In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

* Alternative algebra, an abstract algebra with alternative multiplication

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

Is X a Banach space, the space B ( X ) = B ( X, X ) forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps.

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.

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

In abstract algebra, a field is a commutative ring which contains a multiplicative inverse for every nonzero element, equivalently a ring whose nonzero elements form an abelian group under multiplication.

For any associative algebra A with multiplication, one can construct a Lie algebra L ( A ).

This is called the exponential map, and it maps the Lie algebra into the Lie group G. It provides a diffeomorphism between a neighborhood of 0 in and a neighborhood of e in G. This exponential map is a generalization of the exponential function for real numbers ( because R is the Lie algebra of the Lie group of positive real numbers with multiplication ), for complex numbers ( because C is the Lie algebra of the Lie group of non-zero complex numbers with multiplication ) and for matrices ( because M < sub > n </ sub >( R ) with the regular commutator is the Lie algebra of the Lie group GL < sub > n </ sub >( R ) of all invertible matrices ).

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

If in the above we relax Banach space to normed space the analogous structure is called a normed algebra.

Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.

For example, one cannot define all the trigonometric functions in a Banach algebra without identity.

For example, the spectrum of an element of a complex Banach algebra can never be empty, whereas in a real Banach algebra it could be empty for some elements.

The prototypical example of a Banach algebra is, the space of ( complex-valued ) continuous functions on a locally compact ( Hausdorff ) space that vanish at infinity.

In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.

* In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

The concept of an abelian group is one of the first concepts encountered in undergraduate abstract algebra, with many other basic objects, such as a module and a vector space, being its refinements.

The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets ( for spaces only requiring finite additivity, such as the ba space ).

* 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.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

* Uniform algebra: A Banach algebra that is a subalgebra of C ( X ) with the supremum norm and that contains the constants and separates the points of X ( which must be a compact Hausdorff space ).

* 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.

* The quaternions form a 4-dimensional real Banach algebra, with the norm being given by the absolute value of quaternions.

The norm of a normal element x of a C *- algebra coincides with its spectral radius.

*: The condition number computed with this norm is generally larger than the condition number computed with square-summable sequences, but it can be evaluated more easily ( and this is often the only measurable condition number, when the problem to solve involves a non-linear algebra, for example when approximating irrational and transcendental functions or numbers with numerical methods.

The algebra M < sub > n </ sub >( C ) of n-by-n matrices over C becomes a C *- algebra if we consider matrices as operators on the Euclidean space, C < sup > n </ sup >, and use the operator norm ||.|| on matrices.

The set C of continuous real-valued functions on, together with the supremum norm, is a Banach algebra, ( i. e. an associative algebra and a Banach space such that for all f, g ).

This is an algebra built up from an orthonormal basis of mutually orthogonal vectors under addition and multiplication, p of which have norm + 1 and q of which have norm − 1, with the product rule for the basis vectors

A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx ; or more generally an algebraic structure in which one of the principal binary operations is not commutative ; one also allows additional structures, e. g. topology or norm to be possibly carried by the noncommutative algebra of functions.

* The quaternions form a 4-dimensional unitary associative algebra over the reals ( but not an algebra over the complex numbers, since if complex numbers are treated as a subset of the quaternions, complex numbers and quaternions do not commute ).

In abstract algebra, a field extension L / K is called algebraic if every element of L is algebraic over K, i. e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i. e. which contain transcendental elements, are called transcendental.

Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.

The left and right alternative identities for an algebra are equivalent to

In a unital alternative algebra, multiplicative inverses are unique whenever they exist.

The close relationship of alternative algebras and composition algebras was given by Guy Roos in 2008: He shows ( page 162 ) the relation for an algebra A with unit element e and an involutive anti-automorphism such that a + a * and aa * are on the line spanned by e for all a in A.

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

Note however that both in algebra and model theory the binary operations considered are defined on all of S × S.

Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more.

Many binary operations of interest in both algebra and formal logic are commutative or associative.

Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa.

* Natural Banach function algebra: A uniform algebra whose all characters are evaluations at points of X.

Several elementary functions which 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.

* 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 all Banach extension B of A.