By Theodore W. Palmer

This is often the 1st quantity of a quantity set that gives a latest account of uncomplicated Banach algebra idea together with all recognized effects on common Banach *-algebras. This account emphasizes the function of *-algebraic constitution and explores the algebraic effects that underlie the speculation of Banach algebras and *-algebras. the 1st quantity, which incorporates formerly unpublished effects, is an autonomous, self-contained reference on Banach algebra thought. every one subject is handled within the greatest attention-grabbing generality in the framework of a few classification of complicated algebras instead of topological algebras. Proofs are awarded in entire element at a degree obtainable to graduate scholars. The ebook includes a wealth of ancient reviews, history fabric, examples, really in noncommutative harmonic research, and an intensive bibliography. quantity II is drawing close.

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**Additional resources for Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach Algebras (Encyclopedia of Mathematics and its Applications) **

**Example text**

Since confusion between them sometimes occurs, it is worthwhile to point out the distinction clearly here. We will only describe the case of normed algebras, since the case of normed linear spaces is exactly similar. In the more common and important category, the morphisms are con tinuous algebra homornorphisms. That is,

Hence, under suitable restrictions on the annihilator ideals, the construc tion of V ( A ) provides another way to embed an algebra in a unital algebra and a Banach algebra in a unital Banach algebra. ) Note that A equals T>(A) if and only if A is uni tal. For technical reasons, it is sometimes desirable to replace D { A ) by the poss:bly smaller unital subalgebra V ( A ) z + A . For a discussion of this, see Dauns [1969] and Dauns and Hofmann [1968], [1969]. 5 Definition Let A be a subalgebra of an algebra B.

Operations in this algebra bring out some o f the more subtle structure o f the original algebra A . It is used in most definitions o f K-theory. O f course we could complete M oc(A ) or any other normed algebra. It is worth noting that if A is a C*-algebra, then the completion of M X (A ) has an interesting interpretation. First, M o o (A ) itself is isometrically iso morphic to a dense subalgebra o f the tensor product o f A with the Banach algebra o f all compact operators on separable Hilbert space.