C * algebra

C * -algebras are examined in the mathematical sub-area of functional analysis. They originated in mathematical physics . It is an abstraction of the restricted linear operators on a Hilbert space , so they play a role in the mathematical description of quantum mechanics . C * -algebras are special Banach algebras in which there is a close relationship between algebraic and topological properties; the category of locally compact spaces turns out to be equivalent to the category of commutative C * -algebras, therefore the theory of C * -algebras is also viewed as a non-commutative topology . If such a non-commutative topology is induced by a metric, this is covered by the relatively new research field of non-commutative geometry , which was founded by Alain Connes in the 1990s .

Definition and characteristics

A C * -algebra above the body or is a Banach algebra with an involution with the following properties ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle ^{*}\colon {\mathcal {A}}\ni a\mapsto a^{*}\in {\mathcal {A}}}$

${\displaystyle \forall a\in {\mathcal {A}}:(a^{*})^{*}=a}$	(involutive)
${\displaystyle \forall a,b\in {\mathcal {A}}:(ab)^{*}=b^{*}a^{*}}$	(anti-multiplicative)
${\displaystyle \forall a,b\in {\mathcal {A}}:\forall z,w\in \mathbb {K} :(za+wb)^{*}={\bar {z}}a^{*}+{\bar {w}}b^{*}}$	(semilinear, anti-linear or conjugated linear)
${\displaystyle \forall a\in {\mathcal {A}}:\|a^{*}a\|=\|a\|^{2}}$	(C * property)

From the C * property it follows that the involution is isometric , which together with the first three properties of the C * algebra makes it a Banach - * algebra (= involutive Banach algebra) .

One speaks of a commutative C * -algebra if the multiplication is commutative. Most authors always understand a C * -algebra as a -Banach algebra and write more precisely real C * -algebra, even if -Banach algebras are allowed. ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {R} }$

Standard examples; the theorems of Gelfand-Neumark and Gelfand-Neumark-Segal

→ Main article : Gelfand-Neumark theorem

The best-known example of a C * -algebra is the algebra of the bounded linear operators on a Hilbert space and, more generally, every self-adjoint subalgebra of which is closed in the norm topology . Conversely, according to Gelfand-Neumark-Segal's theorem, every C * -algebra has this form, i.e. it is isomorphic to a norm-closed self-adjoint subalgebra of one . ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$ ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$

The complex-valued, continuous and infinitely vanishing functions on a locally compact Hausdorff space form a commutative C * -algebra with regard to the supremum norm and the complex conjugation as involution . The Gelfand-Naimark theorem states that every commutative C * -algebra is isomorphic to such an algebra of functions. ${\displaystyle {\mathcal {X}}}$ ${\displaystyle {\mathcal {C}}_{0}({\mathcal {X}})}$

Further properties of C * algebras

Homomorphisms between C * algebras

If and are C * -algebras, then a mapping is called * - homomorphism if it is linear, multiplicative and compatible with the involution. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \varphi \colon {\mathcal {A}}\to {\mathcal {B}}}$

Every * -homomorphism is contracting , that is, it is true for anything , and therefore especially continuous . ${\displaystyle \varphi }$ ${\displaystyle \|\varphi (a)\|\leq \|a\|}$ ${\displaystyle a\in {\mathcal {A}}}$

Injective * homomorphisms are automatically isometric , that is, it applies to anything . ${\displaystyle \varphi }$ ${\displaystyle \|\varphi (a)\|=\|a\|}$ ${\displaystyle a\in {\mathcal {A}}}$

Finite-dimensional C * -algebras

The algebras of the complex matrices , which can be identified with the linear operators on , form a C * algebra with the operator norm . One can show that every finite-dimensional C * -algebra is isomorphic to a direct sum of such matrix algebras. ${\displaystyle n\times n}$ ${\displaystyle M_{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

Construction of new C * -algebras from given

• Direct sums , direct products , inductive limits of C * -algebras are again C * -algebras with a suitable standard definition.

• There is a minimum and a maximum possibility of completing tensor products from C * -algebras to C * -algebras, see “ Spatial tensor product ” and “ Maximum tensor product ”.

• In the article on dual C * -algebras are limited products defined.

• A closed two-sided ideal is automatically closed with respect to the involution and the quotient algebra with the quotient norm is again a C * algebra.

• Further C * -algebras can be constructed from a C * -dynamic system , the cross product and the reduced cross product . ${\displaystyle ({\mathcal {A}},{\mathcal {G}},\alpha )}$ ${\displaystyle {\mathcal {A}}\ltimes _{\alpha }{\mathcal {G}}}$ ${\displaystyle {\mathcal {A}}\ltimes _{\alpha r}{\mathcal {G}}}$

Single elements

C * algebras do not have to have a unit element . However, one can always adjoint a unity or, as a replacement for a missing unity, use a restricted approximation of the unity that exists in every C * algebra.

Hilbert space representations

→ Main article : Hilbert space representation

If a Hilbert space is, then a * homomorphism is called a Hilbert space representation or simply representation of . The theory of Hilbert space representations is an important instrument for the further investigation of C * algebras. ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {A}}\rightarrow {\mathcal {B}}({\mathcal {H}})}$ ${\displaystyle {\mathcal {A}}}$

Examples and special cases of C * algebras

• The compact operators on a Hilbert space form a C * subalgebra of . is a closed two-sided ideal in . The quotient algebra is called Calkin's algebra . ${\displaystyle {\mathcal {K}}({\mathcal {H}})}$ ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$ ${\displaystyle {\mathcal {K}}({\mathcal {H}})}$ ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$ ${\displaystyle {\mathcal {B}}({\mathcal {H}})/{\mathcal {K}}({\mathcal {H}})}$
• Von Neumann algebras are strongly closed * subalgebras of . Since the strong operator topology is coarser than the standard topology, the Von Neumann algebras are also closed in the standard topology and therefore in particular C * algebras. ${\displaystyle {\mathcal {B}}({\mathcal {H}})}$
• Group C * algebras (= enveloping C * algebra of group algebra ) ${\displaystyle L^{1}(G)}$
• Dual C * algebras (= C * algebras of compact operators)
• Liminal C * algebras (= CCR algebras)
• Postliminal C * algebras (= GCR algebras or Type IC * algebras), for example the Toeplitz algebra
• UHF algebras , for example the CAR algebra
• AF-C * algebras ,
• Bunce-Deddens algebras
• Nuclear C * algebras ,
• Irrational rotational algebra ,
• Cuntz algebra ,
• Cuntz Warrior Algebra,

Historical remarks

According to Gelfand and Neumark (1943), a B * -algebra is an involutive Banach algebra (with one element 1) with the two properties ${\displaystyle {\mathcal {A}}}$

1. ${\displaystyle \|a^{*}a\|=\|a\|^{2}}$for all , ${\displaystyle a\in {\mathcal {A}}}$
2. ${\displaystyle 1+a^{*}a}$is invertible for each . ${\displaystyle a\in {\mathcal {A}}}$

A C * -algebra was defined as a norm-closed and involution-closed sub-algebra of the algebra of operators on a Hilbert space. Gelfand and Neumark were then able to show that every B * -algebra is a C * -algebra. The redundancy of the second condition, which they had already assumed, could only be shown in the 1950s by M. Fukamiya and I. Kaplansky . The term B * -algebra as an abstractly defined (i.e. not represented on a Hilbert space) algebra has become dispensable by the theorem of Gelfand-Neumark , which is why the term B * -algebra can only be found in older literature.

The name C * -algebra was coined by the publication Irreducible representations of operator algebras (1947) by the mathematician Irving Segal . Possibly the C in C * algebra suggests that C * algebras are a non-commutative analog of the space of continuous functions and the sign * emphasizes the importance of involution. ${\displaystyle C(T)}$

The C * condition for all could be further weakened in the 1960s to for all , which can be derived from the theorem of Vidav-Palmer , which in turn characterizes the C * -algebras among all Banach algebras. However, this weakening of the C * condition does not play a special role in the theory of C * algebras. ${\displaystyle \|a^{*}a\|=\|a\|^{2}}$ ${\displaystyle a\in A}$ ${\displaystyle \|a^{*}a\|=\|a^{*}\|\|a\|}$ ${\displaystyle a\in A}$

Generalizations

In mathematical physics , the term is generalized for the purpose of treating general physical observables in quantum field theory by assuming not Hilbert or Banach spaces, but more general Gelfand space triples (i.e. also allowing distribution-valued functionals and the like).

See also

• Extension (C * -algebra)
• KK theory

literature

• W. Arveson : Invitation to C * -algebras , ISBN 0-387-90176-0
• J. Dixmier : Les C * -algèbres et leurs représentations , Gauthier-Villars, 1969
• RV Kadison , JR Ringrose : Fundamentals of the Theory of Operator Algebras , 1983, ISBN 0-12-393301-3
• Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-12-549450-5
• M. Takesaki: Theory of Operator Algebras I (Springer 1979, 2002)
• I. Khavkine and V. Moretti: Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction , Univ. Trient, 2015, arxiv : 1412.5945

Individual evidence

1. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-12-549450-5 , p. 5.