* C *- algebra

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

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

* 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.

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.

* C *- algebra.

Category: C *- algebras

C *- algebras ( pronounced " C-star ") are an important area of research in functional analysis, a branch of mathematics.

A C *- algebra is a complex algebra A of continuous linear operators on a complex Hilbert space with two additional properties:

It is generally believed that C *- algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables.

These papers considered a special class of C *- algebras which are now known as von Neumann algebras.

Around 1943, the work of Israel Gelfand and Mark Naimark yielded an abstract characterisation of C *- algebras making no reference to operators on a Hilbert space.

C *- algebras are now an important tool in the theory of unitary representations of locally compact groups, and are also used in algebraic formulations of quantum mechanics.

* 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 ).

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 ).

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 '.

The Secretary of the Treasury, upon the concurrence of the Secretary of State, is authorized and directed, out of the sum covered into the Yugoslav Claims Fund pursuant to subsection ( B ) of this section, after completing the payments of such funds pursuant to subsection ( C ) of this Section, to make payment of the balance of any sum remaining in such fund to the Government of the Federal People's Republic of Yugoslavia to the extent required under Article 1 ( C ) of the Yugoslav Claims Agreement of 1948.

It is agreed that any goods delivered or services rendered after the date of this agreement for projects within categories A, B, and C under paragraph 2 above which may later be approved by the United States will be eligible for financing from currency granted or loaned to the Government of India.

With regard to the rupees accruing to uses indicated under Article 2, of the Agreement, the understanding of the Government of the United States of America, with respect to both paragraphs 1 ( B ) and 1 ( C ) of Article 2, is as follows: ( 1 )

The `` C '' club is composed of the men of the College who have won an official letter in Carleton athletics.

Bar `` C '' is 2-3/4'' '' long.

If A is the major axis of an ellipsoid and B and C are the other two axes, the radius of curvature in the ab plane at the end of the axis Af, and the difference in pressure along the A and B axes is Af.

Since the circulating thyroid hormones are the amino acids thyroxine and tri-iodothyronine ( cf. Section C ), it is clear that some mechanism must exist in the thyroid gland for their release from proteins before secretion.

( C ) if Af is the operator induced on Af by T, then the minimal polynomial for Af is Af.

Consider a simple, closed, plane curve C which is a real-analytic image of the unit circle, and which is given by Af.

On C, from the point P at Af to the point Q at Af, we construct the chord, and upon the chord as a side erect a square in such a way that as S approaches zero the square is inside C.

If the vertex is at Af, and if the interior of C is on the left as one moves in the direction of increasing t, then every such corner can be found from the curve obtained by rotating C clockwise through 90-degrees about the vertex.

With the vertex at Af in the C-plane we assume that Af is the parametric location on C of an ordinary intersection Q between C and Af.

But Af is just the curve Af translated without rotation through a small arc, for Af is always obtained by rotating C through exactly 90-degrees.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

* Take the Banach space R < sup > n </ sup > ( or C < sup > n </ sup >) with norm || x ||

* The spectrum of any bounded linear operator on a Banach space is a nonempty compact subset of the complex numbers C. Conversely, any compact subset of C arises in this manner, as the spectrum of some bounded linear operator.

The term-algebra was introduced by C. E. Rickart in 1946 to describe Banach *- algebras that satisfy the condition:

The space C of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm.

Another version of Hahn – Banach theorem states that if V is a vector space over the scalar field K ( either the real numbers R or the complex numbers C ), if is a seminorm, and is a K-linear functional on a K-linear subspace U of V which is dominated by on U in absolute value,

The Banach-Mazur theorem asserts that any separable Banach space is isometrically isomorphic to a closed linear subspace of C ().

* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

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 ).

In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz constant ≤ K is a locally compact convex subset of the Banach space C ( X ).

The Stone – Čech compactification can be used to characterize ( the Banach space of all bounded sequences in the scalar field R or C, with supremum norm ) and its dual space.

As a rule of thumb, any sort of construction that takes as input a fairly general object ( often of an algebraic, or topological-algebraic nature ) and outputs a compact space is likely to use Tychonoff: e. g., the Gelfand space of maximal ideals of a commutative C * algebra, the Stone space of maximal ideals of a Boolean algebra, and the Berkovich spectrum of a commutative Banach ring.

Likewise, the Banach space C () of continuous functions on is not reflexive.

The promised geometric property of reflexive Banach spaces is the following: if C is a closed non-empty convex subset of the reflexive space X, then for every x in X there exists a c in C such that || x − c || minimizes the distance between x and points of C. ( Note that while the minimal distance between x and C is uniquely defined by x, the point c is not.

Let A be a commutative Banach algebra, defined over the field C of complex numbers.