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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### C*-algebra – Wikipedia

Retrieved from ” https: Let H be a separable infinite-dimensional Hilbert space. February Learn how and when to remove this template message. Please help improve this article by adding citations to reliable sources. Riesz extension Catar representation Open mapping Parseval’s identity Schauder fixed-point. In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism.

Articles needing additional references from February All articles needing additional references Wikipedia articles needing clarification from August For separable Hilbert spaces, it is the unique cstaf. Views Read Edit View history.

Unsourced material may be challenged and removed. Kribs, and Raymond Laflamme. This article needs additional citations for verification.

By using this site, you agree to the Terms of Use and Privacy Policy. Let X be a locally compact Hausdorff space. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators.

In the language of K-theorythis vector is the positive cone of the K 0 group of A.

## C^*-Algebra

Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H.

This characterization is one of the cstad for the noncommutative topology and noncommutative geometry programs. Though K H does not have an algebrz 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.

In fact it is sufficient to consider only factor representations, i. More generally, one can consider finite direct sums of matrix algebras.

The involution is pointwise conjugation. From Wikipedia, the free encyclopedia.

The involution is given by the conjugate transpose. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces. K H is a two-sided closed ideal of B H. Volume 2, Number 5, p.

This page was last edited on 27 Julyat This line of research began with Werner Heisenberg ‘s matrix mechanics and in catar more mathematically developed form with Pascual Jordan around Elements of this cone are called non-negative or sometimes positiveeven though this terminology conflicts with its use for elements of R.