aarjaneiro/planetmath huggingface dataset:
locally compact quantum groups: uniform continuity

locally compact quantum groups: uniform continuity

0.1

Uniform continuity over locally compact quantum groups (LCG)

One can consider

locally compact quantum groups

(

L

⁢

C

⁢

G

) to be defined as
a particular case of

locally compact

quantum groupoids

(

L

⁢

C

⁢

Q

⁢

G

n

) when the
object space of the

L

⁢

C

⁢

Q

⁢

G

n

consists of just one object whose elements are, for example,
those of a

(

non-commutative

)

Hopf algebra

. This is also consistent with the definition introduced by Kustermans and Vaes for a locally compact quantum group by including a Haar measure system associated with the

quantum group

.

0.2

Operator system containg the C*-algebra

Let us consider

L

⁢

C

⁢

G

to be a locally compact quantum group. Then consider the space

L

⁢

U

⁢

C

⁢

(

G

)

of left

uniformly continuous

elements in

L

∞

⁢

(

G

)

introduced in ref.

[

1

]

.
The definition according to

V. Runde

(

loc. cit.) covers both the space of left uniformly continuous functions on a

locally compact group

and (Granirer’s) uniformly continuous

functionals

on the Fourier algebra.

With the above definition of

L

⁢

C

⁢

G

, and with

G

being a group, and also the essential data specified in the previous

section

,

L

⁢

U

⁢

C

⁢

(

G

)

is an

operator system containing the C*-

algebra

C

o

⁢

(

G

)

.

References

1

V. Runde. 2008. Uniform continuity over locally compact quantum groups.

http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.2053v4.pdf

(math.OA -arxiv/0802.2053v4).

Title

locally compact quantum groups: uniform continuity

Canonical name

LocallyCompactQuantumGroupsUniformContinuity

Date of creation

2013-03-22 18:21:14

Last modified on

2013-03-22 18:21:14

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Example

Classification

msc 81T05

Classification

msc 57T05

Classification

msc 81R15

Classification

msc 22A22

Classification

msc 81R50

Classification

msc 54E15

Synonym

uniform continuity over topological groups associated with Hopf algebras

Related topic

QuantumGroups

Related topic

C_cG

Related topic

LocallyCompact

Related topic

LocallyCompactGroupoids

Related topic

WeakHopfCAlgebra2

Related topic

LocallyCompactQuantumGroup

Defines

left uniformly continuous functions on a locally compact group

Defines

uniformly continuous functionals on the Fourier algebra

Generated on Sat Feb 10 13:06:36 2018 by

LaTeXML
nuclear C*-algebra

nuclear C*-algebra

Definition 0.1

.

A C*-

algebra

A

is called a

nuclear

C*-algebra if all C*-norms on every algebraic

tensor product

A

⊗

X

, of

A

with any other C*-algebra

X

, agree with, and also equal the spatial C*-norm (

viz

Lance, 1981). Therefore, there is a unique

completion

of

A

⊗

X

to a C*-algebra , for any other C*-algebra

X

.

0.1

Examples of nuclear C*-algebras

•

All

commutative

C*-algebras and all

finite-dimensional

C*-algebras

•

Group C*-algebras of amenable groups

•

Crossed products of strongly amenable C*-algebras by amenable

discrete groups

,

•

Type

1

C*-algebras.

0.2

Exact C*-algebra

In general terms, a

C

*

-algebra is exact if it is

isomorphic

with a

C

*

-subalgebra of some nuclear

C

*

-algebra. The precise definition of an

exact

C

*

-algebra

follows.

Definition 0.2

.

Let

M

n

be a matrix space, let

𝒜

be a general operator space, and also let

ℂ

be a C*-algebra.
A

C

*

-algebra

ℂ

is exact if it is ‘finitely

representable

’ in

M

n

, that is, if for every finite dimensional

subspace

E

in

𝒜

and quantity

e

⁢

p

⁢

s

⁢

i

⁢

l

⁢

o

⁢

n

>

0

, there exists a subspace

F

of some

M

n

, and
also a

linear isomorphism

T

:

E

→

F

such that the

c

⁢

b

-norm

|

T

|

c

⁢

b

⁢

|

T

-

1

|

c

⁢

b

<

1

+

e

⁢

p

⁢

s

⁢

i

⁢

l

⁢

o

⁢

n

.

0.3

Note: A counter-example

A

C

*

-subalgebra of a nuclear C*-algebra

need not be

nuclear.

References

1

E. C. Lance. 1981. Tensor Products and nuclear C*-algebras., in

Operator
Algebras and Applications,

R.V. Kadison, ed., Proceed. Symp. Pure Maths.,

38

: 379-399, part 1.

2

N. P. Landsman. 1998. “Lecture notes on

C

*

-algebras, Hilbert

C

*

-Modules and Quantum Mechanics”, pp. 89

http://planetmath.org/?op=getobj&from=books&id=66

a graduate level preprint discussing general C*-algebras

http://aux.planetmath.org/files/books/66/C*algebrae.ps

in Postscript format.

Title

nuclear C*-algebra

Canonical name

NuclearCalgebra

Date of creation

2013-03-22 18:12:25

Last modified on

2013-03-22 18:12:25

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

63

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81T05

Classification

msc 81R50

Classification

msc 81R15

Synonym

quantum operator algebra

Synonym

C*-algebra

Synonym

C

*

-algebra

Related topic

QuantumOperatorAlgebrasInQuantumFieldTheories

Related topic

NoncommutativeGeometry

Related topic

GroupoidCConvolutionAlgebra

Related topic

GroupoidCDynamicalSystem

Related topic

CAlgebra3

Related topic

CAlgebra

Related topic

QuotientsInCAlgebras

Defines

generated C*-algebra

Defines

exact C^*-algebra

Defines

group C*-algebra

Generated on Sat Feb 10 13:06:40 2018 by

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R-algebroid

R-algebroid

Definition 0.1

.

If

𝖦

is a

groupoid

(for example, regarded as a

category

with all

morphisms

invertible

)
then we can

construct

an

R

-algebroid,

R

⁢

𝖦

as follows. Let us consider first a module over a ring

R

, also called a

R

-module

, that is, a module (

http://planetmath.org/Module

)

M

R

that takes its

coefficients

in a ring

R

. Then, the object set of

R

⁢

𝖦

is the same as that of

𝖦

and

R

⁢

𝖦

⁢

(

b

,

c

)

is the free

R

-module on the set

𝖦

⁢

(

b

,

c

)

, with

composition

given by the usual bilinear rule, extending the composition of

𝖦

.

Definition 0.2

.

Alternatively, one can define

R

¯

⁢

𝖦

⁢

(

b

,

c

)

to be the set of functions

𝖦

⁢

(

b

,

c

)

⟶

R

with

finite support

, and then one defines the

convolution product

as follows:

(

f

*

g

)

⁢

(

z

)

=

∑

{

(

f

⁢

x

)

⁢

(

g

⁢

y

)

∣

z

=

x

∘

y

}

.

(0.1)

Remark 0.1

.

As it is very well known, only the second construction is natural
for the topological case, when one needs to replace the general

concept

of ‘function’ by
the topological-analytical concept of ‘

continuous function

with compact support’ (or alternatively, with ‘

locally
compact

support

’) for all

quantum field theory

(QFT) extended

symmetry

sectors; in this case, one has that

R

≅

ℂ

.
The point made here is that to carry out the usual construction and end up with only an

algebra

rather than an algebroid, is a procedure analogous to replacing a
groupoid

𝖦

by a semigroup

G

′

=

G

∪

{

0

}

in which the
compositions not defined in

G

are defined to be

0

in

G

′

. We
argue that this construction removes the main advantage of
groupoids, namely the presence of the

spatial component

given by the set of objects of the groupoid.

More generally, a

R-category

(

http://planetmath.org/RCategory

) is similarly defined as an

extension

to this R-algebroid
concept.

References

1

R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.

2

G. H. Mosa:

Higher dimensional algebroids and Crossed
complexes

, PhD thesis, University of Wales, Bangor, (1986). (supervised by R. Brown).

Title

R-algebroid

Canonical name

Ralgebroid

Date of creation

2013-03-22 18:14:19

Last modified on

2013-03-22 18:14:19

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

25

Author

bci1 (20947)

Entry type

Definition

Classification

msc 81T10

Classification

msc 81P05

Classification

msc 81T05

Classification

msc 81R10

Classification

msc 81R50

Synonym

groupoid-derived algebroids

Synonym

double groupoid

dual of an algebroid

Related topic

Module

Related topic

RCategory

Related topic

Algebroids

Related topic

HamiltonianAlgebroids

Related topic

RSupercategory

Related topic

SuperalgebroidsAndHigherDimensionalAlgebroids

Related topic

CategoricalAlgebras

Defines

R

-module

Defines

convolution product

Defines

R-algebroid

Generated on Sat Feb 10 13:06:43 2018 by

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approximation theorem for an arbitrary space

approximation theorem for an arbitrary space

Theorem 0.1

.

(

Approximation theorem for an arbitrary

topological space

in terms of the

colimit

of a

sequence

of cellular

inclusions

of

C

⁢

W

-complexes

):

“There is a

functor

Γ

:

𝒉𝑼

⟶

𝒉𝑼

where

hU

is the

homotopy category

for unbased spaces , and a

natural transformation

γ

:

Γ

⟶

I

⁢

d

that asssigns a

C

⁢

W

-complex

Γ

⁢

X

and a

weak equivalence

γ

e

:

Γ

⁢

X

⟶

X

to an arbitrary space

X

, such that the following

diagram

commutes:

Γ

⁢

X

→

Γ

⁢

f

Γ

⁢

Y

⁢

γ

⁢

(

X

)

↓

↓

γ

⁢

(

Y

)

X

⁢

@

>

f

≫

Y

and

Γ

⁢

f

:

Γ

⁢

X

→

Γ

⁢

Y

is unique up to

homotopy equivalence

.”

(viz. p. 75 in ref.

[

1

]

).

Remark 0.1

.

The

C

⁢

W

-complex specified in the
approximation theorem for an arbitrary space (

http://planetmath.org/ApproximationTheoremForAnArbitrarySpace

) is constructed as the colimit

Γ

⁢

X

of a sequence of cellular inclusions of

C

⁢

W

-complexes

X

1

,

…

,

X

n

, so
that one obtains

X

≡

c

⁢

o

⁢

l

⁢

i

⁢

m

⁢

[

X

i

]

. As a

consequence

of J.H.C. Whitehead’s

Theorem

, one also has that:

γ

*

:

[

Γ

X

,

Γ

Y

]

⟶

[

Γ

X

,

Y

]

is an

isomorphism

.

Furthermore, the

homotopy groups

of the

C

⁢

W

-complex

Γ

⁢

X

are the colimits of the
homotopy groups of

X

n

and

γ

n

+

1

:

π

q

⁢

(

X

n

+

1

)

⟼

π

q

⁢

(

X

)

is a group epimorphism.

References

1

May, J.P. 1999,

A Concise Course in Algebraic Topology.

, The University of Chicago Press: Chicago

Title

approximation theorem for an arbitrary space

Canonical name

ApproximationTheoremForAnArbitrarySpace

Date of creation

2013-03-22 18:14:40

Last modified on

2013-03-22 18:14:40

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

43

Author

bci1 (20947)

Entry type

Theorem

Classification

msc 81T25

Classification

msc 81T05

Classification

msc 81T10

Classification

msc 55U15

Classification

msc 57Q05

Classification

msc 57Q55

Classification

msc 55U05

Classification

msc 55U10

Synonym

approximation theorems for topological spaces

Related topic

TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT

Related topic

CWComplex

Related topic

SpinNetworksAndSpinFoams

Related topic

HomotopyCategory

Related topic

WeakHomotopyEquivalence

Related topic

GroupHomomorphism

Related topic

ApproximationTheoremAppliedToWhitneyCrMNSpaces

Defines

unique colimit of a sequence of cellular inclusions of

C

⁢

W

-complexes

Generated on Sat Feb 10 13:06:59 2018 by

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C⁢W-complex approximation of quantum state spaces in QAT

C

⁢

W

-complex approximation of quantum state spaces in QAT

Theorem 1.

Let

[

Q

⁢

F

j

]

j

=

1

,

…

,

n

be a complete sequence of commuting quantum spin ‘foams’
(QSFs) in an arbitrary

quantum state space

(QSS) (

http://planetmath.org/QuantumSpaceTimes

), and let

(

Q

⁢

F

j

,

Q

⁢

S

⁢

S

j

)

be the corresponding

sequence

of pair

subspaces

of QST. If

Z

j

is a sequence of

CW-complexes

such that for any

j

,

Q

⁢

F

j

⊂

Z

j

, then there exists a sequence of

n

-connected models

(

Q

⁢

F

j

,

Z

j

)

of

(

Q

⁢

F

j

,

Q

⁢

S

⁢

S

j

)

and a sequence of induced

isomorphisms

f

*

j

:

π

i

⁢

(

Z

j

)

→

π

i

⁢

(

Q

⁢

S

⁢

S

j

)

for

i

>

n

, together with a sequence of induced

monomorphisms

for

i

=

n

.

Remark 0.1

.

There exist

weak

homotopy equivalences

between each

Z

j

and

Q

⁢

S

⁢

S

j

spaces
in such a sequence. Therefore, there exists a

C

⁢

W

–complex approximation of QSS defined by the sequence

[

Z

j

]

j

=

1

,

…

,

n

of CW-complexes with

dimension

n

≥

2

. This

C

⁢

W

–approximation is
unique up to

regular

homotopy equivalence.

Corollary 2.

The

n

-connected models

(

Q

⁢

F

j

,

Z

j

)

of

(

Q

⁢

F

j

,

Q

⁢

S

⁢

S

j

)

form the

Model Category

of
Quantum Spin Foams (

http://planetmath.org/SpinNetworksAndSpinFoams

)

(

Q

⁢

F

j

)

,

whose

morphisms

are maps

h

j

⁢

k

:

Z

j

→

Z

k

such that

h

j

⁢

k

∣

Q

F

j

=

g

:

(

Q

S

S

j

,

Q

F

j

)

→

(

Q

S

S

k

,

Q

F

k

)

, and also such that the following

diagram

is

commutative

:

Z

j

→

f

j

Q

⁢

S

⁢

S

j

h

j

⁢

k

↓

↓

g

Z

k

⁢

@

>

f

k

≫

Q

⁢

S

⁢

S

k

Furthermore, the maps

h

j

⁢

k

are unique up to the

homotopy

rel

Q

⁢

F

j

, and also rel

Q

⁢

F

k

.

Remark 0.2

.

Theorem 1 complements other data presented in the parent entry on QAT (

http://planetmath.org/QuantumAlgebraicTopology

).

Title

C

⁢

W

-complex approximation of quantum state spaces in QAT

Canonical name

CWcomplexApproximationOfQuantumStateSpacesInQAT

Date of creation

2013-03-22 18:14:37

Last modified on

2013-03-22 18:14:37

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

29

Author

bci1 (20947)

Entry type

Theorem

Classification

msc 81T25

Classification

msc 81T05

Classification

msc 81T10

Synonym

quantum spin networks approximations by

C

⁢

W

-complexes

Related topic

ApproximationTheoremForAnArbitrarySpace

Related topic

HomotopyEquivalence

Related topic

QuantumAlgebraicTopology

Related topic

ApproximationTheoremForAnArbitrarySpace

Related topic

SpinNetworksAndSpinFoams

Related topic

QuantumSpaceTimes

Defines

C

⁢

W

-complex approximation of quantum state spaces in QAT

Generated on Sat Feb 10 13:07:02 2018 by

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Fermat numbers

Fermat numbers

The

n

-th

Fermat number

is defined as:

F

n

=

2

2

n

+

1

.

Fermat incorrectly conjectured that all these numbers were primes,
although he had no proof.
The first 5 Fermat numbers:

3

,

5

,

17

,

257

,

65537

(corresponding to

n

=

0

,

1

,

2

,

3

,

4

) are all primes (so called

Fermat primes

)
Euler was the first to point out the falsity of Fermat’s

conjecture

by proving that

641

is a

divisor

of

F

5

. (In fact,

F

5

=

641

×

6700417

).
Moreover, no other Fermat number is known to be prime for

n

>

4

, so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many

composite

Fermat numbers or not.

One of the famous achievements of Gauss was to prove that the

regular polygon

of

m

sides can be constructed with ruler and

compass

if and only if

m

can be written as

m

=

2

k

⁢

F

r

1

⁢

F

r

2

⁢

⋯

⁢

F

r

t

where

k

≥

0

and the other factors are distinct primes of the form

F

n

(of course,

t

may be

0

here, i.e.

m

=

2

k

is allowed).

There are many interesting

properties

involving Fermat numbers. For instance:

F

m

=

F

0

⁢

F

1

⁢

⋯

⁢

F

m

-

1

+

2

for any

m

≥

1

, which implies that

F

m

-

2

is

divisible

by all smaller Fermat numbers.

The previous formula holds because

F

m

-

2

=

(

2

2

m

+

1

)

-

2

=

2

2

m

-

1

=

(

2

2

m

-

1

-

1

)

⁢

(

2

2

m

-

1

+

1

)

=

(

2

2

m

-

1

-

1

)

⁢

F

m

-

1

and expanding recursively the left factor in the last

expression

gives the desired result.

References.

Krízek, Luca, Somer.

17 Lectures on Fermat Numbers.

CMS Books in Mathematics.

Title

Fermat numbers

Canonical

name

FermatNumbers

Date of creation

2013-03-22 11:42:46

Last modified on

2013-03-22 11:42:46

Owner

drini (3)

Last modified by

drini (3)

Numerical id

30

Author

drini (3)

Entry type

Definition

Classification

msc 81T45

Classification

msc 81T13

Classification

msc 11A51

Classification

msc 20L05

Classification

msc 46L87

Classification

msc 43A35

Classification

msc 43A25

Classification

msc 22D25

Classification

msc 55U40

Classification

msc 18B40

Classification

msc 46L05

Classification

msc 22A22

Classification

msc 81R50

Classification

msc 55U35

Defines

Fermat prime

Generated on Sat Feb 10 13:07:14 2018 by

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dot product

dot product

Let

u

=

(

u

1

,

u

2

,

…

,

u

n

)

and

v

=

(

v

1

,

v

2

,

…

,

v

n

)

two vectors on

k

n

where

k

is a field (like

ℝ

or

ℂ

).
Then we define the

dot product

of the two vectors as:

u

⋅

v

=

u

1

⁢

v

1

+

u

2

⁢

v

2

+

⋯

+

u

n

⁢

v

n

.

Notice that

u

⋅

v

is NOT a vector but a scalar (an element from the field

k

).

If

u

,

v

are vectors in

ℝ

n

and

ϑ

is the angle between them, then we also have

u

⋅

v

=

∥

u

∥

⁢

∥

v

∥

⁢

cos

⁡

ϑ

.

Thus, in this case,

u

⟂

v

if and only if

u

⋅

v

=

0

.

The special case

u

⋅

u

of

scalar product

is the

scalar square

of the vector

u

.  In

ℝ

n

it equals to the square of the length of

u

:

u

⋅

u

=

∥

u

∥

2

Title

dot product

Canonical

name

DotProduct

Date of creation

2013-03-22 11:46:33

Last modified on

2013-03-22 11:46:33

Owner

drini (3)

Last modified by

drini (3)

Numerical id

13

Author

drini (3)

Entry type

Definition

Classification

msc 83C05

Classification

msc 15A63

Classification

msc 14-02

Classification

msc 14-01

Synonym

scalar product

Related topic

CauchySchwarzInequality

Related topic

CrossProduct

Related topic

Vector

Related topic

DyadProduct

Related topic

InvariantScalarProduct

Related topic

AngleBetweenLineAndPlane

Related topic

TripleScalarProduct

Related topic

ProvingThalesTheoremWithVectors

Defines

scalar square

Generated on Sat Feb 10 13:22:16 2018 by

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Mercator projection

Mercator projection

In a Mercator Projection the point on the sphere (of radius R) with

longitude

L

(positive East) and

latitude

λ

(positive North) is mapped to the point in the plane with

coordinates

x

,

y

:

x

=

R

⁢

L

y

=

R

⁢

ln

⁡

(

tan

⁡

(

π

4

+

λ

2

)

)

The Mercator projection satisfies two important properties: it is conformal, that is it preserves angles, and it maps the sphere’s

parallels

into

straight line

segments of length

2

⁢

π

⁢

R

. (A parallel of latitude means a

small circle

comprised of points at a specified latitude).

Starting from these two properties we can derive the Mercator Projection. First note that a parallel of latitude

λ

has length

2

⁢

π

⁢

R

⁢

cos

⁡

(

λ

)

. To make the

projections

of the parallels all the same length a stretching factor in longitude of

1

cos

⁡

(

λ

)

will have to be applied. For the mapping to be conformal, the same stretching factor must be applied in latitude also. Note that the stretching factor varies with

λ

so to map a specified latitude

λ

0

to an

ordinate

y

we must evaluate an integral.

y

=

∫

0

λ

0

(

1

/

cos

⁡

(

λ

)

)

⁢

𝑑

λ

Early mapmakers such as Mercator evaluated this integral numerically to produce what is called a Table of Meridional Parts that can be used to map

λ

0

into y. Later it was noticed that the integral of one over cosine actually has a closed form, leading to the expression for

y

shown above.

Title

Mercator projection

Canonical name

MercatorProjection

Date of creation

2013-03-22 15:19:53

Last modified on

2013-03-22 15:19:53

Owner

acastaldo (8031)

Last modified by

acastaldo (8031)

Numerical id

5

Author

acastaldo (8031)

Entry type

Definition

Classification

msc 86A30

Related topic

RiemannSphere

Related topic

ConformalityOfStereographicProjection

Related topic

InverseGudermannianFunction

Generated on Sat Feb 10 13:36:26 2018 by

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plateau

plateau

A

plateau

of a function is a region where a function has constant value.

More formally, let

U

and

V

be

topological spaces

. A plateau for a scalar function

f

:

U

→

V

is a

path-connected

set of points

P

⊆

U

such that for some

y

we have

∀

p

∈

P

,

f

⁢

(

p

)

=

y

(1)

Please take note that this entry is not authoritative. If you know of a more standard definition of “plateau”, please contribute it, thank you.

Title

plateau

Canonical name

Plateau

Date of creation

2013-03-22 12:59:48

Last modified on

2013-03-22 12:59:48

Owner

bshanks (153)

Last modified by

bshanks (153)

Numerical id

8

Author

bshanks (153)

Entry type

Definition

Classification

msc 90-00

Classification

msc 26B12

Related topic

Extremum

Generated on Sat Feb 10 13:36:40 2018 by

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Beta testing checklist

Beta testing checklist

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\href

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Talk to you again soon! Joe

Title

Beta testing checklist

Canonical name

BetaTestingChecklist

Date of creation

2013-05-16 10:55:58

Last modified on

2013-05-16 10:55:58

Owner

unlord (1)

Last modified by

unlord (1)

Numerical id

2

Author

unlord (1)

Entry type

Topic

Classification

msc 90B18

Generated on Sat Feb 10 13:36:42 2018 by

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