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An involution is a function f : X → X that, when applied twice, brings one back to the starting point. In mathematics, an involution, involutory function, or self-inverse function [1] is a function f that is its own inverse, f(f(x)) = x. for all x in the domain of f. [2] Equivalently, applying f twice produces the original value.
Any commutative *-ring is a *-algebra over itself and, more generally, over any its *-subring. For a commutative *-ring R, its quotient by any its *-ideal is a *-algebra over R. For example, any commutative trivially-*-ring is a *-algebra over its dual numbers ring, a *-ring with non-trivial *, because the quotient by ε = 0 makes the original ...
In this example, a self-adjoint morphism is a symmetric relation. The category Cob of cobordisms is a dagger compact category , in particular it possesses a dagger structure. The category Hilb of Hilbert spaces also possesses a dagger structure: Given a bounded linear map f : A → B {\displaystyle f:A\rightarrow B} , the map f † : B → A ...
A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space ), a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize).
An involution is non-defective, and each eigenvalue equals , so an involution diagonalizes to a signature matrix. A normal involution is Hermitian (complex) or symmetric (real) and also unitary (complex) or orthogonal (real). The determinant of an involutory matrix over any field is ±1. [4]
The algebra E(A) is called the C*-enveloping algebra of the Banach *-algebra A. Of particular importance is the C*-algebra of a locally compact group G. This is defined as the enveloping C*-algebra of the group algebra of G. The C*-algebra of G provides context for general harmonic analysis of G in the case G is non-abelian. In particular, the ...
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A special case of particular importance is the case where is a complete normed *-algebra, that satisfies the C*-identity (‖ ‖ = ‖ ‖ ), which is called a C*-algebra. Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian. [ 1 ]
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