Structure space
Encyclopedia
The structure space of a commutative Banach algebra
is an analog of the spectrum of a C*-algebra. It consists of all multiplicative linear functional
s on the algebra. The Gelfand representation
of the Banach algebra is a map taking the Banach algebra elements to continuous function
s on the structure space.
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
is an analog of the spectrum of a C*-algebra. It consists of all multiplicative linear functional
Linear functional
In linear algebra, a linear functional or linear form is a linear map from a vector space to its field of scalars. In Rn, if vectors are represented as column vectors, then linear functionals are represented as row vectors, and their action on vectors is given by the dot product, or the...
s on the algebra. The Gelfand representation
Gelfand representation
In mathematics, the Gelfand representation in functional analysis has two related meanings:* a way of representing commutative Banach algebras as algebras of continuous functions;...
of the Banach algebra is a map taking the Banach algebra elements to continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s on the structure space.