3.2 C*-algebras

C*-algebras are the backbone of noncommutative geometry, extending the idea of continuous functions on spaces to quantum realms. They provide a framework for studying quantum spaces and their symmetries, allowing us to explore noncommutative worlds.

These algebras come with special properties like involutions and norm conditions. Examples include continuous functions on compact spaces and bounded operators on Hilbert spaces. C*-algebras are crucial for understanding quantum mechanics and other noncommutative phenomena.

Definition of C*-algebras

• C*-algebras are a fundamental concept in noncommutative geometry, providing a framework for studying quantum spaces and their symmetries
• They generalize the notion of continuous functions on a topological space to the noncommutative setting, allowing for the study of "quantum" or "noncommutative" spaces

Banach algebras with involution

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• C*-algebras are Banach algebras (complete normed algebras) equipped with an involution, which is a map $*:A\to A$ satisfying $(a^*)^*=a$, $(ab)^*=b^*a^*$, and $(\lambda a)^*=\overline{\lambda}a^*$ for all $a,b\in A$ and $\lambda\in\mathbb{C}$
• The involution generalizes the concept of complex conjugation and allows for the definition of self-adjoint elements (satisfying $a^*=a$), which play a crucial role in the theory

Norm conditions for C*-algebras

• The norm on a C*-algebra satisfies the C*-identity: $\|a^*a\|=\|a\|^2$ for all $a\in A$
• This condition ensures that the norm is compatible with the involution and leads to many important properties of C*-algebras, such as the continuous functional calculus

Examples of C*-algebras

• The space of continuous functions on a compact Hausdorff space, $C(X)$, with pointwise operations and the supremum norm, is a commutative C*-algebra
• The space of bounded linear operators on a Hilbert space, $B(H)$, with the operator norm and the adjoint operation as the involution, is a noncommutative C*-algebra
• The group C*-algebra $C^*(G)$ associated with a locally compact group $G$ is another important example in noncommutative geometry

Representations of C*-algebras

• Representations allow for the study of C*-algebras through their actions on Hilbert spaces, providing a powerful tool for understanding their structure and properties
• The representation theory of C*-algebras is closely related to the representation theory of locally compact groups and plays a central role in noncommutative geometry

Hilbert space representations

• A Hilbert space representation of a C*-algebra $A$ is a -homomorphism $\pi:A\to B(H)$ from $A$ into the C-algebra of bounded linear operators on a Hilbert space $H$
• Representations allow for the study of C*-algebras through their actions on Hilbert spaces, providing a concrete realization of the abstract algebraic structure

GNS construction

• The Gelfand-Naimark-Segal (GNS) construction associates a Hilbert space representation to each state (positive linear functional of norm 1) on a C*-algebra
• Given a state $\varphi$ on $A$, the GNS construction yields a triple $(H_\varphi,\pi_\varphi,\xi_\varphi)$ consisting of a Hilbert space $H_\varphi$, a representation $\pi_\varphi:A\to B(H_\varphi)$, and a cyclic vector $\xi_\varphi\in H_\varphi$ such that $\varphi(a)=\langle\pi_\varphi(a)\xi_\varphi,\xi_\varphi\rangle$ for all $a\in A$

Irreducible representations

• An irreducible representation of a C*-algebra is a representation $\pi:A\to B(H)$ for which there are no non-trivial closed $\pi(A)$-invariant subspaces of $H$
• Irreducible representations play a crucial role in the study of C*-algebras, as they serve as the building blocks for more general representations

Equivalence of representations

• Two representations $\pi_1:A\to B(H_1)$ and $\pi_2:A\to B(H_2)$ are said to be equivalent if there exists a unitary operator $U:H_1\to H_2$ such that $U\pi_1(a)U^*=\pi_2(a)$ for all $a\in A$
• The equivalence classes of irreducible representations form the spectrum of a C*-algebra, which is a key object in noncommutative geometry

Spectrum of C*-algebras

• The spectrum of a C*-algebra is a generalization of the space of characters (multiplicative linear functionals) of a commutative Banach algebra, providing a noncommutative analog of the Gelfand transform
• Understanding the spectrum is crucial for studying the structure and properties of C*-algebras in noncommutative geometry

Spectrum of an element

• The spectrum of an element $a$ in a C*-algebra $A$ is the set $\sigma(a)=\{\lambda\in\mathbb{C}:a-\lambda 1\text{ is not invertible in }A\}$
• The spectrum generalizes the notion of eigenvalues to the noncommutative setting and plays a fundamental role in the analysis of C*-algebras

Spectral radius formula

• The spectral radius of an element $a$ in a C*-algebra $A$ is defined as $r(a)=\sup\{|\lambda|:\lambda\in\sigma(a)\}$
• The spectral radius formula states that $r(a)=\lim_{n\to\infty}\|a^n\|^{1/n}$, connecting the spectral properties of an element with its norm

Continuous functional calculus

• For a normal element $a$ (satisfying $a^*a=aa^*$) in a C*-algebra $A$, the continuous functional calculus allows for the definition of $f(a)$ for any continuous function $f$ on the spectrum $\sigma(a)$
• This powerful tool enables the transfer of functional analytic properties from the commutative setting to the noncommutative realm

Spectral theorem for commutative C*-algebras

• The spectral theorem states that any commutative C*-algebra $A$ is isometrically *-isomorphic to the algebra of continuous functions on its spectrum, $C(\text{spec}(A))$
• This result highlights the deep connection between commutative C*-algebras and topological spaces, serving as a foundation for noncommutative geometry

Ideals in C*-algebras

• Ideals in C*-algebras play a crucial role in understanding their structure and in the construction of quotient algebras, which are essential in noncommutative geometry
• The study of ideals and their properties provides insight into the representation theory and the classification of C*-algebras

Left, right, and two-sided ideals

• A left (resp. right) ideal in a C*-algebra $A$ is a closed subspace $I\subset A$ satisfying $AI\subset I$ (resp. $IA\subset I$)
• A two-sided ideal is a subspace that is both a left and a right ideal
• Closed two-sided ideals are particularly important, as they allow for the construction of quotient C*-algebras

Closed ideals and quotient algebras

• For a closed two-sided ideal $I$ in a C*-algebra $A$, the quotient algebra $A/I$ is the C*-algebra obtained by considering the cosets $a+I$ with the induced operations and norm
• Quotient algebras allow for the study of C*-algebras by "dividing out" certain substructures, which is a powerful technique in noncommutative geometry

Primitive ideals and primitive spectrum

• A primitive ideal in a C*-algebra $A$ is the kernel of an irreducible representation of $A$
• The primitive spectrum, $\text{Prim}(A)$, is the set of primitive ideals of $A$ with the Jacobson topology
• The primitive spectrum is a key object in the representation theory of C*-algebras and plays a role analogous to the maximal ideal space in commutative Banach algebras

Prime ideals and prime spectrum

• A prime ideal in a C*-algebra $A$ is a proper two-sided ideal $P$ such that whenever $I,J$ are two-sided ideals with $IJ\subset P$, then either $I\subset P$ or $J\subset P$
• The prime spectrum of $A$ is the set of prime ideals with the Jacobson topology
• Prime ideals and the prime spectrum are important in the study of the ideal structure of C*-algebras and in their classification

Positivity in C*-algebras

• Positivity is a fundamental concept in C*-algebras, generalizing the notion of non-negative functions in the commutative setting
• The study of positive elements and positive maps is essential in noncommutative geometry, as it provides a framework for understanding the order structure and the probabilistic aspects of quantum systems

Positive elements and positive functionals

• An element $a$ in a C*-algebra $A$ is called positive if it is self-adjoint and its spectrum is contained in the non-negative real numbers
• A linear functional $\varphi:A\to\mathbb{C}$ is called positive if $\varphi(a)\geq 0$ for all positive elements $a\in A$
• Positive elements form a cone in the real vector space of self-adjoint elements, and positive functionals are crucial in the definition of states on C*-algebras

States and pure states

• A state on a C*-algebra $A$ is a positive linear functional $\varphi:A\to\mathbb{C}$ with $\|\varphi\|=1$
• A state $\varphi$ is called pure if it cannot be written as a non-trivial convex combination of other states
• States correspond to the probabilistic interpretation of quantum systems, with pure states representing the extremal points of the convex set of states

Completely positive maps

• A linear map $\Phi:A\to B$ between C*-algebras is called positive if it maps positive elements to positive elements
• $\Phi$ is called completely positive if the amplified maps $\Phi\otimes\text{id}_{M_n}:A\otimes M_n\to B\otimes M_n$ are positive for all $n\in\mathbb{N}$, where $M_n$ denotes the C*-algebra of $n\times n$ complex matrices
• Completely positive maps are essential in the study of quantum channels and the dynamics of quantum systems

Stinespring's dilation theorem

• Stinespring's dilation theorem states that any completely positive map $\Phi:A\to B(H)$ can be written as $\Phi(a)=V^*\pi(a)V$, where $\pi:A\to B(K)$ is a *-representation and $V:H\to K$ is a bounded linear operator
• This result provides a powerful tool for studying completely positive maps and has numerous applications in quantum information theory and noncommutative geometry

Tensor products of C*-algebras

• Tensor products allow for the construction of new C*-algebras from given ones, which is essential in noncommutative geometry and quantum field theory
• There are several different tensor product constructions for C*-algebras, each with its own properties and applications

Algebraic tensor products

• The algebraic tensor product of two C*-algebras $A$ and $B$ is the tensor product of $A$ and $B$ as complex vector spaces, equipped with the unique *-algebra structure satisfying $(a\otimes b)(c\otimes d)=ac\otimes bd$ and $(a\otimes b)^*=a^*\otimes b^*$
• The algebraic tensor product is the starting point for the construction of the various C*-algebraic tensor products

Spatial tensor products

• The spatial tensor product (or minimal tensor product) of two C*-algebras $A$ and $B$, denoted by $A\otimes_{\text{min}}B$, is the completion of the algebraic tensor product with respect to the spatial norm, which is the norm induced by the representations of $A$ and $B$ on Hilbert spaces
• The spatial tensor product is the smallest among all C*-algebraic tensor products and has the universal property for *-representations of the algebraic tensor product

Maximal tensor products

• The maximal tensor product of two C*-algebras $A$ and $B$, denoted by $A\otimes_{\text{max}}B$, is the completion of the algebraic tensor product with respect to the maximal C*-norm, which is the largest C*-norm on the algebraic tensor product
• The maximal tensor product has the universal property for -homomorphisms from $A$ and $B$ into a common C-algebra

Nuclear C*-algebras

• A C*-algebra $A$ is called nuclear if the spatial and maximal tensor products of $A$ with any other C*-algebra coincide
• Nuclear C*-algebras form a large and well-behaved class of algebras, which includes all commutative C*-algebras and all finite-dimensional C*-algebras
• The notion of nuclearity is crucial in the classification theory of C*-algebras and has deep connections with the approximation properties of the underlying quantum spaces

K-theory of C*-algebras

• K-theory is a powerful tool in noncommutative geometry, providing invariants that capture important structural properties of C*-algebras
• The K-theory groups encode information about the projections and unitaries in a C*-algebra, which are the noncommutative analogs of vector bundles and invertible functions, respectively

K0 and K1 groups

• The K0-group of a C*-algebra $A$, denoted by $K_0(A)$, is the Grothendieck group of the monoid of equivalence classes of projections in the matrix algebras over $A$
• The K1-group of $A$, denoted by $K_1(A)$, is the group of equivalence classes of unitaries in the matrix algebras over $A$, with the direct sum as the group operation
• The K0 and K1 groups are abelian groups and are functorial with respect to -homomorphisms between C-algebras

Projections and unitaries

• A projection in a C*-algebra $A$ is a self-adjoint element $p$ satisfying $p^2=p$
• A unitary in $A$ is an element $u$ satisfying $u^*u=uu^*=1$
• Projections and unitaries are the building blocks of K-theory and play a central role in the study of the structure and classification of C*-algebras

Bott periodicity

• Bott periodicity is a fundamental result in K-theory, stating that there are natural isomorphisms $K_0(A)\cong K_1(SA)$ and $K_1(A)\cong K_0(SA)$, where $SA$ denotes the suspension of the C*-algebra $A$
• This periodicity allows for the computation of the K-theory groups of a C*-algebra in terms of the K-theory groups of its suspension, providing a powerful computational tool

Classification of C*-algebras

• The K-theory groups, along with other invariants such as the trace space and the Cuntz semigroup, are used in the classification of C*-algebras
• The Elliott classification program seeks to classify simple separable nuclear C*-algebras by their K-theoretic and tracial data
• The classification of C*-algebras is a central problem in noncommutative geometry and has deep connections with the structure and properties of the underlying quantum spaces

Applications of C*-algebras

• C*-algebras have numerous applications in various branches of mathematics and physics, providing a unifying framework for the study of quantum systems and noncommutative spaces
• The theory of C*-algebras is a fundamental tool in noncommutative geometry, operator algebras, and quantum physics

Quantum mechanics and noncommutative geometry

• C*-algebras provide a natural framework for the mathematical description of quantum systems, with self-adjoint elements representing observables and states corresponding to positive linear functionals
• Noncommutative geometry, pioneered by Alain Connes, uses C*-algebras to generalize the tools of differential geometry to noncommutative spaces, such as the space of leaves of a foliation or the noncommutative tori

Operator algebras and von Neumann algebras

• C*-algebras are closely related to von Neumann algebras, which are *-subalgebras of $B(H)$ that are closed in the weak operator topology
• The study of operator algebras, including C*-algebras and von Neumann algebras, is a rich and active area of research, with deep connections to various branches of mathematics and physics

Group C*-algebras and crossed products

• Given a locally compact group $G$, the group C*-algebra $C^*(G)$ is a C*-algebra that encodes the unitary representation theory of