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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February Learn how and when to remove this template message. Retrieved from ” https: Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators.

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

The involution is pointwise conjugation. This article needs additional citations for verification.

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

The involution is given by the conjugate transpose. Though K H does not have an identity element, a sequential approximate identity for K H can be developed. K H is a two-sided closed ideal alfebra B H. This page was last edited on 27 Julyat This line algera 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.

Elements of this cone are called non-negative or sometimes positiveeven though this terminology conflicts with its use for elements of R. Articles needing additional references from February All articles needing additional references Wikipedia articles needing clarification from August Csgar H be a separable infinite-dimensional Hilbert space.

Let X be a locally compact Hausdorff space. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces.

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## C^*-Algebra

csar Kribs, and Raymond Laflamme. Riesz extension Riesz 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. More generally, one can consider finite direct sums of matrix algebras.

## C*-algebra

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Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H. In the language of K-theorythis vector is the positive cone of the K 0 group of A.

Volume 2, Number 5, p.