C*-algebra

2020 Mathematics Subject Classification: Primary: 46L05 [MSN][ZBL]

$$ \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left\|#1\right\|} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\Ah}{A_{\text{h}}} \newcommand{\Cstar}{C^*\!} $$ A Banach algebra $A$ over the field of complex numbers, with an involution $x \rightarrow x^*$, $x \in A$, such that the norm and the involution are connected by the relation $\norm{x^* x} = \norm{x}^2$ for any element $x \in A$. $\Cstar\!$-algebras were introduced in 1943 [GeNe] under the name of totally regular rings; they are also known under the name of $B^*$-algebras. The most important examples of $\Cstar$-algebras are:

1) The algebra $C_0(X)$ of continuous complex-valued functions on a locally compact Hausdorff space $X$ which tend towards zero at infinity (i.e. continuous functions $f$ on $X$ such that, for any $\epsilon > 0$, the set of points $x \in X$ which satisfy the condition $\abs{f(x)} \geq \epsilon$ is compact in $X$); $C_0(X)$ has the uniform norm $$ \norm{f} = \sup_{x \in X} \abs{f(x)}. $$ The involution in $C_0(X)$ is defined as transition to the complex-conjugate function: $f^*(x) = \overline{f(x)}$. Any commutative $\Cstar$-algebra $A$ is isometrically and symmetrically isomorphic (i.e. is isomorphic as a Banach algebra $A$ with involution) to the $\Cstar$-algebra $C_0(X)$, where $X$ is the space of maximal ideals of $A$ endowed with the Gel'fand topology [GeNe], [Na], [Di].

2) The algebra $L(H)$ of all bounded linear operators on a Hilbert space $H$, considered with respect to the ordinary linear operations and operator multiplication. The involution in $L(H)$ is defined as transition to the adjoint operator, and the norm is defined as the ordinary operator norm.

A subset $M \subset A$ is said to be self-adjoint if $M = M^*$, where $M^* = \set{x^* : x \in M}$. Any closed self-adjoint subalgebra $B$ of a $\Cstar$-algebra $A$ is a $\Cstar$-algebra with respect to the linear operations, multiplication, involution, and norm taken from $A$; $B$ is said to be a $\Cstar$-subalgebra of $A$. Any $\Cstar$-algebra is isometrically and symmetrically isomorphic to a $\Cstar$-subalgebra of some $\Cstar$-algebra of the form $L(H)$. Any closed two-sided ideal $I$ in a $\Cstar$-algebra is self-adjoint (thus $I$ is a $\Cstar$-subalgebra of $A$), and the quotient algebra $A/I$, endowed with the natural linear operations, multiplication, involution, and quotient space norm, is a $\Cstar$-algebra. The set $K(H)$ of completely-continuous linear operators on a Hilbert space $H$ is a closed two-sided ideal in $L(H)$. If $A$ is a $\Cstar$-algebra and $\tilde{A}$ is the algebra with involution obtained from $A$ by addition of a unit element, there exists a unique norm on $\tilde{A} $ which converts $\tilde{A}$ into a $\Cstar$-algebra and which extends the norm on $A$. Moreover, the operations of bounded direct sum and tensor product [Di], [Sa] have been defined for $\Cstar$-algebras.

As in all symmetric Banach algebras with involution, in a $\Cstar$-algebra $A$ it is possible to define the following subsets: the real linear space $\Ah$ of Hermitian elements; the set of normal elements; the multiplicative group $U$ of unitary elements (if $A$ contains a unit element); and the set $A^+$ of positive elements. The set $A^+$ is a closed cone in $\Ah$, $A^+ \cap (-A)^+ = \set{0}$, $A^+ - A^+ = \Ah$, and the cone $A^+$ converts $\Ah$ into a real ordered vector space. If $A$ contains a unit element $1$, then $1$ is an interior point of the cone $A^+ \subset \Ah$. A linear functional $f$ on $A$ is called positive if $f(x) \geq 0 $ for all $x \in A^+$; such a functional is continuous. If $x \in B $, where $B$ is a $\Cstar$-subalgebra of $A$, the spectrum of $x$ in $B$ coincides with the spectrum of $x$ in $A$. The spectrum of a Hermitian element is real, the spectrum of a unitary element lies on the unit circle, and the spectrum of a positive element is non-negative. A functional calculus for the normal elements of a $\Cstar$-algebra has been constructed. Any $\Cstar$-algebra $A$ has an approximate unit, located in the unit ball of $A$ and formed by positive elements of $A$. If $I$, $J$ are closed two-sided ideals in $A$, then $(I+J)$ is a closed two-sided ideal in $A$ and $(I+J)^+ = I^+ + J^+$. If $I$ is a closed two-sided ideal in $J$ and $J$ is a closed two-sided ideal in $A$, then $I$ is a closed two-sided ideal in $A$. Any closed two-sided ideal is the intersection of the primitive two-sided ideals in which it is contained; any closed left ideal in $A$ is the intersection of the maximal regular left ideals in which it is contained.

Any $^*$-isomorphism of a $\Cstar$-algebra is isometric. Any $^*$-isomorphism $\pi$ of a Banach algebra $B$ with involution into a $\Cstar$-algebra $A$ is continuous, and $\norm{\pi(x)} \leq \norm{x}$ for all $x \in B$. In particular, all representations of a Banach algebra with involution (i.e. all $^*$-homomorphisms of $B$ into a $\Cstar$-algebra of the form $L(H)$) are continuous. The theory of representations of $\Cstar$-algebras forms a significant part of the theory of $\Cstar$-algebras, and the applications of the theory of $\Cstar$-algebras are related to the theory of representations of $\Cstar$-algebras. The properties of representations of $\Cstar$-algebras make it possible to construct for each $\Cstar$-algebra $A$ a topological space $\hat{A}$, called the spectrum of the $\Cstar$-algebra $A$, and to endow this space with a Mackey–Borel structure. In the general case, the spectrum of a $\Cstar$-algebra does not satisfy any separation axiom, but is a locally compact Baire space.

A $\Cstar$-algebra $A$ is said to be a CCR-algebra (respectively, a GCR-algebra) if the relation $\pi(A) = K(H_\pi)$ (respectively, $\pi(A) \supset K(H_\pi)$) is satisfied for any non-null irreducible representation $\pi$ of the $\Cstar$-algebra $A$ in a Hilbert space $H$.

A $\Cstar$-algebra $A$ is said to be an NGCR-algebra if $A$ does not contain non-zero closed two-sided GCR-ideals (i.e. ideals which are GCR-algebras). Any $\Cstar$-algebra contains a maximal two-sided GCR-ideal $I$, and the quotient algebra $A/I$ is an NGCR-algebra. Any GCR-algebra contains an increasing family of closed two-sided ideals $I_\alpha$, indexed by ordinals $\alpha$, $\alpha \leq \rho $, such that $I_\rho = A$, $I_1=\set{0}$, $I_{\alpha+1}/I_\alpha$ is a CCR-algebra for all $\alpha < \rho$, and $I_\alpha = \bigcup_{\alpha^\prime < \alpha} I_{\alpha^\prime}$ for limit ordinals $\alpha$. The spectrum of a GCR-algebra contains an open, everywhere-dense, separable, locally compact subset.

A $\Cstar$-algebra $A$ is said to be a $\Cstar$-algebra of type I if, for any representation $\pi$ of the $\Cstar$-algebra $A$ in a Hilbert space $H_\pi$, the von Neumann algebra generated by the family $\pi(A)$ in $H_\pi$ is a type I von Neumann algebra. For a $\Cstar$-algebra, the following conditions are equivalent: a) $A$ is a $\Cstar$-algebra of type I; b) $A$ is a GCR-algebra; and c) any quotient representation of the $\Cstar$-algebra $A$ is a multiple of the irreducible representation. If $A$ satisfies these conditions, then: 1) two irreducible representations of the $\Cstar$-algebra $A$ are equivalent if and only if their kernels are identical; and 2) the spectrum of the $\Cstar$-algebra $A$ is a $T_0$-space. If $A$ is a separable $\Cstar$-algebra, each of the conditions 1) and 2) is equivalent to the conditions a)–c). In particular, each separable $\Cstar$-algebra with a unique (up to equivalence) irreducible representation, is isomorphic to the $\Cstar$-algebra $K(H)$ for some Hilbert space $H$.

Let $A$ be a $\Cstar$-algebra, and let $P$ be a set of elements $x \in A$ such that the function $\pi \rightarrow \mathrm{Tr}\,\pi(x)$ is finite and continuous on the spectrum of $A$. If the linear envelope of $P$ is everywhere dense in $A$, then $A$ is said to be a $\Cstar$-algebra with continuous trace. The spectrum of such a $\Cstar$-algebra is separable and, under certain additional conditions, a $\Cstar$-algebra with a continuous trace may be represented as the algebra of vector functions on its spectrum $\hat{A}$ [Di].

Let $A$ be a $\Cstar$-algebra, let $F$ be the set of positive linear functionals on $A$ with norm no greater than $1$ and let $P(A)$ be the set of non-zero boundary points of the convex set $F$. Then $P(A)$ will be the set of pure states of $A$. Let $B$ be a $\Cstar$-subalgebra of $A$. If $A$ is a GCR-algebra and if $B$ separates the points of the set $P(A)\cup\set{0}$, i.e. for any $f_1, f_2 \in P(A)\cup\set{0}$, $f_1 \neq f_2$, there exists an $x \in B$ such that $f_1(x) \neq f_2(x)$, then $B=A$ (the Stone–Weierstrass theorem). If $A$ is any $\Cstar$-algebra and $B$ separates the points of the set $\overline{P(A)}\cup\set{0}$, then $B = A$.

The second dual space $A^{**}$ of a $\Cstar$-algebra $A$ is obviously provided with a multiplication converting $A^{**}$ into a $\Cstar$-algebra isomorphic to some von Neumann algebra; this algebra is named the von Neumann algebra enveloping the $\Cstar$-algebra [Di], [Sa].

The theory of $\Cstar$-algebras has numerous applications in the theory of representations of groups and symmetric algebras [Di], the theory of dynamical systems [Sa], statistical physics and quantum field theory [Ru], and also in the theory of operators on a Hilbert space [Do].

Comments

If $A$ over $\C$ is an algebra with involution, i.e. if there is an operation $^* : A \rightarrow A$ satisfying $(\lambda x + \mu y)^* = \bar{\lambda}x^* + \bar{\mu}y^*$, $x^{**}=x$, $(xy)^* = y^* x^*$, the Hermitian, normal and positive elements are defined as follows. The element $x$ is a Hermitian element if $x = x^*$; it is a normal element if $xx^* = x^*x$ and it is a positive element if $x = y^*y$ for some $y \in A$. An element $u$ is a unitary element if $uu^*=1$. An algebra with involution is also sometimes called a symmetric algebra (or symmetric ring), cf., e.g., [Na]. However, this usage conflicts with the concept of a symmetric algebra as a special kind of Frobenius algebra, cf. Frobenius algebra.

Recent discoveries have revealed connections with, and applications to, algebraic topology. If $X$ is a compact metrizable space, a group, $\mathrm{Ext}(X)$, can be formed from $\Cstar$-extensions of the compact operators by $C(X)$, $$ K(H) \rightarrow \epsilon \rightarrow C(X). $$ In [BrDoFi], $\mathrm{Ext}(X)$ is shown to be a homotopy invariant functor of $X$ which may be identified with the topological $K$-homology group, $K_1(X)$. In [At] M.F. Atiyah attempted to make a description of $K$-homology, $K_*(X)$, in terms of elliptic operators [Do2], p. 58. In [Ka], [Ka2] G.G. Kasparov developed a solution to this problem. Kasparov and others have used the equivariant version of Kasparov $K$-theory to prove the strong Novikov conjecture on higher signatures in many cases (see [Bl], pp. 309-314).

In addition, deep and novel connections between $K$-theory and operator algebras (cf. Operator ring) were recently discovered by A. Connes [Co]. Finally, V.F.R. Jones [Jo] has exploited operator algebras to provide invariants of topological knots (cf. Knot theory).

Further details on recent developments may be found in [Bl], [Do2].

References

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