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What's a good intro text on (noncommutative) C* Algebras?
A noncommutative C* algebra is a mathematical structure that combines the concepts of a Banach algebra and a Hilbert space. It is a collection of elements that can be added, multiplied, and scaled, and has a norm defined on it. The key feature of a noncommutative C* algebra is that its elements do not necessarily commute under multiplication, unlike in a commutative algebra.
Noncommutative C* algebras have a wide range of applications in mathematics and physics. They are used in the study of operator algebras, functional analysis, and quantum mechanics. They also have applications in signal processing, control theory, and differential equations.
A noncommutative C* algebra can be constructed by defining a set of elements, specifying the operations of addition, multiplication, and scalar multiplication, and defining a norm on the set that satisfies certain properties. Another way to construct a noncommutative C* algebra is by taking a commutative C* algebra and introducing noncommutativity through a suitable representation.
Some important properties of noncommutative C* algebras include the existence of a norm, the closure of the algebra under addition and multiplication, and the existence of an identity element. They also have a Banach algebra structure, meaning that they satisfy certain algebraic and topological properties.
Noncommutative C* algebras play a crucial role in the mathematical formulation of quantum mechanics. They are used to represent observables in quantum systems, and the noncommutativity of their elements reflects the uncertainty principle in quantum mechanics. Noncommutative C* algebras are also used in the study of quantum field theory and other areas of theoretical physics.