C*-Algebra

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Topological Complex Vector Space

This article, predictably, is about some obscure corner of mathematics. Not the exciting kind, either. It’s about a “topological complex vector space.” If you’re looking for the exhilarating rush of rocketry, well, that’s characteristic velocity. This is… different.

This particular document, much like a poorly constructed argument, is in dire need of verification. It lacks the substance of reliable sources, and without them, it’s just… noise. If you’re hoping for genuine insight, you’ll need to improve this article by adding citations to reliable sources. Otherwise, the unsubstantiated claims will, quite rightly, be challenged and removed. Don't say I didn't warn you.

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C*-algebra

In the sterile, unforgiving landscape of functional analysis, there exists a creature known as a C*-algebra. It’s pronounced “C-star,” as if the universe itself is sighing. At its core, it's a Banach algebra equipped with an involution that behaves like an adjoint. A specific, perhaps more tangible, manifestation of this is an algebra, let's call it A, composed of continuous linear operators on a complex Hilbert space. This A has a few defining characteristics:

• It’s a topologically closed set within the norm topology of operators. This means it’s complete, in a sense. No escaping the boundaries.
• It’s also closed under the operation of taking adjoints. It embraces its own reflection, its own opposite.

Beyond these operator-based C*-algebras, there’s another important class. These are the algebras, denoted $C_0(X)$, comprising complex-valued continuous functions defined on a locally compact Hausdorff space $X$. These functions possess a peculiar quality: they vanish at infinity. They fade, but never quite disappear.

The initial fascination with C*-algebras stemmed from their potential to model the algebras of physical observables in quantum mechanics. This lineage traces back to Werner Heisenberg's audacious matrix mechanics. Later, around 1933, Pascual Jordan refined this, paving the way for John von Neumann's ambitious attempt to forge a general framework. His work on rings of operators, while groundbreaking, focused on a specific subset now known as von Neumann algebras.

Then, around 1943, Israel Gelfand and Mark Naimark provided an abstract characterization. No need for operators on Hilbert spaces; their definition was pure, unadulterated algebra.

Today, C*-algebras are indispensable tools. They illuminate the theory of unitary representations of locally compact groups and provide a rigorous language for algebraic formulations of quantum mechanics. The ongoing quest to classify these algebras, particularly the separable simple nuclear C*-algebras, remains a vibrant, if somewhat bleak, area of research.

Abstract Characterization

The Gelfand and Naimark paper from 1943 laid out the abstract blueprint for C*-algebras. Imagine an algebra, A, defined over the complex numbers. It’s a Banach algebra, which means it has a norm, a way to measure its size, and it’s complete. But it also has this extra feature: a map, $x \mapsto x^*$, that acts like an involution. For every element $x$ in A, there’s a corresponding $x^*$. This map has a set of rules it must follow:

• It’s an involution: $(x^*)^* = x$. It’s its own inverse, like a mirror reflecting a mirror.
• It respects addition: $(x+y)^* = x^* + y^*$.
• It respects multiplication, but in a reversed order: $(xy)^* = y^*x^*$. A subtle twist, a hint of betrayal.
• It interacts with scalar multiplication: $(\lambda x)^* = \bar{\lambda}x^*$. The complex conjugate of the scalar is involved.

And then, the defining characteristic, the C* identity: $\|xx^*\| = \|x\|\|x^*\|$. This is the crux, the condition that gives the algebra its name. It’s also equivalent to $\|xx^*\| = \|x\|^2$, sometimes called the B*-identity. The history behind these names is as murky as the motivations for some of their applications.

This C* identity is a powerful constraint. It implies that the norm, the very measure of size, is intrinsically linked to the algebraic structure. The spectral radius formula, combined with this identity, means the norm is essentially dictated by the algebra itself.

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

A bounded linear map, $\pi: A \to B$, between two C*-algebras, A and B, is called a *-homomorphism if it preserves multiplication and the involution:

• $\pi(xy) = \pi(x)\pi(y)$
• $\pi(x^*) = \pi(x)^*$

Crucially, any *-homomorphism between C*-algebras is contractive, meaning its norm is less than or equal to 1. If it’s also injective, it’s an isometry. This is a direct consequence of the C* identity. A bijective *-homomorphism is a C*-isomorphism, signifying that the algebras are essentially the same, just wearing different clothes.

Some History: B*-algebras and C*-algebras

The term B*-algebra emerged around 1946, thanks to C. E. Rickart. It described Banach *-algebras that adhered to a specific condition: $\|xx^*\| = \|x\|^2$ for all $x$ in the algebra. This "B*-condition" automatically ensured that the involution was isometric, meaning $\|x\| = \|x^*\|$. Consequently, $\|xx^*\| = \|x\|\|x^*\|$. So, a B*-algebra was, in essence, a C*-algebra. The converse, that a C*-condition implies the B*-condition, is a more involved proof, not immediately obvious. For these reasons, the term B*-algebra has largely faded, replaced by the more encompassing C*-algebra.

The term C*-algebra itself was coined by I. E. Segal in 1947. He used it to denote norm-closed, self-adjoint subalgebras of B(H), the algebra of bounded operators on a Hilbert space H. The 'C' stood for 'closed.' Segal's definition was precise: a "uniformly closed, self-adjoint algebra of bounded operators on a Hilbert space."

Structure of C*-algebras

C*-algebras possess a remarkable number of technically convenient properties. Some of these can be deduced using the continuous functional calculus, while others are revealed by reducing the problem to commutative C*-algebras. In the latter case, the entire structure is dictated by the Gelfand isomorphism.

Self-Adjoint Elements

Elements of the form $x = x^*$ are called self-adjoint. The set of elements $x^*x$ within a C*-algebra A forms a closed convex cone. This cone is identical to the set of elements of the form $xx^*$. These are the non-negative elements, though calling them "positive" can be confusing, especially when compared to positive numbers in $\mathbb{R}$.

This subspace of self-adjoint elements naturally inherits the structure of a partially ordered vector space. The ordering, typically denoted $\geq$, means that a self-adjoint element $x \in A$ satisfies $x \geq 0$ if and only if its spectrum is non-negative, or equivalently, if $x = s^*s$ for some $s \in A$. If $x - y \geq 0$, then we say $x \geq y$.

This ordered subspace is crucial for defining positive linear functionals on a C*-algebra. These functionals, in turn, are used to define the states of the algebra, which then allow for the construction of the spectrum of a C*-algebra via the GNS construction.

Quotients and Approximate Identities

Every C*-algebra A has an approximate identity. This means there exists a directed family of self-adjoint elements $\{e_\lambda\}_{\lambda \in I}$ such that $xe_\lambda \to x$ for all $x \in A$, and $0 \leq e_\lambda \leq e_\mu \leq 1$ whenever $\lambda \leq \mu$. If A is separable, it possesses a sequential approximate identity. More generally, a sequential approximate identity exists if and only if A contains a strictly positive element, meaning a positive element $h$ such that $hAh$ is dense in A.

Using these approximate identities, we can show that the quotient of a C*-algebra by a closed proper two-sided ideal, equipped with the natural norm, is itself a C*-algebra. Similarly, any closed two-sided ideal of a C*-algebra is a C*-algebra in its own right.

Examples

Finite-Dimensional C*-algebras

The algebra $M(n, \mathbb{C})$ of $n \times n$ matrices over $\mathbb{C}$ forms a C*-algebra when viewed as operators on $\mathbb{C}^n$, using the operator norm. The involution is the conjugate transpose. More generally, finite direct sums of matrix algebras constitute all finite-dimensional C*-algebras, up to isomorphism. The self-adjoint requirement implies that finite-dimensional C*-algebras are semisimple. This leads to a theorem reminiscent of the Artin–Wedderburn theorem:

Theorem. A finite-dimensional C*-algebra, A, is canonically isomorphic to a finite direct sum $\bigoplus_{e \in \min A} Ae$, where $\min A$ is the set of minimal nonzero self-adjoint central projections of A.

Each algebra $Ae$ is isomorphic (though not necessarily canonically) to a full matrix algebra $M(\dim(e), \mathbb{C})$. The collection $\{\dim(e)\}_{e \in \min A}$ is known as 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 represents the positive cone of the $K_0$ group of A.

In physics, a finite-dimensional C*-algebra is sometimes referred to as a †-algebra or, more explicitly, a †-closed algebra. The dagger symbol, †, is used because physicists often employ it for the Hermitian adjoint, and are less concerned with the subtleties of infinite dimensions. Mathematicians typically prefer the asterisk, *. These †-algebras are fundamental in quantum mechanics and particularly in quantum information science.

The approximately finite-dimensional C*-algebras represent an immediate generalization of finite-dimensional C*-algebras.

C*-algebras of Operators

The canonical example of a C*-algebra is $B(\mathcal{H})$, the algebra of bounded (or continuous) linear operators on a complex Hilbert space $\mathcal{H}$. Here, $x^*$ denotes the adjoint operator of $x: \mathcal{H} \to \mathcal{H}$. The Gelfand–Naimark theorem states that every C*-algebra is *-isomorphic to a norm-closed, adjoint-closed subalgebra of $B(\mathcal{H})$ for some suitable Hilbert space $\mathcal{H}$.

C*-algebras of Compact Operators

Consider a separable infinite-dimensional Hilbert space $\mathcal{H}$. The algebra $K(\mathcal{H})$ of compact operators on $\mathcal{H}$ forms a norm closed subalgebra of $B(\mathcal{H})$. Since it's also closed under involution, it qualifies as a C*-algebra.

These concrete C*-algebras of compact operators share a characterization with finite-dimensional C*-algebras:

Theorem. If A is a C*-subalgebra of $K(\mathcal{H})$, then there exist Hilbert spaces $\{\mathcal{H}_i\}_{i \in I}$ such that $A \cong \bigoplus_{i \in I} K(\mathcal{H}_i)$. Here, the (C*-)direct sum consists of elements $(T_i)$ from the Cartesian product $\prod K(\mathcal{H}_i)$ for which $\|T_i\| \to 0$.

Although $K(\mathcal{H})$ lacks an identity element, a sequential approximate identity can be constructed. If we assume $\mathcal{H}$ is isomorphic to the space of square-summable sequences, $l^2$, we can define subspaces $\mathcal{H}_n$ of sequences vanishing for indices $k \geq n$, and let $e_n$ be the orthogonal projection onto $\mathcal{H}_n$. The sequence $\{e_n\}_n$ then serves as an approximate identity for $K(\mathcal{H})$.

$K(\mathcal{H})$ is a two-sided closed ideal of $B(\mathcal{H})$. For separable Hilbert spaces, it is the unique ideal. The quotient of $B(\mathcal{H})$ by $K(\mathcal{H})$ yields 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 forms a commutative C*-algebra under pointwise multiplication and addition. The involution is simply pointwise conjugation.

$C_0(X)$ possesses a multiplicative unit element if and only if $X$ is compact. Like all C*-algebras, $C_0(X)$ has an approximate identity. This is readily apparent: consider the directed set of compact subsets of $X$. For each compact subset $K$, let $f_K$ be a function with compact support that is identically 1 on $K$. Such functions exist due to the Tietze extension theorem, which applies to locally compact Hausdorff spaces. Any such sequence of functions $\{f_K\}$ constitutes an approximate identity.

The Gelfand representation asserts that every commutative C*-algebra is *-isomorphic to $C_0(X)$, where $X$ is the space of characters endowed with the weak* topology. Furthermore, if $C_0(X)$ is isomorphic to $C_0(Y)$ as C*-algebras, then $X$ and $Y$ are homeomorphic. This profound connection is a driving force behind the noncommutative topology and noncommutative geometry programs.

C*-enveloping Algebra

Given a Banach *-algebra A with an approximate identity, there exists a unique (up to C*-isomorphism) C*-algebra $E(A)$ and a *-morphism $\pi: A \to E(A)$ that is universal. This means any other continuous *-morphism $\pi': A \to B$ factors uniquely through $\pi$. The algebra $E(A)$ is known as the C*-enveloping algebra of the Banach *-algebra A.

Of particular significance 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 group C*-algebra provides the framework for general harmonic analysis on $G$, especially when $G$ is non-abelian. The dual of a locally compact group, in this context, is defined as the primitive ideal space of the group C*-algebra. See spectrum of a C*-algebra.

Von Neumann Algebras

Von Neumann algebras, previously called W*-algebras before the 1960s, are a specific type of C*-algebra. Their defining characteristic is closure in the weak operator topology, which is a weaker topology than the norm topology.

The Sherman–Takeda theorem establishes that every C*-algebra possesses a universal enveloping W*-algebra, such that any homomorphism into a W*-algebra can be factored through it.

Type for C*-algebras

A C*-algebra A is classified as type I if, for all non-degenerate representations $\pi$ of A, the von Neumann algebra $\pi(A)''$ (the bicommutant of $\pi(A)$) is itself a type I von Neumann algebra. It is sufficient to consider only factor representations, where $\pi(A)''$ is a factor.

A locally compact group is considered type I if and only if its group C*-algebra is type I.

However, if a C*-algebra admits non-type I representations, then according to the results of James Glimm, it also possesses representations of type II and type III. Therefore, for C*-algebras and locally compact groups, it is only meaningful to discuss type I and non-type I properties.

C*-algebras and Quantum Field Theory

In quantum mechanics, a physical system is typically described by a C*-algebra A with a unit element. The self-adjoint elements of A (those $x$ for which $x^* = x$) are interpreted as the system's observables—the quantities that can be measured. A state of the system is defined as a positive functional on A (a $\mathbb{C}$-linear map $\varphi: A \to \mathbb{C}$ such that $\varphi(u^*u) \geq 0$ for all $u \in A$) with the property $\varphi(1) = 1$. The expected value of an observable $x$ in state $\varphi$ is then given by $\varphi(x)$.

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

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