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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Let H be a separable infinite-dimensional Hilbert space.

## C^*-Algebra

Please help improve this article by adding citations to reliable sources. Kribs, and Raymond Laflamme.

Let X be a locally compact Hausdorff space. February Learn how and when to remove this template message. From Wikipedia, the free encyclopedia.

For separable Hilbert spaces, it is the unique ideal. Though K H does not have an identity element, a sequential approximate identity for K H can be developed. They are required to be closed in the weak operator topologywhich is weaker than the norm topology.

More generally, one can consider finite direct sums of matrix algebras.

The involution is pointwise conjugation. This page was last edited on 27 Julyat Views Read Edit View history.

The involution is given by the conjugate transpose. By using this site, you agree to the Terms of Use and Privacy Policy.

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

In the latter case, we can use the fact cstwr the structure of these is completely determined by the Gelfand isomorphism. Retrieved from ” https: Volume 2, Number 5, p. Articles needing additional references from February All articles needing additional references Wikipedia articles needing clarification from August Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point.

This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs.

Elements of this cone are called non-negative or sometimes positiveeven though this terminology conflicts with its use for csttar of R. In the language of K-theorythis vector is the positive cone of the K 0 group of A. This article needs additional citations aglebra verification.

This line of research began with Werner Heisenberg ‘s matrix mechanics and in a more mathematically developed form with Pascual Jordan around Such functions exist by the Tietze extension theorem alegbra applies to locally compact Hausdorff spaces.

Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H. Unsourced material may be challenged and removed. K H is a two-sided closed ideal of B H. In fact it is sufficient to consider only factor representations, i.

Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators.