*-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

Crook

Crook is another name for criminal.

Crook or Crooks may also refer to:

Places

Surnames

Films

Crook (film)

Crook is a 2010 Bollywood adult thriller film popularly known by the title of It's Good to be Bad!. The film stars Emraan Hashmi and Neha Sharma in the lead. It is directed by Mohit Suri and produced by Mukesh Bhatt. It was released on 8 October 2010. Before the release, the film was given an 'A' certificate from the Indian Censor Board, due to the erotic scenes between Emraan Hashmi and Shella Allen. Mostly shot in Australia and South Africa, the film is based on the controversy regarding the allegedly racial attacks on Indian students in Australia between 2007 and 2010. The film met mixed responses upon its release and went on to become a box office flop.

Story

Crook (music)

A crook, also sometimes called a shank, is an exchangeable segment of tubing in a natural horn (or other brass instrument, such as a natural trumpet) which is used to change the length of the pipe, altering the fundamental pitch and harmonic series which the instrument can sound, and thus the key in which it plays.

Master crook and coupler system

Early horns had unalterable lengths and permanently attached mouthpieces. This presented problems in concert situations. A different horn was required for different keys, and the instrument could not be tuned. Around 1700 the Leichnamschneider brothers in Vienna developed a horn with a removable mouthpiece which could be connected to a short piece of tubing, called a master crook. Additional pieces, couplers, of different length were inserted between the master crook and the body of the horn to change the horn's length, and thus the pitch. Fine tuning was done with even shorter segments called tuning bits. This simple and relatively inexpensive solution remained in use even into the 19th century. Charles Tully's Tutor for the French Horn, published in London, recommended this system for beginners as late as 1840.