*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings R and A, where R is commutative and A has the structure of an associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.
Terminology
*-ring
In mathematics, a *-ring is a ring with a map * : A → A that is an antiautomorphism and an involution.
More precisely, * is required to satisfy the following properties:
for all x, y in A.
This is also called an involutive ring, involutory ring, and ring with involution. Note that the third axiom is actually redundant, because the second and fourth axioms imply 1* is also a multiplicative identity, and identities are unique.
Algebra
Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).
†-algebra
A †-algebra (or, more explicitly, a †-closed algebra) is the name occasionally used in physics for a finite-dimensional C*-algebra. The dagger, †, is used in the name because physicists typically use the symbol to denote a hermitian adjoint, and are often not worried about the subtleties associated with an infinite number of dimensions. (Mathematicians usually use the asterisk, *, to denote the hermitian adjoint.) †-algebras feature prominently in quantum mechanics, and especially quantum information science.
References
Podcasts:
Algebra
ALBUMS
- Feed the Ego released: 2014
- Polymorph released: 2012
Algebra
ALBUMS
- Higher and Higher: A Tribute to the Moody Blues released: 2005
- Fanfare for the Pirates: A Tribute to ELP released: 1998
- Zarathustra's Revenge released: 1997
- The River of Constant Change released: 1996
- Harbour of Joy: A Tribute to Camel released: 1996
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by Camp
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by Soul Hooligan
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by A-Camp
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Algebra
by: CardigansWhy can't you just Forget about algebra It's all about you now And all your talk Of logic and formula Could never help you now Not anymore
Cause you were always On the run From the darkness In your heart So you wear it On the outside Of your chest
I have taken The liberty To tell your ghost to go Bribing them with Sunlight and sympathy They promised not to show For a while
Cause you were always The little boy Who couldn't keep it To yourself So your heart Is on the outside Of your chest
At the speed of light You moved inside my home Nothing is alright If you are still alone
And your heart Is greater than The sum of you And everyone
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