Banach algebra

Banach algebras (according to Stefan Banach ) are mathematical objects of functional analysis that generalize some known function spaces and operator algebras on the basis of essential common properties, e.g. B. spaces of continuous or integrable functions or algebras of continuous linear operators on Banach spaces .

A Banach algebra is a vector space in which a multiplication and a norm are also defined in such a way that certain compatibility conditions are met.

definition

A vector space over the field or the real or complex numbers with a norm and a product is a Banach algebra if: ${\displaystyle ({\mathcal {A}},+)}$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \left\|\cdot \right\|}$ ${\displaystyle \circ \colon {\mathcal {A}}\times {\mathcal {A}}\to {\mathcal {A}}}$

• ${\displaystyle ({\mathcal {A}},+,\left\|\cdot \right\|)}$is a Banach space , i.e. a completely normalized vector space ,
• ${\displaystyle ({\mathcal {A}},+,\circ )}$is an associative - algebra , ${\displaystyle \mathbb {K} }$
• ${\displaystyle \|A\circ B\|\leq \|A\|\cdot \|B\|}$for everyone , d. H. the norm is sub-multiplicative . ${\displaystyle A,B\in {\mathcal {A}}}$

As is common in algebra, the symbol for the product is often left out, only in the case of convolution the symbol or is often used. If one only demands that it is a normalized space, that is, one waives the completeness, one obtains the more general concept of normalized algebra . ${\displaystyle *}$ ${\displaystyle \star }$ ${\displaystyle ({\mathcal {A}},+,\left\|\cdot \right\|)}$

Special classes of Banach algebras

Banach - * - algebra or involutive Banach algebra

A Banach - * - algebra (over ) is a Banach algebra over together with an involution such that ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle ^{*}\colon {\mathcal {A}}\to {\mathcal {A}},\,a\mapsto a^{*}}$

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

In other words, a Banach - * - algebra is a Banach algebra and at the same time a * -algebra with an isometric involution. Some authors leave out the condition of isometry and then speak of a Banach - * algebra with isometric involution . Most naturally occurring involutions on Banach algebras, however, are isometric.

C * algebras and Von Neumann algebras

→ Main articles : C * algebra  and Von Neumann algebra

The Banach algebra of continuous linear operators over a Hilbert space motivates the following definition: A Banach algebra on which a semilinear anti-multiplicative involution is given is called C * -algebra if the so-called C * -condition is fulfilled: ${\displaystyle B(H)}$ ${\displaystyle H}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle *\colon {\mathcal {A}}\to {\mathcal {A}},x\mapsto x^{*}}$

• ${\displaystyle \|x^{*}x\|=\|x\|^{2}}$ for all ${\displaystyle x\in {\mathcal {A}}}$

Such Banach algebras can be represented on Hilbert spaces . They are then in a certain topology in the Operators algebra over the Hilbert space completed , so they are called Von Neumann algebras .

Examples

• Each Banach space is multiplied by zero, i.e. H. = 0 for all elements of the Banach space, to a Banach algebra. ${\displaystyle xy}$ ${\displaystyle x,y}$

• Be a compact space and the space of continuous functions . With the point-wise operations and the involution defined by ( complex conjugation ) and the supremum norm , it becomes a commutative C ^* -algebra. The space of limited complex-valued functions on a topological space (which is equivalent by means of Stone-Čech compactification ) or the space of C ₀ functions , the continuous functions on a locally compact space that vanish at infinity, can also be considered. ${\displaystyle K}$ ${\displaystyle {\mathcal {C}}(K)}$ ${\displaystyle f\colon K\to \mathbb {C} }$ ${\displaystyle f^{*}(x):={\overline {f(x)}}}$ ${\displaystyle \|f\|:=\sup _{x\in K}|f(x)|}$ ${\displaystyle {\mathcal {C}}(K)}$

• Let be the unit circle in . Let it be algebra with continuous functions that are holomorphic in the interior of D. With the point-wise operations and the involution defined by ( complex conjugation ) and the supremum norm, a commutative Banach - * algebra, which is not a C ^* algebra. This Banach algebra is also called the disk algebra . ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A(D)}$ ${\displaystyle f\colon D\to \mathbb {C} }$ ${\displaystyle f^{*}(z):={\overline {f({\overline {z}})}}}$ ${\displaystyle A(D)}$

• If a Banach space, then the algebra of the continuous, linear operators is a Banach algebra, which in the case is not commutative. If a Hilbert space is , then it is a C ^* -algebra. ${\displaystyle V}$ ${\displaystyle B(V)}$ ${\displaystyle V}$ ${\displaystyle \dim V>1}$ ${\displaystyle V}$ ${\displaystyle B(V)}$

• The trace class and the Hilbert-Schmidt class , or more generally the shadow classes , are examples of non-commutative Banach- * algebras that are not C ^* algebras.

• In the harmonic analysis , the Banach - * algebras , i.e. the convolutionalgebras over a locally compact group, are considered. ${\displaystyle L^{1}(G)}$ ${\displaystyle G}$

• H * -algebras are involutive Banach algebras, which are also Hilbert spaces, together with an additional condition that links the involution with the Hilbert space structure.

Basics

Some basics of the theory of Banach algebras are discussed, which show an interplay between algebraic and topological properties.

The one element

Many of the above examples are Banach algebras without a unit element . If a unity is nevertheless required, one can be adjoint . In many cases there are approximations of one in these Banach algebras ; this is a topological construct that is often a substitute for the missing unit element. This is especially true for C * algebras and group algebras . ${\displaystyle L^{1}(G)}$

The group of invertible elements

If a Banach algebra with one element is 1, the group of invertible elements is open . If namely is invertible and with , then it is also invertible, because it is easy to consider that converges and the inverse is closed. Furthermore, the inversion is continuous as a mapping on the group of invertible elements. Hence it is a topological group . ${\displaystyle A}$ ${\displaystyle A^{\times }}$ ${\displaystyle b\in A}$ ${\displaystyle a\in A}$ ${\displaystyle \|a-b\|<{\tfrac {1}{\|b^{-1}\|}}}$ ${\displaystyle a}$ ${\displaystyle \textstyle b^{-1}\sum _{n=0}^{\infty }((b-a)b^{-1})^{n}}$ ${\displaystyle a}$ ${\displaystyle a\mapsto a^{-1}}$ ${\displaystyle A^{\times }}$

The spectrum

In linear algebra , the set of eigenvalues ​​of a matrix plays an important role in the study of the matrices; H. of the elements of the Banach algebra . This generalizes to the term spectrum: ${\displaystyle B({\mathbb {K} }^{n})}$

Be a -Banach algebra with one element. For the range of , , compact and after the Gelfand-Mazur theorem not empty. The formula applies to the spectral radius . This formula is astonishing because the spectral radius is a purely algebraic quantity that only uses the concept of invertibility, the right-hand side of the spectral radius formula, on the other hand, is given by the standard of the Banach algebra. ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle a\in A}$ ${\displaystyle a}$ ${\displaystyle \sigma (a):=\{\lambda \in {\mathbb {C} }:a-\lambda \cdot 1\notin A^{\times }\}}$ ${\displaystyle r(a):=\sup\{|\lambda \mid \lambda \in \sigma (a)\}}$ ${\displaystyle \textstyle r(a)=\lim _{n\to \infty }\|a^{n}\|^{1/n}}$

For the remainder of this section, be commutative with one element. The set of all multiplicative functionals is referred to as the spectrum of , or according to Gelfand also as the Gelfand spectrum or Gelfand space of . The spectrum of is a compact space and the Gelfand transformation imparts a homomorphism of to in the Banach algebra of continuous complex-valued functions . A continuous function is assigned to each element , whereby . The spectrum of an element and the spectrum of the algebra are then related via the formula . This is explained in the article about the Gelfand transform. ${\displaystyle A}$ ${\displaystyle X_{A}}$ ${\displaystyle A\to \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A\rightarrow C(X_{A})}$ ${\displaystyle A}$ ${\displaystyle X_{A}}$ ${\displaystyle a\in A}$ ${\displaystyle {\hat {a}}\colon X_{A}\to \mathbb {C} }$ ${\displaystyle {\hat {a}}(\varphi )=\varphi (a)}$ ${\displaystyle a\in A}$ ${\displaystyle \sigma (a)={\hat {a}}(X_{A})}$

Maximum ideals

Let be a commutative -Banach algebra with one element. Is , then is a maximal ideal (with codimension 1). Conversely, if a maximal ideal is, the closure is a real ideal because of the openness of the group of invertible elements, so it must apply. Then the quotient algebra is a Banach algebra, which is a body , and this must be isomorphic according to the Gelfand-Mazur theorem . Therefore the quotient mapping is a multiplicative functional with a core . If one denotes the set of maximum ideals with , one has a bijective mapping: ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \varphi \in X_{A}}$ ${\displaystyle \ker(\varphi )\subset A}$ ${\displaystyle M\subset A}$ ${\displaystyle {\overline {M}}}$ ${\displaystyle {\overline {M}}=M}$ ${\displaystyle A/M}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A\rightarrow A/M\cong \mathbb {C} }$ ${\displaystyle M}$ ${\displaystyle \operatorname {Max} (A)}$

${\displaystyle X_{A}\to \operatorname {Max} (A),\,\,\,\varphi \mapsto \ker(\varphi )}$

There is thus a bijective relationship between the subset of the dual space and the purely algebraically defined set of maximum ideals. ${\displaystyle X_{A}}$

Applications

• Banach algebras etc. are used. a. in operator theory , as it is e.g. B. is used in quantum field theory.
• There is also the extension to Von Neumann algebras and Hilbert modules and the abstract K and KK theory , which is also referred to as non-commutative geometry .
• For the investigation of locally compact groups, one uses the Banach algebras and the group C * algebras in harmonic analysis . ${\displaystyle L^{1}(G)}$ ${\displaystyle C^{*}(G)}$

literature

• FF Bonsall , J. Duncan: Complete Normed Algebras (= results of mathematics and their border areas. NF vol. 80). Springer, Berlin et al. 1973, ISBN 3-540-06386-2 .
• Richard V. Kadison , John R. Ringrose : Fundamentals of the Theory of Operator Algebras. Special Topics. Academic Press, New York NY et al .;
  • Volume 1: Elementary Theory (= Pure and applied mathematics. Vol. 100, 1). 1983, ISBN 0-12-393301-3 ;
  • Volume 2: Advanced Theory (= Pure and applied mathematics. Vol. 100, 2). 1986, ISBN 0-12-393302-1 .
• Masamichi Takesaki: Theory of Operator Algebras I. Springer, Berlin 1979, ISBN 3-540-90391-7 (2nd printing of the 1st edition. (= Encyclopaedia of Mathematical Sciences. Vol. 124 = Encyclopaedia of Mathematical Sciences. Operator Algebras and Non -Commutative Geometry. Vol. 5). Springer, New York et al. 2002, ISBN 3-540-42248-X ).