# nLab C-star-algebra

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

### Euclidean QFT

#### Noncommutative geometry

noncommutative geometry

(geometry $\leftarrow$ Isbell duality $\to$ algebra)

# Contents

## Definitions

### Abstract $C^\ast$-algebras

###### Definition

A $C^*$-algebra is a Banach algebra $(A, {\|-\|})$ over a topological field $K$ (often the field $K \coloneqq \mathbb{C}$ of complex numbers) equipped with an involution $(-)^\ast$ compatible with complex conjugation if appropriate (that is: a Banach star-algebra) that satisfies the $C^*$-identity

${\|{A^* A}\|} = {\|{A^*}\|} \, {\|{A}\|}$

or equivalently the $B^*$-identity

${\|{A^* A}\|} = {\|{A}\|^2} \,.$

A homomorphism of $C^\ast$-algebras is a map that preserves all this structure. For this it is sufficient for it to be a star-algebra homomorphisms.

$C^\ast$-algebras with these homomorphisms form a category C*Alg.

###### Remark

Often one sees the definition without the clause (which should be in the definition of Banach $*$-algebra) that the involution is an isometry (so that ${\|A^*\|} = {\|A\|}$, which is key for the equivalence of the $B^*$ and $C^*$ identities). This follows easily from the $B^*$-identity, while it follows from the $C^*$-identity after some difficulty.

###### Remark

There are different concepts for the tensor product of $C^*$-algebras, see for example at spatial tensor product.

###### Remark

$C^*$-algebras equipped with the action of a group by automorphisms of the algebra are called C-star-systems .

### Concrete $C^\ast$-algebras and $C^\ast$-representations

###### Definition

Given a complex Hilbert space $H$, a concrete $C^*$-algebra on $H$ is a $*$-subalgebra of the algebra of bounded operators on $H$ that is closed in the norm topology.

###### Definition

A representation of a $C^*$-algebra $A$ on a Hilbert space $H$ is a $*$-homomorphism from $A$ to the algebra of bounded operators on $H$.

###### Remark

It is immediate that concrete $C^*$-algebras correspond precisely to faithful representations of abstract $C^*$-algebras. It is an important theorem that every $C^*$-algebra has a faithful representation; that is, every abstract $C^*$-algebra is isomorphic to a concrete $C^*$-algebra.

###### Remark

The original definition of the term ‘$C^*$-algebra’ was in fact the concrete notion; the ‘C’ stood for ‘closed’. Furthermore, the original term for the abstract notion was ‘$B^*$-algebra’ (where the ‘B’ stood for ‘Banach’). However, we now usually interpret ‘$C^*$-algebra’ abstractly. (Compare ‘$W^*$-algebra’ and ‘von Neumann algebra’.)

### In $\dagger$-compact categories

The notion of $C^*$-algebra can be abstracted to the general context of symmetric monoidal †-categories, which serves to illuminate their role in quantum mechanics in terms of †-compact categories.

For a discussion of this in the finite-dimensional case see for instance (Vicary).

## Properties

### GNS construction

The GNS construction shows that every abstract $C^*$-algebra over the complex numbers as a concrete $C^*$-algebra: a subalgebra of an algebra of bounded operators on some Hilbert space.

### Gelfand duality

Gelfand duality says that every (unital) commutative $C^*$-algebra over the complex numbers is that of complex-valued continuous functions from some compact Hausdorff topological space: there is an equivalence of categories $C^* CAlg_{com} \simeq$ Top${}_{cpt}$.

Accordingly one may think of the study of non-commutative $C^\ast$-algebras as non-commutative topology.

### General

###### Proposition

For $A$ and $B$ two $C^\ast$-algebras and $f : A \to B$ a star-algebra homomorphism the set-theoretic image $f(A) \subset B$ is a $C^\ast$-subalgebra of $B$, hence is also the image of $f$ in $C^\ast Alg$.

###### Corollary

There is a functor

$\mathcal{C} : C^\ast Alg \to Poset$

to the category Poset of posets, which sends each $A \in C^\ast Alg$ to its poset of commutative subalgebras $\mathcal{C}(A)$ and sends each morphism $f : A \to B$ to the functor $\mathcal{C}(f) : \mathcal{C}(A) \to \mathcal{C}(B)$ which sends a commutative subalgebra $C \subset A$ to $f(C) \subset B$.

### Construction as groupoid convolution algebras

Many $C^\ast$-algebras arise as groupoid algebras of Lie groupoids. See at groupoid algebra - References - For smooth geometry

### Homotopy theory

There is homotopy theory of $C^\ast$-algebras, being a non-commutative generalization of that of Top. (e.g. Uuye 12). For more see at homotopical structure on C*-algebras.

## Examples

###### Example

Any algebra $M_n(A)$ of matrices with coefficients in a $C^\ast$-algebra is again a $C^\ast$-algebra. In particular $M_n(\mathbb{C})$ is a $C^\ast$-algebra for all $n \in \mathbb{N}$.

###### Example

For $A$ a $C^\ast$-algebra and for $X$ a locally compact Hausdorff topological space, the set of continuous functions $X \to A$ which vanish at infinity is again a $C^\ast$-algebra by extending all operations pointwise. (This algebra is unital precisely if $A$ is and if $X$ is a compact topological space.)

This algebra is denoted

$C_0(X,A) \in C^\ast Alg \,.$

If $A = \mathbb{C}$ then one usually just writes

$C_0(X) \coloneqq C_0(X, \mathbb{C}) \,.$

This are the $C^\ast$-algebras to which the Gelfand duality theorem applies.

###### Example

A uniformly hyperfinite algebra is in particular a $C^\ast$-algebra, by definition.

###### Example

A von Neumann algebra is in particular a $C^\ast$-algebra, by definition.

## References

A standard textbook reference is chapter 4 in volume 1 of

• Richard Kadison, John Ringrose, Fundamentals of the theory of operator algebras Academic Press, (1983)

An exposition that explicitly gives Gelfand duality as an equivalence of categories and introduces all the notions of category theory necessary for this statement is in

• Ivo Dell’Ambrogio, Categories of $C^\ast$-algebras (pdf)

A characterizations of injections of commutative sub-$C^*$-algebras – hence of the poset of commutative subalgebras of a $C^*$-algebra – is in

General properties of the category of $C^\ast$-algebras are discussed in

Specifically pullback and pushout of $C^\ast$-algebras is discussed in

• Gerd Petersen, Pullback and pushout constructions in $C^\ast$-algebra theory (pdf)

The homotopy theory of $C^\ast$-algebras (a category of fibrant objects-structure on $C^\ast Alg$) is discussed in

• Otgonbayar Uuye, Homotopy theory for $C^\ast$-algebras (arXiv:1011.2926)

For more along such lines see the references at KK-theory and E-theory.

Revised on January 14, 2014 13:37:17 by Urs Schreiber (89.204.153.51)