be absolutely summing

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Absolutely summing operators

A continuous seminorm p on a LCS X is called an absolutely summing seminorm if there exists a σ(X 0, X )–closed equicontinuous subset B of X 0 and a positive Radon measure µ on B such that Z p(x) ≤ |hx, x0i|dµ(x0), ∀x ∈ X . B Let Pas(X ) be the locally convex (Hausdorff) topology on X generated by the family of all absolutely summing seminorms on X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies if

T ∈ O(Pas)(X , Y ) = L(XPas , Y ).

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Absolutely summing operators

A continuous seminorm p on a LCS X is called an absolutely summing seminorm if there exists a σ(X 0, X )–closed equicontinuous subset B of X 0 and a positive Radon measure µ on B such that Z p(x) ≤ |hx, x0i|dµ(x0), ∀x ∈ X . B Let Pas(X ) be the locally convex (Hausdorff) topology on X generated by the family of all absolutely summing seminorms on X . A continuous operator T from a LCS X into a LCS Y is said to be absolutely summing

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Absolutely summing operators

A continuous seminorm p on a LCS X is called an absolutely summing seminorm if there exists a σ(X 0, X )–closed equicontinuous subset B of X 0 and a positive Radon measure µ on B such that Z p(x) ≤ |hx, x0i|dµ(x0), ∀x ∈ X . B Let Pas(X ) be the locally convex (Hausdorff) topology on X generated by the family of all absolutely summing seminorms on X . A continuous operator T from a LCS X into a LCS Y is said to be absolutely summing if

T ∈ O(Pas)(X , Y ) = L(XPas , Y ).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Denote by P = O(Pas) the injective ideal of all absolutely summing operators between LCS’s, and b b by P = O (Pas) the injective ideal of prenuclear–bounded operators

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

In case X and Y are Banach spaces, T is absolutely summing if and only if T sends every weakly summable series in X to an absolutely summable series in Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b b by P = O (Pas) the injective ideal of prenuclear–bounded operators

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

In case X and Y are Banach spaces, T is absolutely summing if and only if T sends every weakly summable series in X to an absolutely summable series in Y .

Denote by P = O(Pas) the injective ideal of all absolutely summing operators between LCS’s, and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

In case X and Y are Banach spaces, T is absolutely summing if and only if T sends every weakly summable series in X to an absolutely summable series in Y .

Denote by P = O(Pas) the injective ideal of all absolutely summing operators between LCS’s, and b b by P = O (Pas) the injective ideal of prenuclear–bounded operators

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn,

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn,

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn,

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

T = Σnλnan ⊗ yn,

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn,

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn, Denote by N the ideal of all nuclear operators between LCS’s.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn, Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn, Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s Nuclear operators and nuclear spaces

A continuous operator T from a LCS X into a LCS Y is said to be nuclear if there exist a (λn) in `1, an equicontinuous 0 sequence {an} in X and a sequence {yn} contained in an infracomplete bounded disk B in Y such that

T = Σnλnan ⊗ yn, Denote by N the ideal of all nuclear operators between LCS’s. It is more or less classical that P = Prup, N = Ninf and P3 ⊂ N ⊂ P . B B B B B A LCS X is said to be nuclear if every continuous seminorm p on X is absolutely summing. 0 A LCS Y is said to be co–nuclear if its strong dual Yβ is nuclear.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 2 Qp ∈ P(X , Xp) for every continuous seminorm p on X . 3 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ P(Xq, Xp).

4 idX ∈ P(X , X ). 5 P(X , Y ) = L(X , Y ) for every LCS Y . 6 P(X , N) = L(X , N) for every normed space N. 7 X is a P–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

Let X be a LCS. The following are all equivalent. 1 X is a nuclear space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 3 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ P(Xq, Xp).

4 idX ∈ P(X , X ). 5 P(X , Y ) = L(X , Y ) for every LCS Y . 6 P(X , N) = L(X , N) for every normed space N. 7 X is a P–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

Let X be a LCS. The following are all equivalent. 1 X is a nuclear space.

2 Qp ∈ P(X , Xp) for every continuous seminorm p on X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 4 idX ∈ P(X , X ). 5 P(X , Y ) = L(X , Y ) for every LCS Y . 6 P(X , N) = L(X , N) for every normed space N. 7 X is a P–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

Let X be a LCS. The following are all equivalent. 1 X is a nuclear space.

2 Qp ∈ P(X , Xp) for every continuous seminorm p on X . 3 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ P(Xq, Xp).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 5 P(X , Y ) = L(X , Y ) for every LCS Y . 6 P(X , N) = L(X , N) for every normed space N. 7 X is a P–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

Let X be a LCS. The following are all equivalent. 1 X is a nuclear space.

2 Qp ∈ P(X , Xp) for every continuous seminorm p on X . 3 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ P(Xq, Xp).

4 idX ∈ P(X , X ).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 6 P(X , N) = L(X , N) for every normed space N. 7 X is a P–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

Let X be a LCS. The following are all equivalent. 1 X is a nuclear space.

2 Qp ∈ P(X , Xp) for every continuous seminorm p on X . 3 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ P(Xq, Xp).

4 idX ∈ P(X , X ). 5 P(X , Y ) = L(X , Y ) for every LCS Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 7 X is a P–topological space.

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

Let X be a LCS. The following are all equivalent. 1 X is a nuclear space.

2 Qp ∈ P(X , Xp) for every continuous seminorm p on X . 3 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ P(Xq, Xp).

4 idX ∈ P(X , X ). 5 P(X , Y ) = L(X , Y ) for every LCS Y . 6 P(X , N) = L(X , N) for every normed space N.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

Let X be a LCS. The following are all equivalent. 1 X is a nuclear space.

2 Qp ∈ P(X , Xp) for every continuous seminorm p on X . 3 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qpq ∈ P(Xq, Xp).

4 idX ∈ P(X , X ). 5 P(X , Y ) = L(X , Y ) for every LCS Y . 6 P(X , N) = L(X , N) for every normed space N. 7 X is a P–topological space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep).

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s ... and more equivalences

1 X is a nuclear space. 2 Qep ∈ N(X , Xep) for every continuous seminorm p on X . 3 N(X , F ) = L(X , F ) for every Banach space F . 4 ∀ continuous seminorm p on X , ∃ continuous seminorm q ≥ p such that Qepq ∈ N(Xeq, Xep). 5 ∀ 0–neighborhood V in X , ∃ 0–neighborhood U ⊆ V such that the canonical map X 0(V ◦) → X 0(U◦) is nuclear. 6 X is a N–topological space. b b 7 L (X , Y ) ∩ P(X , Y ) = P (X , Y ) for every LCS Y . b b 8 P (X , Y ) = L (X , Y ) for every LCS Y . b 9 L (X , Y ) ⊆ P(X , Y ) for every LCS Y . b 10 Kp(X , Y ) ⊆ P (X , Y ) for every LCS Y .

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Full paper of this talk can be found in http:\\www.math.nsysu.edu.tw\ ∼wong Thank you!

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

1 Y.-C. Wong and N.-C. Wong, “Topologies and bornologies determined by operator ideals,” Math Ann, 282, 587–614, 1988. 2 N.-C. Wong and Y.-C. Wong, “The bornologically surjective hull of an operator ideal on locally convex spaces,” Math. Nachr. 60, 265–275, 1993. 3 N.-C. Wong, “Topologies and bornologies determined by operator ideals, II,” Studia Math. 111, 153–162, 1994. 4 N.-C. Wong, “Triangle of operators, topology, bornology”, in the Proceedings of the 3rd International Congress of Chinese Mathematicans (2004), 395-421, AMS/IP, Studies in Advanced Math. 42, 2008.

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Thank you!

Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

1 Y.-C. Wong and N.-C. Wong, “Topologies and bornologies determined by operator ideals,” Math Ann, 282, 587–614, 1988. 2 N.-C. Wong and Y.-C. Wong, “The bornologically surjective hull of an operator ideal on locally convex spaces,” Math. Nachr. 60, 265–275, 1993. 3 N.-C. Wong, “Topologies and bornologies determined by operator ideals, II,” Studia Math. 111, 153–162, 1994. 4 N.-C. Wong, “Triangle of operators, topology, bornology”, in the Proceedings of the 3rd International Congress of Chinese Mathematicans (2004), 395-421, AMS/IP, Studies in Advanced Math. 42, 2008. Full paper of this talk can be found in http:\\www.math.nsysu.edu.tw\ ∼wong

??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies Abstract Outline Preliminaries Schwartz spaces and co-Schwartz spaces General setting Nuclear spaces and co-nuclear spaces Applications to operator algebras Applications to LCS’s

1 Y.-C. Wong and N.-C. Wong, “Topologies and bornologies determined by operator ideals,” Math Ann, 282, 587–614, 1988. 2 N.-C. Wong and Y.-C. Wong, “The bornologically surjective hull of an operator ideal on locally convex spaces,” Math. Nachr. 60, 265–275, 1993. 3 N.-C. Wong, “Topologies and bornologies determined by operator ideals, II,” Studia Math. 111, 153–162, 1994. 4 N.-C. Wong, “Triangle of operators, topology, bornology”, in the Proceedings of the 3rd International Congress of Chinese Mathematicans (2004), 395-421, AMS/IP, Studies in Advanced Math. 42, 2008. Full paper of this talk can be found in http:\\www.math.nsysu.edu.tw\ ∼wong Thank you! ??? %%% Ngai-Ching Wong The triangle of operators, topologies, bornologies