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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This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs. Views Read Edit View history.

Let X be a locally compact Hausdorff space. Kribs, and Raymond Laflamme.

### C-star-algebra in nLab

February Learn how and when to remove this template message. This page was last edited on 27 Julyat K H is a two-sided closed ideal of B H. They are required to be closed in the weak operator topologywhich is weaker than the norm topology. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces.

Elements of this cone cstsr called non-negative or sometimes positiveeven though this terminology conflicts with its use for elements of R. Volume 2, Number 5, p.

Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H. This article needs additional citations for verification.

The involution is given by the conjugate transpose. Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators.

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

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

Though K H does not have an identity element, a sequential approximate identity for K H can be developed. 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 ideal. Retrieved from ” https: Let H be a separable infinite-dimensional Hilbert space. By using this site, you agree to the Terms of Use and Privacy Policy.

In the latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism. In the language of K-theorythis vector is the positive cone of the K 0 group of A.

## C*-algebra

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The involution is pointwise conjugation.