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-algebra

(Definition)

[still under construction]

Definition

-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 geometry, 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, -algebras play a decisive role in formulations of quantum statistical mechanics and quantum field theory.

The defining property of these algebras is that the norm and the involution are related in a very special way.

Definition 1 - A -algebra $\mathcal{A}$ is a Banach *-algebra such that $\Vert a^*a\Vert = \Vert a\Vert ^2$ for all $a \in \mathcal{A}$ .

The equality in Definition 1 is sometimes called the axiom. It turns out that one can weaken this condition and still specify the same class of algebras.

Definition 2 - A -algebra $\mathcal{A}$ is a Banach algebra with an antilinear involution such that $\Vert a\Vert ^2 \leq \Vert a^* a\Vert$ for all $a \in \mathcal{A}$ .

Definition 3 - A -algebra $\mathcal{A}$ is a Banach algebra with an antilinear involution such that $\Vert a^* a\Vert = \Vert a^*\Vert \Vert a\Vert$

C* Norm

-algebras are a very peculiar type of topological algebras. The axiom, deceptively simple, imposes severe restrictions on the the algebraic and topological structure of a -algebra.

A most striking consequence of the axiom is that the norm is solely determined by the algebraic structure of the algebra. More specifically,

$\displaystyle \Vert a\Vert = \sqrt{R_{\sigma}(a^*a)}$

where $R_{\sigma}(x)$ denotes the spectral radius of the element $x \in \mathcal{A}$ . For

algebras with an identity element

we can specify even further: the norm of an element $a \in \mathcal{A}$ is determined by

$\displaystyle \Vert a\Vert^2 = \sup \{ \vert\lambda\vert: \lambda \in \mathbb{C} \;$ and $\displaystyle \; a^*a- \lambda e \;$ is not invertible $\displaystyle \}$

This also implies that the norm in a -algebra is unique, in the sense that there is no other norm in the algebra that satisfies that axiom, i.e. that turns the algebra into a -algebra. This is a stark contrast to the case of general normed algebras, where one may find many norms which are compatible with the algebraic structure.

Moreover, the norm occupies a unique place amongst the possible norms for an involutive algebra. Suppose that $\mathcal{A}$ is a algebra with norm $\Vert \cdot\Vert _{C^*}$ . If $\Vert \cdot\Vert _{B}$ is any other norm for which $\mathcal{A}$ is a Banach *-algebra, then we must have

$\displaystyle \Vert a\Vert _{C^*} \leq \Vert a\Vert _B \,, \qquad \forall a \in \mathcal{A}$

Hence we see that the

norm enjoys an extremal property -- it is the smallest possible norm for which $\mathcal{A}$ is a Banach *-algebra.

There are many other surprising consequences of the axiom, like: *-homomorphisms between -algebras are automatically continuous and every -algebra is semi-simple, which again are not true for general involutive algebras.

Elements of a C*-algebra

Like in involutory rings, there are some special elements in -algebras that deserve some attention. We recall some definitions here:

Let $\mathcal{A}$ be a -algebra with identity element . An element $a \in \mathcal{A}$ is said to be

self-adjoint if
unitary if
positive if for some element $b \in \mathcal{A}$

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 -algebra as complex numbers (or as functions with values in $\mathbb{C}$ ) and the involution as complex conjugation.

In this frame, self-adjoint elements correspond to real numbers (real functions), unitary elements correspond to complex numbers of norm 1 (functions whose values are in the unit circle in $\mathbb{C}$ ) and positive elements correspond to positive real numbers (functions that take only positive values).

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 -algebra analogous to some decompositions in complex analysis. For instance, every element in a -algebra has a unique decomposition of the form

$\displaystyle a = x + i y$

where

are self-adjoint. This is similar to the decomposition of a complex number in its real and imaginary parts (or of a complex valued function in its real and imaginary parts).

Moreover, every self-adjoint element is of the form

$\displaystyle a = x - y$

where

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

For example, -algebras happen to have a natural partial ordering. One can define an ordering by declaring that when 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 -algebras.

Commutative and Noncommutative C* algebras

Examples

Having discussed these algebras in general terms, it is high time that we illustrate the definition with some examples. Therefore, in this section, we shall focus on three representative examples of classes of -algebras.

Example 1

As our first class of examples, we consider algebras of functions. Let be a locally compact Haussdorff topological space and let be the algebra of continuous functions from to $\mathbb{C}$ whose modulus is bounded. As our involution operation, we take pointwise complex conjugation and as our norm we take the norm of uniform convergence:

$\displaystyle \Vert f\Vert = \sup_{p \in T} \vert f(p)\vert$

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 Haussdorff topological space is continuous.

Example 2

As our second class of examples, we consider operator algebras. Let be a complex Hilbert space with inner product and let be the algebra of bounded operators on . As our involution, we take the adjoint operation and as our norm we take the operator norm:

$\displaystyle \Vert T\Vert = \sup_{\Vert\xi\Vert = 1} \Vert T\xi\Vert$

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.

Example 3

Algebra of compact operators on a Hilbert space

Example 4

$L^{\infty}(X)$

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" $C^*$

-algebra" is owned by asteroid. [ full author list (3) | owner history (2) ]

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See Also: group $C^*$

-algebra, von Neumann algebra

Also defines:

axiom

Attachments:: bounded operators on a Hilbert space form a -algebra (Result) by HkBst
example of Banach algebra which is not a -algebra for any involution (Example) by asteroid
special elements in a -algebra and their spectral properties (Definition) by asteroid
norm and spectral radius in -algebras (Theorem) by asteroid
-algebra homomorphisms are continuous (Theorem) by asteroid
equivalence of definitions of -algebra (Theorem) by rspuzio

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Cross-references: compact operators, functional analysis, operator norm, adjoint, bounded operators, inner product, limit, uniform convergence, pointwise, operation, bounded, modulus, classes, focus, section, terms, positive linear functionals, sequences, monotonic, monotonic functions, partial ordering, theory, negative, self-adjoint element, complex, imaginary parts, decompositions, numbers, multiplication, addition, closed under, unit circle, real functions, real numbers, frame, complex conjugation, functions, complex numbers, complex analysis, interpretations, definitions, involutory rings, semi-simple, continuous, *-homomorphisms, normed algebras, implies, even, identity element, spectral radius, algebra, consequence, topological algebras, equality, involution, norm, algebras, defining property, knots, topological spaces, locally compact, representations, Lie group, Hilbert spaces, operators, Banach algebras, involutive
There are 37 references to this entry.

This is version 22 of $C^*$

-algebra, born on 2002-08-23, modified 2008-02-09.
Object id is 3334, canonical name is CAlgebra.
Accessed 4489 times total.

Classification:

AMS MSC:	46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
	46L87 (Functional analysis :: Selfadjoint operator algebras :: Noncommutative differential geometry)

Pending Errata and Addenda

None.[ View all 4 ]

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