Finite-dimensional C*-algebras 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 : A = \bigoplus_{e \in \min A } A e 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 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.
C*-algebras of operators 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.
C*-algebras of compact operators 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 {
Hi}
i∈
I such that : A \cong \bigoplus_{i \in I } K(H_i), where the (C*-)direct sum consists of elements (
Ti) of the Cartesian product Π
K(
Hi) with ||
Ti|| → 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
Hn be the subspace of sequences of
l2 which vanish for indices
k ≥
n and let
en be the orthogonal projection onto
Hn. The sequence {
en}
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.
Commutative C*-algebras Let
X be a
locally compact Hausdorff space. The space 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 C_0(X) under pointwise multiplication and addition. The involution is pointwise conjugation. C_0(X) has a multiplicative unit element if and only if X is compact. As does any C*-algebra, C_0(X) has an
approximate identity. In the case of C_0(X) this is immediate: consider the directed set of compact subsets of X, and for each compact K let f_K be a function of compact support which is identically 1 on K. Such functions exist by the
Tietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions \{f_K\} is an approximate identity. The
Gelfand representation states that every commutative C*-algebra is *-isomorphic to the algebra C_0(X), where X is the space of
characters equipped with the
weak* topology. Furthermore, if C_0(X) is
isomorphic to C_0(Y) as C*-algebras, it follows that X and Y are
homeomorphic. This characterization is one of the motivations for the
noncommutative topology and
noncommutative geometry programs.
C*-enveloping algebra 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 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 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. == Type for C*-algebras ==