* 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.

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 '.

When he started school at the age of five-and-a-half, he could not understand why the alphabet begins with the letter A, instead of C, as in the scale.

( C ) Decisions of a general kind are made by the central command.

-- Fetch me the copies of everything B and C companies have requisitioned in the last six months.

( C ) tapping the energies of the entire population.

( C ) the need to allocate country resources.

Meaningful policies include: ( A ) kinds of cars the state should own, ( B ) when cars should be traded, ( C ) the need and assignment of vehicles, ( D ) use of cars in lieu of mileage allowances, ( E ) employees taking cars home, and ( F ) need for liability insurance on state automobiles.

and ( C ) to finance, for not more than three years beyond the end of said period, such activities as are required to correlate, coordinate, and round out the results of studies and research undertaken pursuant to this Act: Provided, That funds available in any one year for research and development may, subject to the approval of the Secretary of State to assure that such activities are consistent with the foreign policy objectives of the United States, be expended in cooperation with public or private agencies in foreign countries in the development of processes useful to the program in the United States: And provided further, That every such contract or agreement made with any public or private agency in a foreign country shall contain provisions effective to insure that the results or information developed in connection therewith shall be available without cost to the United States for the use of the United States throughout the world and for the use of the general public within the United States.

There shall be deducted from the amount of each payment made pursuant to subsection ( C ) of Section 8, as reimbursement for the expenses incurred by the United States, an amount equal to 5 per centum of such payment.

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.

Inventory and evaluate wildlife habitat resources in cooperation with other Federal agencies and with the States in which National Forests and Grasslands are located, as a basis for orderly development of wildlife habitat improvement and coordination programs, including ( A ) big-game, gamebird, and small-game habitat surveys and investigations on the 186 million acres of National Forests and Grasslands, ( B ) fishery habitat surveys and investigations on the 81,000 miles of National Forest fishing streams and nearly 3 million acres of lakes and impoundments, and ( C ) participation in planning, inspection, and control phases of all habitat improvement, land and water use projects conducted on National Forest lands by States, other Federal agencies, and private groups to assure that projects will benefit wildlife and be in harmony with other resource values.

( C ) quicker and more effective control action in the initial stages to prevent a large-scale epidemic.

For the year 2000 this means servicing ( A ) the protection requirements of a watershed producing at least 200 million acre-feet of water each year, ( B ) recreation and wildlife resources used each year by 635 million visitors, ( C ) a timber resource supporting an annual cut of 21 billion board feet, and ( D ) 60 million acres of rangelands.

With respect to Article 2,, paragraphs 1 ( B ) and 1 ( C ): Uses of Section 104 ( E ) and Section 104 ( G ) rupees: The Government of India will use the amount of rupees granted or loaned to it by the United States pursuant to paragraphs 1 ( B ) and 1 ( C ) for projects to promote economic development with emphasis upon the agricultural sector including food reserve storage structures and facilities as may from time to time be agreed upon by the authorized representatives of the United States and the authorized representatives of the Government of India, in the following sectors: A.

C.

The Government of India further agrees in cooperation with the Government of the United States, to coordinate the use of grant and loan funds provided for in paragraphs 1 ( B ) and 1 ( C ) of Article 2, with such direct dollar assistance as may be made available by the Government of the United States of America, so that both sources of financing may be channeled to specific and clearly identifiable economic development programs and projects.

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.

Local currency will be advanced or reimbursed to the Government of India for financing agreed projects under paragraphs 1 ( B ) and 1 ( C ) of Article 2, of the Agreement upon the presentation of such documentation as the United States may specify.

The Government of India shall maintain or cause to be maintained books and records adequate to identify the goods and services financed for agreed projects pursuant to paragraphs 1 ( B ) and 1 ( C ) of Article 2, of the Agreement, to disclose the use thereof in the projects and to record the progress of the projects ( including the cost thereof ).

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

Aside from this, the average portion contains some protein, an appreciable amount of vitamins A and C -- about one-tenth of the minimum daily requirement, and about a third of the official vitamin E requirement.

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

To assemble lid, press clay piece C in cavity of wooden design head.

Score plain side of C and leave in mold.