Subalgebra of banach algebra 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. If e is semisimple, and a subalgebra a of cx, e is a banach algebra under some norm. Representable extensions of positive functionals and. Also, the non unital commutative complex banach algebra can be embed ded as a subalgebra of c. Banach algebras indian institute of technology hyderabad. 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 known extension theorem for pure states on a banach algebra with isometric involution is shown to hold for the wider class of banach algebras with arbitrary, possibly discontinuous, involutions. It is clear that the norm on a is submultiplicative.

The algebra cx, e is an evalued function algebra on x in the strong sense with the norm. 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. Note that all these examples are commutative banach algebras. Also, the non unital commutative complex banach algebra can be embed ded as a subalgebra of c0. The result presented here are needed in section 7 as well as in chapter v.

Maximal ideal space of a quotient or banach subalgebra. This particular paper focuses on gelfand theory the relation between mul tiplicative linear functionals on a commutative banach algebra and. A banach algebra that is a subalgebra of the complex algebra cx with the supremum norm and that contains the constants and separates the. 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. Banach algebras in this section we discuss an important concept in functional analysis. A complex banach algebra a is iso metrically isomorphic to the algebra ck,c, of the continuous functions k c, for. Here are some examples of banach algebra that arise as a space of functions on a topological space. A complete normed algebra is called a banach algebra. 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.

Introduction to normed algebras and their representations. Chapter 1 banach algebras graduate school of mathematics. A norm closed, selfadjoint, subalgebra a of bh algebra of bounded operators on a hilbert space. 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 real banach algebra ais iso metrically isomorphic to the algebra ck. If ais an algebra and kkis a norm on asatisfying kabk kakkbk. Willis, introduction to banach algebras, operators, and harmonic analysis. Banach algebras of vectorvalued functions request pdf. Let k be either r or c a normed algebra over k is an algebra a over k, which also carries a norm k.