# ${C}^{*}$-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 ${C}^{\mathrm{*}}$-algebra $\mathrm{A}$ is a Banach *-algebra such that
$\mathrm{\parallel}{a}^{\mathrm{*}}\mathit{}a\mathrm{\parallel}\mathrm{=}{\mathrm{\parallel}a\mathrm{\parallel}}^{\mathrm{2}}$ for all $a\mathrm{\in}\mathrm{A}$*.

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 ${C}^{\mathrm{*}}$-algebra $\mathrm{A}$ is a Banach algebra ^{} with an antilinear
involution $\mathrm{*}$ such that ${\mathrm{\parallel}a\mathrm{\parallel}}^{\mathrm{2}}\mathrm{\le}\mathrm{\parallel}{a}^{\mathrm{*}}\mathit{}a\mathrm{\parallel}$ for all $a\mathrm{\in}\mathrm{A}$.*

Definition 3 - *A ${C}^{\mathrm{*}}$-algebra $\mathrm{A}$ is a Banach algebra with an antilinear
involution $\mathrm{*}$ such that $\mathrm{\parallel}{a}^{\mathrm{*}}\mathit{}a\mathrm{\parallel}\mathrm{=}\mathrm{\parallel}{a}^{\mathrm{*}}\mathrm{\parallel}\mathit{}\mathrm{\parallel}a\mathrm{\parallel}$*

## 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,

$$\parallel a\parallel =\sqrt{{R}_{\sigma}({a}^{*}a)}$$ |

where ${R}_{\sigma}(x)$ denotes the spectral radius of the element $x\in \mathcal{A}$. For ${C}^{*}$ algebras with an identity element^{} $e$ we can specify even further: the norm of an element $a\in \mathcal{A}$ is determined by

$${\parallel a\parallel}^{2}=sup\{|\lambda |:\lambda \in \u2102\text{and}{a}^{*}a-\lambda e\text{is not invertible}\}$$ |

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 $\mathcal{A}$ is a ${C}^{*}$ algebra with norm $\parallel \cdot {\parallel}_{{C}^{*}}$. If $\parallel \cdot {\parallel}_{B}$ is any other norm for which $\mathcal{A}$ is a Banach *-algebra, then we must have

$${\parallel a\parallel}_{{C}^{*}}\le {\parallel a\parallel}_{B},\forall a\in \mathcal{A}$$ |

Hence we see that the ${C}^{*}$ 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 ${C}^{*}$ 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 $\mathcal{A}$ be a ${C}^{*}$-algebra with identity element $e$. An element $a\in \mathcal{A}$ is said to be

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 $\u2102$ 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 $\u2102$ and positive elements^{} correspond to positive functions (functions with values in ${\mathbb{R}}_{\mathrm{\U0001d7d8}}^{+}$).

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

$$a=x+iy$$ |

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
$\u2102$. For the involution operation,
we take pointwise complex conjugation and for the norm we take the
norm of uniform convergence:

$$\parallel f\parallel =\underset{x\in X}{sup}|f(x)|$$ |

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 $X\to \u2102$ 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 $\u27e8\cdot ,\cdot \u27e9$ 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:

$$\parallel T\parallel =\underset{\parallel \xi \parallel =1}{sup}\parallel T\xi \parallel $$ |

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 $(X,\U0001d505,\mu )$ be a measure space. The space ${L}^{\mathrm{\infty}}(X)$ (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 $\parallel \cdot {\parallel}_{\mathrm{\infty}}$. It can be readily verified that, under these operations and norm, ${L}^{\mathrm{\infty}}(X)$ is a ${C}^{*}$-algebra.

The algebras ${L}^{\mathrm{\infty}}(X)$ 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 |