Here are some examples of banach algebra that arise as a space of functions on a topological space. The real analogue to the above theorem is segals theorem. If e is semisimple, and a subalgebra a of cx, e is a banach algebra under some norm. A real banach algebra ais iso metrically isomorphic to the algebra ck. Introduction to normed algebras and their representations. It is clear that the norm on a is submultiplicative. Remark if iis a twosided ideal of a, then aiis also an algebra, since it is both a ring and a vector space and xy xy xy x y and similarly xy x y.
I tried to find a definition of subalgebra but the only definition i found was in murphy who defines it to be itself a normed algebra with the norm got by restriction. A complete normed algebra is called a banach algebra. The algebra cx, e is an evalued function algebra on x in the strong sense with the norm. A banach algebra a with an involution with ka a k ka k2 for a 2a. This particular paper focuses on gelfand theory the relation between mul tiplicative linear functionals on a commutative banach algebra and. Banach algebras in this section we discuss an important concept in functional analysis. If e is a complex banach space, then be, the set of bounded linear operators on e is a unital banach algebra when equipped with the usual linear structure and. Let k be either r or c a normed algebra over k is an algebra a over k, which also carries a norm k. In 7 we prove essential continuity and boundedness properties of the elements of. A banach algebra is a complex banach space atogether with an associative and distributive multiplication such that ab ab ab and kabkkakkbk for all a. A banach algebra that is a subalgebra of the complex algebra cx with the supremum norm and that contains the constants and separates the. Also, the non unital commutative complex banach algebra can be embed ded as a subalgebra of c.
Chapter 1 banach algebras graduate school of mathematics. Banach algebras of vectorvalued functions request pdf. Now suppose that a is a banach algebra, and that an is the set of elements of a of dimension no greater than the integer re. A norm closed, selfadjoint, subalgebra a of bh algebra of bounded operators on a hilbert space. Willis, introduction to banach algebras, operators, and harmonic analysis. A banach algebra that is a subalgebra of the complex algebra cx with the supremum norm and that contains the constants and separates the points of x which must be a compact hausdorff space. Also, the non unital commutative complex banach algebra can be embed ded as a subalgebra of c0. A complex banach algebra a is iso metrically isomorphic to the algebra ck,c, of the continuous functions k c, for.