Each C *- algebra, Ae, is isomorphic ( in a noncanonical way ) to the full matrix algebra M < sub > dim ( e )</ sub >( C ).

The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields.

* The Lie algebra of the general linear group GL < sub > n </ sub >( R ) of invertible matrices is the vector space M < sub > n </ sub >( R ) of square matrices with the Lie bracket given by

# 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.

# If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.

The exponential map from the Lie algebra M < sub > n </ sub >( R ) of the general linear group GL < sub > n </ sub >( R ) to GL < sub > n </ sub >( R ) is defined by the usual power series:

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 ).

C < sup >∞</ sup >( M ) is a real associative algebra for the pointwise product and sum of functions and scalar multiplication.

In linear algebra, a symmetric real matrix M is said to be positive definite if z < sup > T </ sup > Mz is positive, for all non-zero column vectors z of n real numbers ; where z < sup > T </ sup > denotes the transpose of z.

* In abstract algebra, Ass ( M ) denotes the collection of all associated primes of a module M

This is simply a restatement of the following fact from linear algebra: for two square matrices M and N, M M < sup >*</ sup >

For an abstract vector space V ( rather than the concrete vector space ), or more generally a module M over a ring R, with the endomorphism algebra End ( M ) ( algebra of linear operators on M ) replacing the algebra of matrices, the analog of scalar matrices are scalar transformations.

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.

* 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 ).

If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

* The three-dimensional Euclidean space R < sup > 3 </ sup > with the Lie bracket given by the cross product of vectors becomes a three-dimensional Lie algebra.

* The Heisenberg algebra H < sub > 3 </ sub >( R ) is a three-dimensional Lie algebra with elements:

* The Lie algebra of the vector space R < sup > n </ sup > is just R < sup > n </ sup > with the Lie bracket given by = 0.

For example, the orthogonal group O < sub > n </ sub >( R ) consists of matrices A with AA < sup > T </ sup > = 1, so the Lie algebra consists of the matrices m with ( 1 + εm )( 1 + εm )< sup > T </ sup > = 1, which is equivalent to m + m < sup > T </ sup > = 0 because ε < sup > 2 </ sup > = 0.

* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.

For example, the Lie algebra of SU ( n ) is written as.

In particular, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra The associative algebra A is called an enveloping algebra of the Lie algebra L ( A ).

This is a matrix Lie group, and its Lie algebra consists of all n × n matrices with real entries and trace 0.

* The real vector space of all n × n skew-hermitian matrices is closed under the commutator and forms a real Lie algebra denoted.

This is the Lie algebra of the unitary group U ( n ).

If G is any subgroup of GL < sub > n </ sub >( R ), then the exponential map takes the Lie algebra of G into G, so we have an exponential map for all matrix groups.

The Clifford algebra Cℓ < sub > n </ sub >( C ) is algebraically isomorphic to the algebra of complex matrices, if is even ; or the algebra of two copies of the matrices, if is odd.

* C *- algebra

The complex conjugation being an involution, is in fact a C *- algebra.

More generally, every C *- algebra is a Banach algebra.

* 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 ).

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

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

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 ).

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 in A is closed, 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.

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 ).

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 )).

As it is now known that all B *- algebras are C *- algebras ( and vice versa ), the term B *- algebra is no longer widely used.

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.

Because of this, the term B * algebra is rarely used in current terminology, and has been replaced by the ( overloading of ) the term ' C * algebra '.