C^*-algebra

C

*

-algebra

1

Definition

C

*

-

algebras

are a type of

involutive Banach algebras

which arise
in the study of

operators

on

Hilbert spaces

,

Lie group
representations

,

locally compact topological spaces

, knots,

noncommutative

, among other
topics in mathematics and theoretical physics . Their study was initiated in the 1930’s with the purpose of axiomatizing quantum mechanics, and still today,

C

*

-algebras play a decisive role in formulations of quantum statistical mechanics and quantum .

The defining property of these algebras is
that the norm and the

involution

are related in a very special way.

Definition 1

-

A

C

*

-algebra

A

is a Banach *-algebra such that

∥

a

*

⁢

a

∥

=

∥

a

∥

2

for all

a

∈

A

.

The

equality

in Definition 1 is sometimes called the

C

*

axiom

. It turns out that one can weaken this condition and still specify
the same of algebras.

Definition 2

-

A

C

*

-algebra

A

is a

Banach algebra

with an antilinear
involution

*

such that

∥

a

∥

2

≤

∥

a

*

⁢

a

∥

for all

a

∈

A

.

Definition 3

-

A

C

*

-algebra

A

is a Banach algebra with an antilinear
involution

*

such that

∥

a

*

⁢

a

∥

=

∥

a

*

∥

⁢

∥

a

∥

2

C* Norm

C

*

-algebras are a very peculiar type of

topological algebras

. The

C

*

axiom, deceptively , imposes severe on the the

algebraic

and
topological structure of a

C

*

-algebra.

A most striking

consequence

of the

C

*

axiom is that the norm is solely determined by the

algebraic structure

of the algebra. More specifically,

∥

a

∥

=

R

σ

⁢

(

a

*

⁢

a

)

where

R

σ

⁢

(

x

)

denotes the spectral radius of the

element

x

∈

𝒜

. For

C

*

algebras with an

identity element

e

we can specify even further: the norm of an element

a

∈

𝒜

is determined by

∥

a

∥

2

=

sup

⁡

{

|

λ

|

:

λ

∈

ℂ

⁢

and

⁢

a

*

⁢

a

-

λ

⁢

e

⁢

is not invertible

}

This also implies that the norm in a

C

*

-algebra is unique, in the sense that there is no other norm in the algebra that

satisfies

that

C

*

axiom, i.e. that turns the algebra into a

C

*

-algebra. This is a stark contrast to the case
of general normed algebras, where one may find many norms which are
with the algebraic structure.

Moreover, the

C

*

norm occupies a unique amongst the possible
norms for an

involutive algebra

. Suppose that

𝒜

is a

C

*

algebra
with norm

∥

⋅

∥

C

*

. If

∥

⋅

∥

B

is
any other norm for which

𝒜

is a Banach *-algebra, then we must have

∥

a

∥

C

*

≤

∥

a

∥

B

,

∀

a

∈

𝒜

Hence we see that the

C

*

norm enjoys an extremal property — it
is the smallest possible norm for which

𝒜

is a Banach *-algebra.

There are many other surprising consequences of the

C

*

axiom, like: *-

homomorphisms

between

C

*

-algebras are automatically

continuous

and every

C

*

-algebra is semi-simple, which again are not true for general involutive algebras.

3

Elements of a C*-algebra

Like in involutory rings, there are some special elements in

C

*

-algebras that deserve some attention. We recall some definitions here:

Let

𝒜

be a

C

*

-algebra with identity element

e

. An element

a

∈

𝒜

is said to be

•

self-adjoint

if

a

*

=

a

•

unitary

if

a

*

⁢

a

=

a

⁢

a

*

=

e

•

positive

if

a

=

b

*

⁢

b

for some element

b

∈

𝒜

It is many times useful to have some

interpretation

for this elements. One of this interpretations comes from complex analysis: we regard the elements of a

C

*

-algebra as

functions

with values in

ℂ

and the involution as complex conjugation.

In this frame,

self-adjoint elements

correspond to real functions,

unitary elements

correspond to functions whose values lie in the

unit circle

in

ℂ

and

positive elements

correspond to positive functions (functions with values in

ℝ

𝟘

+

).

It is easily seen that self-adjoint elements are

closed under

addition

, multiplication and multiplication by

real numbers

. It can be proven the same for positive elements (with multiplication by positive numbers).

There are some decompositions of elements in a

C

*

-algebra analogous to some decompositions in complex analysis. For

instance

, every element

a

in a

C

*

-algebra has a unique decomposition of the form

a

=

x

+

i

⁢

y

where

x

,

y

are self-adjoint. This is

similar

to the decomposition of a complex valued function in its real and imaginary parts.

Moreover, every self-adjoint element

a

is of the form

a

=

x

-

y

where

x

,

y

are positive elements. This is similar to the decomposition of real valued functions in its positive and negative parts.

There are many other aspects of the theory of

C

*

-algebras for which this kind of interpretation proves to be very insightful.

For example,

C

*

-algebras happen to have a natural

partial ordering

. One can define an
ordering by declaring that

x

>

y

when

x

-

y

is positive. Given this ordering, one can then
speak of such things as

monotonic functions

,

monotonic sequences

,
and

positive linear functionals

on the algebra. These notions, in
turn, prove to be extremely useful in the study of

C

*

-algebras.

4

Examples

Having discussed these algebras in general terms, it is high time
that we illustrate the definition with some examples.

Example 1

As our first class of examples, we consider algebras of functions.
Let

X

be a

compact

Hausdorff topological space

and let

C

⁢

(

X

)

be the algebra of

continuous functions

from

X

to

ℂ

. For the involution

operation

,
we take

pointwise

complex conjugation and for the norm we take the
norm of uniform convergence:

∥

f

∥

=

sup

x

∈

X

⁡

|

f

⁢

(

x

)

|

It is a routine matter to check that the norm and involution
satisfy the appropriate algebraic requirements. Completeness
under this norm follows from the fact that the uniform limit
of continuous functions on a

locally compact

Hausdorff topological
space is continuous.

More generally, instead of a

compact space

, we can take a

locally compact Hausdorff space

X

and consider the algebra

C

0

⁢

(

X

)

of continuous functions

X

→

ℂ

that

vanish at infinity

, endowed with the same norm and involution. These are important examples of

C

*

-algebras.

Example 2

As our second class of examples, we consider operator algebras.
Let

H

be a complex Hilbert space with

inner product

⟨

⋅

,

⋅

⟩

and
let

B

⁢

(

H

)

be the algebra of

bounded operators

on

H

. For the
involution, we take the

adjoint

operation and as a norm we
take the usual

operator norm

:

∥

T

∥

=

sup

∥

ξ

∥

=

1

⁡

∥

T

⁢

ξ

∥

Again, it is straightforward to verify that the norm and
involution satisfy the appropriate algebraic requirements,
as is done in an attachment to this entry. Completeness
under the norm follows from a well-known

theorem

of

functional analysis

.

5

Commutative and noncommutative C*-algebras

The algebras

C

0

⁢

(

X

)

in Example 1 above are more than just an example. In fact, all

commutative

C

*

-algebras are *-

isomorphic

to

C

0

⁢

(

X

)

for some locally compact Hausdorff space

X

. Moreover,

X

is compact if and only if the

C

*

-algebra has an identity element. This is the

content

of the

Gelfand-Naimark theorem

.

Furthermore, there is a correspondence between properties of the

topological space

and properties of the

C

*

-algebra. For example: a

compactification

of the space corresponds to a

unitization

of the

C

*

-algebra; the space is

connected

if and only if the

C

*

-algebra has no non-trivial

projections

, among many other interesting correspondences.

For this reason, the theory of (noncommutative)

C

*

-algebras is many times called

noncommutative topology

(click on the link for more information).

The second example is also more than just an example of

C

*

-algebras. In fact, by the

Gelfand-Naimark representation theorem

, all

C

*

-algebras are *-isomorphic to a norm closed *-

subalgebra

of

B

⁢

(

H

)

, for some Hilbert space

H

.

Note, however, that this does not provide a “classification” of

C

*

-algebras since we do not know in general what are the closed *-subalgebras of

B

⁢

(

H

)

. This is merely a (very-important) structural theorem. The classification problem for

C

*

-algebras is still open.

6

Additional Examples

Example 3

Compact operators

in a Hilbert space

H

form a closed ideal of

B

⁢

(

H

)

. Moreover, this ideal is also closed for the involution of operators. Hence, the algebra of compact operators,

K

⁢

(

H

)

, is a

C

*

-algebra.

Example 4

Let

(

X

,

𝔅

,

μ

)

be a measure space. The space

L

∞

⁢

(

X

)

(

http://planetmath.org/LpSpace

) is an algebra under pointwise operations. We can define an involution again by complex conjugation and we consider the essential supremum norm

∥

⋅

∥

∞

. It can be readily verified that, under these operations and norm,

L

∞

⁢

(

X

)

is a

C

*

-algebra.

The algebras

L

∞

⁢

(

X

)

are also particularly important since they are examples of

von Neumann algebras

, which are a specific kind of

C

*

-algebras.

Title

C

*

-algebra

Canonical name

Calgebra

Date of creation

2013-03-22 12:57:55

Last modified on

2013-03-22 12:57:55

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

34

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46L05

Classification

msc 46L87

Synonym

C*-algebra

Synonym

C* algebra

Related topic

GroupCAlgebra

Related topic

VonNeumannAlgebra

Related topic

NoncommutativeGeometry

Related topic

GroupoidCConvolutionAlgebra

Related topic

GroupoidCDynamicalSystem

Related topic

CAlgebra3

Related topic

NuclearCAlgebra

Related topic

HomomorphismsOfCAlgebrasAreContinuous

Related topic

ContinuousLinearMapping

Related topic

OperatorNorm

Related topic

C_cG

Related topic

UniformContinuityOverLocallyCompa

Defines

C

*

axiom

Generated on Fri Feb 9 21:43:00 2018 by

LaTeXML