Copyright © Philip M. Parker, INSEAD. Terms of Use.

C*-ALGEBRA

Specialty Definition: C-star-algebra

(From Wikipedia, the free Encyclopedia)

C^*-algebras are studied in functional analysis and are used in some formulations of quantum mechanics. A C^*-algebra A is a Banach algebra over the field of complex numbers, together with a map ^* : A -> A called involution which has the following properties:
(x + y)^* = x^* + y^* for all x, y in A
(λ x)^* = λ^* x^* for every λ in C and every x in A; here, λ^* stands for the complex conjugation of λ.
(xy)^* = y^* x^* for all x, y in A
(x^*)^* = x for all x in A
||x x^*|| = ||x||² for all x in A.

C* algebras are also * algebras. If the last property is omitted, we speak of a B^*-algebra.

*-Homormorphisms and *-Isomorphisms
A map f : A -> B between B^*-algebras A and B is called a *-homomorphism if
f is C-linear
f(xy) = f(x)f(y) for x and y in A
f(x^*) = f(x)^* for x in A
Such a map f is automatically continuous. If f is bijective, then its inverse is also a *-homorphism and f is called a *-isomorphism and A and B are called *-isomorphic. In that case, A and B are for all practical purposes identical; they only differ in the notation of their elements.

Examples of C^*-algebras
The algebra of n-by-n matrices over C becomes a C^*-algebra if we use the matrix norm ||.||₂ arising as the operator norm from the Euclidean norm on Cⁿ. The involution is given by the conjugate transpose.
The motivating example of a C^*-algebra is the algebra of continuous linear operators defined on a complex Hilbert space H; here x^* denotes the adjoint operator of the operator x : H -> H. In fact, every C^*-algebra is *-isomorphic to a closed subalgebra of such an operator algebra for a suitable Hilbert space H; this is the content of the Gelfand-Naimark theorem.
An example of a commutative C^*-algebra is the algebra C(X) of all complex-valued continuous functions defined on a compact Hausdorff space X. Here the norm of a function is the supremum of its absolute value, and the star operation is complex conjugation. Every commutative C^*-algebra with unit element is *-isomorphic to such an algebra C(X) using the Gelfand representation.
If one starts with a locally compact Hausdorff space X and considers the complex-valued continuous functions on X that vanish at infinity (defined in the article on local compactness), then these form a commutative C^*-algebra C₀(X); if X is not compact, then C₀(X) does not have a unit element. Again, the Gelfand representation shows that every commutative C^*-algebra is *-isomorphic to an algebra of the form C₀(X).

W* algebras
W* algebras are a special kind of C* algebra.

C^*-algebras and quantum field theory
In quantum field theory, one typically describes a physical system with a C^*-algebra A with unit element; the self-adjoint elements of A (elements x with x^* = x) are thought of as the observables, the measurable quantities, of the system. A state of the system is defined as a positive functional on A (a C-linear map φ : A -> C with φ(u u^*) > 0 for all u∈A) such that φ(1) = 1. The expected value of the observable x, if the system is in state φ, is then φ(x).
See Local quantum physics. See also algebra, associative algebra, * algebra, B* algebra.
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "C-star-algebra."

Anagrams: C*-ALGEBRA

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "a-a-b-c-e-g-l-r"
	-1 letter: algebra.
	-2 letters: aglare, alegar, arable, garble, laager.
	-3 letters: abler, acerb, algae, areal, areca, argal, argle, bagel, baler, barge, belga, blare, blear, brace, cabal, caber, cable, cager, carle, clear, craal, gable, galea, glace, glare, gleba, graal, grace, labra, lacer, lager, large, regal.
	-4 letters: able, acre, agar, ager, alae, alar, alba, alec, alga, area, baal, bale.
	Words containing the letters "a-a-b-c-e-g-l-r"
	+1 letter: algebraic, cablegram.
	+2 letters: cablegrams, chargeable.
	+3 letters: berascaling, rebalancing.
	+4 letters: blackguarded, rechargeable, tabernacling.
	+5 letters: algebraically, dischargeable, overbalancing, recalibrating.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.

Alternative Orthography: C*-ALGEBRA

Hexadecimal (or equivalents, 770AD-1900s) (references)

43 2A 2D 41 4C 47 45 42 52 41

Leonardo da Vinci (1452-1519; backwards) (references)

Binary Code (1918-1938, probably earlier) (references)

01000011 00101010 00101101 01000001 01001100 01000111 01000101 01000010 01010010 01000001

HTML Code (1990) (references)

C * - A L G E B R A

ISO 10646 (1991-1993) (references)

0043 002A 002D 0041 004C 0047 0045 0042 0052 0041

Encryption (beginner's substitution cypher): (references)

37121535464139365235

INDEX

1. Anagrams
2. Orthography
3. Bibliography

Copyright © Philip M. Parker, INSEAD. Terms of Use.