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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In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism. By using this site, you agree to the Terms of Use and Privacy Policy. February Learn how and when algenra remove this template message. The involution is given by the conjugate transpose. Views Read Edit View history.

Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point. Articles needing additional references from February All articles needing additional references Wikipedia articles needing clarification from August More generally, one can consider finite direct sums of matrix algebras.

Kribs, and Raymond Laflamme.

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

### C-star-algebra in nLab

This page was last edited on 27 Julyat Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H. Volume 2, Number 5, p. Elements of this cone algerba called non-negative or sometimes positiveeven though this terminology conflicts with its use for elements of R.

K H is a two-sided closed ideal of B H. This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs.

In the language of K-theorythis vector is the positive cone of the K 0 group of A. Unsourced material may be challenged and removed. Though K H does not have an identity element, a sequential algeba identity for K H can be developed. Let X be a locally compact Hausdorff space.

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

Please help improve this article by adding citations to reliable sources. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces.

This article needs additional citations for verification. Let H be a aalgebra infinite-dimensional Hilbert space. They are required to be closed in the weak operator topologywhich is weaker than the norm topology.

For separable Hilbert spaces, it is the unique ideal.

## C^*-Algebra

This line of research began with Werner Heisenberg ‘s matrix mechanics and in a more mathematically developed form with Pascual Jordan around In fact it is sufficient to consider only factor representations, i. From Wikipedia, the free encyclopedia.

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