# Banach algebra

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In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, that is, a normed space that is complete in the metric induced by the norm. The norm is required to satisfy

## Contents

${\displaystyle \forall x,y\in A:\|x\,y\|\ \leq \|x\|\,\|y\|.}$

This ensures that the multiplication operation is continuous.

A Banach algebra is called unital if it has an identity element for the multiplication whose norm is 1, and commutative if its multiplication is commutative. Any Banach algebra ${\displaystyle A}$ (whether it has an identity element or not) can be embedded isometrically into a unital Banach algebra ${\displaystyle A_{e}}$ so as to form a closed ideal of ${\displaystyle A_{e}}$. Often one assumes a priori that the algebra under consideration is unital: for one can develop much of the theory by considering ${\displaystyle A_{e}}$ and then applying the outcome in the original algebra. However, this is not the case all the time. For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

The theory of real Banach algebras can be very different from the theory of complex Banach algebras. For example, the spectrum of an element of a nontrivial complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.

Banach algebras can also be defined over fields of p-adic numbers. This is part of p-adic analysis.

## Examples

The prototypical example of a Banach algebra is ${\displaystyle C_{0}(X)}$, the space of (complex-valued) continuous functions on a locally compact (Hausdorff) space that vanish at infinity. ${\displaystyle C_{0}(X)}$ is unital if and only if X is compact. The complex conjugation being an involution, ${\displaystyle C_{0}(X)}$ is in fact a C*-algebra. More generally, every C*-algebra is a Banach algebra.

• The set of real (or complex) numbers is a Banach algebra with norm given by the absolute value.
• The set of all real or complex n-by-n matrices becomes a unital Banach algebra if we equip it with a sub-multiplicative matrix norm.
• Take the Banach space Rn (or Cn) with norm ||x|| = max |xi| and define multiplication componentwise: (x1,...,xn)(y1,...,yn) = (x1y1,...,xnyn).
• The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.
• The algebra of all bounded real- or complex-valued functions defined on some set (with pointwise multiplication and the supremum norm) is a unital Banach algebra.
• The algebra of all bounded continuous real- or complex-valued functions on some locally compact space (again with pointwise operations and supremum norm) is a Banach algebra.
• The algebra of all continuous linear operators on a Banach space E (with functional composition as multiplication and the operator norm as norm) is a unital Banach algebra. The set of all compact operators on E is a Banach algebra and closed ideal. It is without identity if dim E = ∞. [1]
• If G is a locally compact Hausdorff topological group and μ is its Haar measure, then the Banach space L1(G) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy(g) = ∫ x(h) y(h−1g) dμ(h) for x, y in L1(G). [2]
• Uniform algebra: A Banach algebra that is a subalgebra of the complex algebra C(X) with the supremum norm and that contains the constants and separates the points of X (which must be a compact Hausdorff space).
• Natural Banach function algebra: A uniform algebra all of whose characters are evaluations at points of X.
• C*-algebra: A Banach algebra that is a closed *-subalgebra of the algebra of bounded operators on some Hilbert space.
• Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution of measures. [2]

## Counterexamples

The algebra of the quaternions ${\displaystyle \mathbb {H} }$ is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex numbers.

## Properties

Several elementary functions that are defined via power series may be defined in any unital Banach algebra; examples include the exponential function and the trigonometric functions, and more generally any entire function. (In particular, the exponential map can be used to define abstract index groups.) The formula for the geometric series remains valid in general unital Banach algebras. The binomial theorem also holds for two commuting elements of a Banach algebra.

The set of invertible elements in any unital Banach algebra is an open set, and the inversion operation on this set is continuous, (and hence is a homeomorphism) so that it forms a topological group under multiplication. [3]

If a Banach algebra has unit 1, then 1 cannot be a commutator; i.e., ${\displaystyle xy-yx\neq \mathbf {1} }$ for any x, y  A. This is because xy and yx have the same spectrum except possibly 0.

The various algebras of functions given in the examples above have very different properties from standard examples of algebras such as the reals. For example:

• Every real Banach algebra that is a division algebra is isomorphic to the reals, the complexes, or the quaternions. Hence, the only complex Banach algebra that is a division algebra is the complexes. (This is known as the Gelfand–Mazur theorem.)
• Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions. [4]
• Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
• Every commutative real unital Noetherian Banach algebra (possibly having zero divisors) is finite-dimensional.
• Permanently singular elements in Banach algebras are topological divisors of zero, i.e., considering extensions B of Banach algebras A some elements that are singular in the given algebra A have a multiplicative inverse element in a Banach algebra extension B. Topological divisors of zero in A are permanently singular in any Banach extension B of A.

## Spectral theory

Unital Banach algebras over the complex field provide a general setting to develop spectral theory. The spectrum of an element x  A, denoted by ${\displaystyle \sigma (x)}$, consists of all those complex scalars λ such that x  λ1 is not invertible in A. The spectrum of any element x is a closed subset of the closed disc in C with radius ||x|| and center 0, and thus is compact. Moreover, the spectrum ${\displaystyle \sigma (x)}$ of an element x is non-empty and satisfies the spectral radius formula:

${\displaystyle \sup\{|\lambda |:\lambda \in \sigma (x)\}=\lim _{n\to \infty }\|x^{n}\|^{1/n}.}$

Given x  A, the holomorphic functional calculus allows to define ƒ(x) A for any function ƒ holomorphic in a neighborhood of ${\displaystyle \sigma (x).}$ Furthermore, the spectral mapping theorem holds:

${\displaystyle \sigma (f(x))=f(\sigma (x)).}$ [5]

When the Banach algebra A is the algebra L(X) of bounded linear operators on a complex Banach space X (e.g., the algebra of square matrices), the notion of the spectrum in A coincides with the usual one in operator theory. For ƒ C(X) (with a compact Hausdorff space X), one sees that:

${\displaystyle \sigma (f)=\{f(t):t\in X\}.}$

The norm of a normal element x of a C*-algebra coincides with its spectral radius. This generalizes an analogous fact for normal operators.

Let A be a complex unital Banach algebra in which every non-zero element x is invertible (a division algebra). For every a A, there is λ C such that a  λ1 is not invertible (because the spectrum of a is not empty) hence a = λ1 : this algebra A is naturally isomorphic to C (the complex case of the Gelfand–Mazur theorem).

## Ideals and characters

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A. Since a maximal ideal ${\displaystyle {\mathfrak {m}}}$ in A is closed, ${\displaystyle A/{\mathfrak {m}}}$ is a Banach algebra that is a field, and it follows from the Gelfand–Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ(A) of all nonzero homomorphisms from A to C. The set Δ(A) is called the "structure space" or "character space" of A, and its members "characters."

A character χ is a linear functional on A that is at the same time multiplicative, χ(ab) = χ(a) χ(b), and satisfies χ(1) = 1. Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (i.e., operator norm) of a character is one. Equipped with the topology of pointwise convergence on A (i.e., the topology induced by the weak-* topology of A), the character space, Δ(A), is a Hausdorff compact space.

For any xA,

${\displaystyle \sigma (x)=\sigma ({\hat {x}})}$

where ${\displaystyle {\hat {x}}}$ is the Gelfand representation of x defined as follows: ${\displaystyle {\hat {x}}}$ is the continuous function from Δ(A) to C given by ${\displaystyle {\hat {x}}(\chi )=\chi (x).}$ The spectrum of ${\displaystyle {\hat {x}},}$ in the formula above, is the spectrum as element of the algebra C(Δ(A)) of complex continuous functions on the compact space Δ(A). Explicitly,

${\displaystyle \sigma ({\hat {x}})=\{\chi (x):\chi \in \Delta (A)\}}$.

As an algebra, a unital commutative Banach algebra is semisimple (i.e., its Jacobson radical is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when A is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between A and C(Δ(A)) . [lower-alpha 1]

## Banach *-algebras

A Banach *-algebra A is a Banach algebra over the field of complex numbers, together with a map * : AA that has the following properties:

1. (x*)* = x for all x in A (so the map is an involution).
2. (x + y)* = x* + y* for all x, y in A.
3. ${\displaystyle (\lambda x)^{*}={\bar {\lambda }}x^{*}}$ for every λ in C and every x in A; here, ${\displaystyle {\bar {\lambda }}}$ denotes the complex conjugate of λ.
4. (xy)* = y* x* for all x, y in A.

In other words, a Banach *-algebra is a Banach algebra over ${\displaystyle \mathbb {C} }$ that is also a *-algebra.

In most natural examples, one also has that the involution is isometric, that is,

||x*|| = ||x|| for all x in A.

Some authors include this isometric property in the definition of a Banach *-algebra.

A Banach *-algebra satisfying ||x* x|| = ||x*|| ||x|| is a C*-algebra.

## Notes

1. Proof: Since every element of a commutative C*-algebra is normal, the Gelfand representation is isometric; in particular, it is injective and its image is closed. But the image of the Gelfand representation is dense by the Stone–Weierstrass theorem.

## Related Research Articles

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication, and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multiplication operations together give A the structure of a vector space over K. In this article we will also use the term K-algebra to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.

In mathematics, specifically in functional analysis, a C-algebra is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.

In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.

In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.

In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.

In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of bounded operators on a Hilbert space. This result was proven by Israel Gelfand and Mark Naimark in 1943 and was a significant point in the development of the theory of C*-algebras since it established the possibility of considering a C*-algebra as an abstract algebraic entity without reference to particular realizations as an operator algebra.

In mathematics, the Gelfand representation in functional analysis has two related meanings:

In functional analysis, a branch of mathematics, a compact operator is a linear operator , where are normed vector spaces, with the property that maps bounded subsets of to relatively compact subsets of . Such an operator is necessarily a bounded operator, and so continuous. Some authors require that are Banach, but the definition can be extended to more general spaces.

In functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus, which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function ss2 to T yields the operator T2. Using the functional calculus for larger classes of functions, we can for example define rigorously the "square root" of the (negative) Laplacian operator −Δ or the exponential

The spectrum of a linear operator that operates on a Banach space consists of all scalars such that the operator does not have a bounded inverse on . The spectrum has a standard decomposition into three parts:

In mathematics, ergodic flows occur in geometry, through the geodesic and horocycle flows of closed hyperbolic surfaces. Both of these examples have been understood in terms of the theory of unitary representations of locally compact groups: if Γ is the fundamental group of a closed surface, regarded as a discrete subgroup of the Möbius group G = PSL(2,R), then the geodesic and horocycle flow can be identified with the natural actions of the subgroups A of real positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow as the flow built from an invertible ergodic transformation on a measure space using a ceiling function. In the case of geodesic flow, the ergodic transformation can be understood in terms of symbolic dynamics; and in terms of the ergodic actions of Γ on the boundary S1 = G / AN and G / A = S1 × S1 \ diag S1. Ergodic flows also arise naturally as invariants in the classification of von Neumann algebras: the flow of weights for a factor of type III0 is an ergodic flow on a measure space.

In functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a matrix acting on a finite-dimensional vector space; in Hilbert space, compact operators are precisely the closure of finite-rank operators in the topology induced by the operator norm. As such, results from matrix theory can sometimes be extended to compact operators using similar arguments. By contrast, the study of general operators on infinite-dimensional spaces often requires a genuinely different approach.

In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states that a Banach algebra with unit over the complex numbers in which every nonzero element is invertible is isometrically isomorphic to the complex numbers, i. e., the only complex Banach algebra that is a division algebra is the complex numbers C.

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers. This space, denoted by , is a vector space with respect to the pointwise addition of functions and scalar multiplication by constants. It is, moreover, a normed space with norm defined by

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L2(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.

In mathematics, Jordan operator algebras are real or complex Jordan algebras with the compatible structure of a Banach space. When the coefficients are real numbers, the algebras are called Jordan Banach algebras. The theory has been extensively developed only for the subclass of JB algebras. The axioms for these algebras were devised by Alfsen, Schultz & Størmer (1978). Those that can be realised concretely as subalgebras of self-adjoint operators on a real or complex Hilbert space with the operator Jordan product and the operator norm are called JC algebras. The axioms for complex Jordan operator algebras, first suggested by Irving Kaplansky in 1976, require an involution and are called JB* algebras or Jordan C* algebras. By analogy with the abstract characterisation of von Neumann algebras as C* algebras for which the underlying Banach space is the dual of another, there is a corresponding definition of JBW algebras. Those that can be realised using ultraweakly closed Jordan algebras of self-adjoint operators with the operator Jordan product are called JW algebras. The JBW algebras with trivial center, so-called JBW factors, are classified in terms of von Neumann factors: apart from the exceptional 27 dimensional Albert algebra and the spin factors, all other JBW factors are isomorphic either to the self-adjoint part of a von Neumann factor or to its fixed point algebra under a period two *-anti-automorphism. Jordan operator algebras have been applied in quantum mechanics and in complex geometry, where Koecher's description of bounded symmetric domains using Jordan algebras has been extended to infinite dimensions.

In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of

This is a glossary for the terminology in a mathematical field of functional analysis.

## References

1. Conway 1990 , Example VII.1.8.
2. Conway 1990 , Example VII.1.9.
3. Conway 1990 , Theorem VII.2.2.
4. García, Miguel Cabrera; Palacios, Angel Rodríguez (1995). "A New Simple Proof of the Gelfand-Mazur-Kaplansky Theorem". Proceedings of the American Mathematical Society. 123 (9): 2663–2666. doi:10.2307/2160559. ISSN   0002-9939.
5. Takesaki 1979 , Proposition 2.8.