In that setting one can use birational geometry, techniques from number theory, Galois theory and commutative algebra, and close analogues of the methods of algebraic topology, all in an integrated way.

The class of all unital associative R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg.

* An example of a non-unitary associative algebra is given by the set of all functions f: R → R whose limit as x nears infinity is zero.

for all x and y in the algebra.

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.

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.

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I.

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

* 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 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 closed ideal in this 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 ).

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

* 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.

* 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.

* C *- algebra: A Banach algebra that is a closed *- subalgebra of the algebra of bounded operators on some Hilbert space.

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 the operator theory.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

The Lie algebra of any compact Lie group ( very roughly: one for which the symmetries form a bounded set ) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones.

Here C < sub > b </ sub >( X ) denotes the C *- algebra of all continuous bounded functions on X with sup-norm.

* Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix " ' ", and governed by the axioms x ' x = 1, x ( x ' y )

* A bounded distributive lattice with an involution satisfying De Morgan's laws ( i. e. a De Morgan algebra ), additionally satisfying the inequality x ∧− x ≤ y ∨− y.

The space of bounded linear operators B ( X ) on a Banach space X is an example of a unital Banach algebra.

This extends the definition for bounded linear operators B ( X ) on a Banach space X, since B ( X ) is a Banach algebra.

* A Heyting algebra is a Cartesian closed ( bounded ) lattice.

* Positive element of a C *- algebra ( such as a bounded linear operator ) whose spectrum consists of positive real numbers

In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism f: X → Y is the quotient of Y by the image of f. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient.

This suggests that one might define a noncommutative Riemannian manifold as a spectral triple ( A, H, D ), consisting of a representation of a C *- algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that is bounded for all a in some dense subalgebra of A.

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

The Gelfand representation or Gelfand isomorphism for a commutative C *- algebra with unit is an isometric *- isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of A with the weak * topology.

The continuous complex-valued functions on a compact Hausdorff topological space form a commutative C *- algebra.

By the Gelfand theorem, a commutative C *- algebra is isomorphic to the C *- algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C *- algebra up to homeomorphism.

Informally, C can be regarded as the *- algebra of continuous complex-valued functions over the compact matrix quantum group, and u can be regarded as a finite-dimensional representation of the compact matrix quantum group.

For any locally compact Hausdorff topological space X, the space C < sub > 0 </ sub >( X ) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C *- algebra:

Note that A is unital if and only if X is compact, in which case C < sub > 0 </ sub >( X ) is equal to C ( X ), the algebra of all continuous complex-valued functions on X.

Under this duality, every compact Hausdorff space is associated with the algebra of continuous complex-valued functions on, and every commutative C *- algebra is associated with the space of its maximal ideals.

Using the Haar measure, one can define a convolution operation on the space C < sub > c </ sub >( G ) of complex-valued continuous functions on G with compact support ; C < sub > c </ sub >( G ) can then be given any of various norms and the completion will be a group algebra.

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.

where is the Gelfand representation of x defined as follows: is the continuous function from Δ ( A ) to C given by The spectrum of 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 ).

*: 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.

* Clone ( algebra ), a collection of functions with certain properties

For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.

Algebra of continuous functions: a contravariant functor from the category of topological spaces ( with continuous maps as morphisms ) to the category of real associative algebras is given by assigning to every topological space X the algebra C ( X ) of all real-valued continuous functions on that space.

Set theory has come to play the role of a foundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects ( for example, numbers and functions ) from all the traditional areas of mathematics ( such as algebra, analysis and topology ) in a single theory, and provides a standard set of axioms to prove or disprove them.

Because abstract algebra studies sets endowed with operations that generate interesting structure or properties on the set, functions which preserve the operations are especially important.

# Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket = XY − YX, because the Lie bracket of any two derivations is a derivation.

In abstract algebra, one distinguishes between polynomials and polynomial functions.

The Stone – Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval, an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from more general subalgebras of C ( X ) is investigated.

The Stone – Weierstrass theorem is a vital result in the study of the algebra of continuous functions on a compact Hausdorff space.