C*-algebra

We begin with the abstract characterization of C*-algebras given in the 1943 paper by Gelfand and Naimark.

A C*-algebra, A, is a Banach algebra over the field of complex numbers, together with a map ${\textstyle x\mapsto x^{*}}$  for ${\textstyle x\in A}$  with the following properties:

• It is an involution, for every x in A:

${\displaystyle x^{**}=(x^{*})^{*}=x}$ 

• For all x, y in A:

${\displaystyle (x+y)^{*}=x^{*}+y^{*}}$ 

${\displaystyle (xy)^{*}=y^{*}x^{*}}$ 

• For every complex number ${\displaystyle \lambda \in \mathbb {C} }$  and every x in A:

${\displaystyle (\lambda x)^{*}={\overline {\lambda }}x^{*}.}$ 

• For all x in A:

${\displaystyle \|xx^{*}\|=\|x\|\|x^{*}\|.}$ 

Remark. The first four identities say that A is a *-algebra. The last identity is called the C* identity and is equivalent to:

${\displaystyle \|xx^{*}\|=\|x\|^{2},}$ 

which is sometimes called the B*-identity. For history behind the names C*- and B*-algebras, see the history section below.

The C*-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies that the C*-norm is uniquely determined by the algebraic structure:

${\displaystyle \|x\|^{2}=\|x^{*}x\|=\sup\{|\lambda |:x^{*}x-\lambda \,1{\text{ is not invertible}}\}.}$ 

A bounded linear map, π : A → B, between C*-algebras A and B is called a *-homomorphism if

• For x and y in A

${\displaystyle \pi (xy)=\pi (x)\pi (y)\,}$ 

• For x in A

${\displaystyle \pi (x^{*})=\pi (x)^{*}\,}$ 

In the case of C*-algebras, any *-homomorphism π between C*-algebras is contractive, i.e. bounded with norm ≤ 1. Furthermore, an injective *-homomorphism between C*-algebras is isometric. These are consequences of the C*-identity.

A bijective *-homomorphism π is called a C*-isomorphism, in which case A and B are said to be isomorphic.

The term B*-algebra was introduced by C. E. Rickart in 1946 to describe Banach *-algebras that satisfy the condition:

• ${\displaystyle \lVert xx^{*}\rVert =\lVert x\rVert ^{2}}$  for all x in the given B*-algebra. (B*-condition)

This condition automatically implies that the *-involution is isometric, that is, ${\displaystyle \lVert x\rVert =\lVert x^{*}\rVert }$ . Hence, ${\displaystyle \lVert xx^{*}\rVert =\lVert x\rVert \lVert x^{*}\rVert }$ , and therefore, a B*-algebra is also a C*-algebra. Conversely, the C*-condition implies the B*-condition. This is nontrivial, and can be proved without using the condition ${\displaystyle \lVert x\rVert =\lVert x^{*}\rVert }$ .^[1] For these reasons, the term B*-algebra is rarely used in current terminology, and has been replaced by the term 'C*-algebra'.

The term C*-algebra was introduced by I. E. Segal in 1947 to describe norm-closed subalgebras of B(H), namely, the space of bounded operators on some Hilbert space H. 'C' stood for 'closed'.^[2][3] In his paper Segal defines a C*-algebra as a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space".^[4]

C*-algebras have a large number of properties that are technically convenient. Some of these properties can be established by using the continuous functional calculus or by reduction to commutative C*-algebras. In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism.

Self-adjoint elements are those of the form ${\displaystyle x=x^{*}}$ . The set of elements of a C*-algebra A of the form ${\displaystyle x^{*}x}$  forms a closed convex cone. This cone is identical to the elements of the form ${\displaystyle xx^{*}}$ . Elements of this cone are called non-negative (or sometimes positive, even though this terminology conflicts with its use for elements of ${\displaystyle \mathbb {R} }$ )

The set of self-adjoint elements of a C*-algebra A naturally has the structure of a partially ordered vector space; the ordering is usually denoted ${\displaystyle \geq }$ . In this ordering, a self-adjoint element ${\displaystyle x\in A}$  satisfies ${\displaystyle x\geq 0}$  if and only if the spectrum of ${\displaystyle x}$  is non-negative, if and only if ${\displaystyle x=s^{*}s}$  for some ${\displaystyle s\in A}$ . Two self-adjoint elements ${\displaystyle x}$  and ${\displaystyle y}$  of A satisfy ${\displaystyle x\geq y}$  if ${\displaystyle x-y\geq 0}$ .

This partially ordered subspace allows the definition of a positive linear functional on a C*-algebra, which in turn is used to define the states of a C*-algebra, which in turn can be used to construct the spectrum of a C*-algebra using the GNS construction.

Any C*-algebra A has an approximate identity. In fact, there is a directed family {e_λ}_λ∈I of self-adjoint elements of A such that

${\displaystyle xe_{\lambda }\rightarrow x}$ 

${\displaystyle 0\leq e_{\lambda }\leq e_{\mu }\leq 1\quad {\mbox{ whenever }}\lambda \leq \mu .}$ 

In case A is separable, A has a sequential approximate identity. More generally, A will have a sequential approximate identity if and only if A contains a strictly positive element, i.e. a positive element h such that hAh is dense in A.

Using approximate identities, one can show that the algebraic quotient of a C*-algebra by a closed proper two-sided ideal, with the natural norm, is a C*-algebra.

Similarly, a closed two-sided ideal of a C*-algebra is itself a C*-algebra.

The algebra M(n, C) of n × n matrices over C becomes a C*-algebra if we consider matrices as operators on the Euclidean space, Cⁿ, and use the operator norm ||·|| on matrices. The involution is given by the conjugate transpose. More generally, one can consider finite direct sums of matrix algebras. In fact, all C*-algebras that are finite dimensional as vector spaces are of this form, up to isomorphism. The self-adjoint requirement means finite-dimensional C*-algebras are semisimple, from which fact one can deduce the following theorem of Artin–Wedderburn type:

Theorem. A finite-dimensional C*-algebra, A, is canonically isomorphic to a finite direct sum

${\displaystyle A=\bigoplus _{e\in \min A}Ae}$ 

where min A is the set of minimal nonzero self-adjoint central projections of A.

Each C*-algebra, Ae, is isomorphic (in a noncanonical way) to the full matrix algebra M(dim(e), C). The finite family indexed on min A given by {dim(e)}_e is called the dimension vector of A. This vector uniquely determines the isomorphism class of a finite-dimensional C*-algebra. In the language of K-theory, this vector is the positive cone of the K₀ group of A.

A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics^[5] for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a Hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the Hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.

An immediate generalization of finite dimensional C*-algebras are the approximately finite dimensional C*-algebras.

The prototypical example of a C*-algebra is the algebra B(H) of bounded (equivalently continuous) linear operators defined on a complex Hilbert space H; here x* denotes the adjoint operator of the operator x : H → H. In fact, every C*-algebra, A, is *-isomorphic to a norm-closed adjoint closed subalgebra of B(H) for a suitable Hilbert space, H; this is the content of the Gelfand–Naimark theorem.

Let H be a separable infinite-dimensional Hilbert space. The algebra K(H) of compact operators on H is a norm closed subalgebra of B(H). It is also closed under involution; hence it is a C*-algebra.

Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional C*-algebras:

Theorem. If A is a C*-subalgebra of K(H), then there exists Hilbert spaces {H_i}_i∈I such that

${\displaystyle A\cong \bigoplus _{i\in I}K(H_{i}),}$ 

where the (C*-)direct sum consists of elements (T_i) of the Cartesian product Π K(H_i) with ||T_i|| → 0.

Though K(H) does not have an identity element, a sequential approximate identity for K(H) can be developed. To be specific, H is isomorphic to the space of square summable sequences l²; we may assume that H = l². For each natural number n let H_n be the subspace of sequences of l² which vanish for indices k ≥ n and let e_n be the orthogonal projection onto H_n. The sequence {e_n}_n is an approximate identity for K(H).

K(H) is a two-sided closed ideal of B(H). For separable Hilbert spaces, it is the unique ideal. The quotient of B(H) by K(H) is the Calkin algebra.

Let X be a locally compact Hausdorff space. The space ${\displaystyle C_{0}(X)}$  of complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness) forms a commutative C*-algebra ${\displaystyle C_{0}(X)}$  under pointwise multiplication and addition. The involution is pointwise conjugation. ${\displaystyle C_{0}(X)}$  has a multiplicative unit element if and only if ${\displaystyle X}$  is compact. As does any C*-algebra, ${\displaystyle C_{0}(X)}$  has an approximate identity. In the case of ${\displaystyle C_{0}(X)}$  this is immediate: consider the directed set of compact subsets of ${\displaystyle X}$ , and for each compact ${\displaystyle K}$  let ${\displaystyle f_{K}}$  be a function of compact support which is identically 1 on ${\displaystyle K}$ . Such functions exist by the Tietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions ${\displaystyle \{f_{K}\}}$  is an approximate identity.

The Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra ${\displaystyle C_{0}(X)}$ , where ${\displaystyle X}$  is the space of characters equipped with the weak* topology. Furthermore, if ${\displaystyle C_{0}(X)}$  is isomorphic to ${\displaystyle C_{0}(Y)}$  as C*-algebras, it follows that ${\displaystyle X}$  and ${\displaystyle Y}$  are homeomorphic. This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs.

Given a Banach *-algebra A with an approximate identity, there is a unique (up to C*-isomorphism) C*-algebra E(A) and *-morphism π from A into E(A) that is universal, that is, every other continuous *-morphism π ' : A → B factors uniquely through π. The algebra E(A) is called the C*-enveloping algebra of the Banach *-algebra A.

Of particular importance is the C*-algebra of a locally compact group G. This is defined as the enveloping C*-algebra of the group algebra of G. The C*-algebra of G provides context for general harmonic analysis of G in the case G is non-abelian. In particular, the dual of a locally compact group is defined to be the primitive ideal space of the group C*-algebra. See spectrum of a C*-algebra.

Von Neumann algebras, known as W* algebras before the 1960s, are a special kind of C*-algebra. They are required to be closed in the weak operator topology, which is weaker than the norm topology.

The Sherman–Takeda theorem implies that any C*-algebra has a universal enveloping W*-algebra, such that any homomorphism to a W*-algebra factors through it.

A C*-algebra A is of type I if and only if for all non-degenerate representations π of A the von Neumann algebra π(A)″ (that is, the bicommutant of π(A)) is a type I von Neumann algebra. In fact it is sufficient to consider only factor representations, i.e. representations π for which π(A)″ is a factor.

A locally compact group is said to be of type I if and only if its group C*-algebra is type I.

However, if a C*-algebra has non-type I representations, then by results of James Glimm it also has representations of type II and type III. Thus for C*-algebras and locally compact groups, it is only meaningful to speak of type I and non type I properties.

In quantum mechanics, one typically describes a physical system with a C*-algebra A with unit element; the self-adjoint elements of A (elements x with x* = x) are thought of as the observables, the measurable quantities, of the system. A state of the system is defined as a positive functional on A (a C-linear map φ : A → C with φ(u*u) ≥ 0 for all u ∈ A) such that φ(1) = 1. The expected value of the observable x, if the system is in state φ, is then φ(x).

This C*-algebra approach is used in the Haag–Kastler axiomatization of local quantum field theory, where every open set of Minkowski spacetime is associated with a C*-algebra.

• Banach algebra
• Banach *-algebra
• *-algebra
• Hilbert C*-module
• Operator K-theory
• Operator system, a unital subspace of a C*-algebra that is *-closed.
• Gelfand–Naimark–Segal construction
• Jordan operator algebra

• Arveson, W. (1976), An Invitation to C*-Algebra, Springer-Verlag, ISBN 0-387-90176-0. An excellent introduction to the subject, accessible for those with a knowledge of basic functional analysis.
• Connes, Alain (1994), Non-commutative geometry, ISBN 0-12-185860-X. This book is widely regarded as a source of new research material, providing much supporting intuition, but it is difficult.
• Dixmier, Jacques (1969), Les C*-algèbres et leurs représentations, Gauthier-Villars, ISBN 0-7204-0762-1. This is a somewhat dated reference, but is still considered as a high-quality technical exposition. It is available in English from North Holland press.
• Doran, Robert S.; Belfi, Victor A. (1986), Characterizations of C*-algebras: The Gelfand-Naimark Theorems, CRC Press, ISBN 978-0-8247-7569-8.
• Emch, G. (1972), Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, ISBN 0-471-23900-3. Mathematically rigorous reference which provides extensive physics background.
• A.I. Shtern (2001) [1994], "C*-algebra", Encyclopedia of Mathematics, EMS Press
• Sakai, S. (1971), C*-algebras and W*-algebras, Springer, ISBN 3-540-63633-1.
• Segal, Irving (1947), "Irreducible representations of operator algebras", Bulletin of the American Mathematical Society, 53 (2): 73–88, doi:10.1090/S0002-9904-1947-08742-5.