A commutative Banach algebra is a Banach algebra A with the property that ab = ba for all a, b ∈ A Examples and are of commutative. Banach. Of course, if A is a normed algebra, then the norm induces a metric on A which Similarly weak star topology on A∗ is generated by the sets. *-SJbalgebra A of B (H) which is closed in tIE nonn tOIDlogy is a C*-algebra. E.g.: . A C*-algebra A is unital if A has a unit 1 A i otherwise, A is nonunital. I.

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The involution is given by the conjugate transpose.

### C*-algebra – Wikipedia

Let X be a locally compact Hausdorff space. Let H be a separable infinite-dimensional Hilbert space. They csgar required to be closed in the weak operator topologywhich is weaker than the norm topology. Though K H does not have an identity element, a sequential approximate identity for K H can be developed. Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point.

This line of research began with Werner Heisenberg ‘s matrix mechanics and in cstad more mathematically developed form with Alhebra Jordan around Articles needing additional references from February All articles needing additional references Wikipedia articles needing clarification from August In fact it is sufficient to consider only factor representations, i.

K H is a two-sided closed ideal of B H.

Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H. Such algehra exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces. More generally, one can consider finite direct sums of matrix algebras.

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For separable Hilbert spaces, it is the unique ideal. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators.

This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs. In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism. Retrieved from apgebra https: Volume 2, Number 5, p.

From Wikipedia, the free encyclopedia. In the language of K-theorythis vector is the positive cone of the K 0 group of A.

### C^*-Algebra — from Wolfram MathWorld

This page was last edited on 27 Julyat By using this site, you agree to the Terms of Use and Privacy Policy. Elements of this cone are called non-negative or sometimes positiveeven xstar this terminology conflicts with its use for elements of R.

February Learn how and when to remove this template message. Kribs, and Raymond Laflamme.

## C*-algebra

This article needs additional citations for verification. Unsourced material may be challenged and removed. The involution is pointwise conjugation.