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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Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces. For separable Hilbert spaces, it is the unique ideal.
From Wikipedia, the free encyclopedia. More generally, one can consider finite direct sums of matrix algebras. K H is a two-sided closed ideal of B H. Kribs, and Raymond Laflamme.
Please help improve this article by adding citations to reliable sources. Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point.
They are required to be closed in the weak operator topologywhich is weaker than the norm topology. This article needs additional citations for verification. Let H be a separable infinite-dimensional Hilbert space.
Unsourced material may be challenged and removed. The involution is given by the conjugate transpose.
C*-algebra
This characterization is one of the motivations for the noncommutative topology and noncommutative geometry programs. Articles needing additional references from February All articles needing additional references Wikipedia articles needing clarification from August Volume 2, Number 5, p.
This line of research began with Werner Heisenberg ‘s matrix mechanics and in ccstar 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. Let X be a locally compact Hausdorff space. The involution is pointwise conjugation.
C^*-Algebra — from Wolfram MathWorld
In fact it is sufficient to consider only factor representations, i. February Learn how and when to remove this template message. Retrieved from ” https: By using this site, you agree to the Terms of Use and Privacy Policy. 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. In the language of K-theorythis vector is the positive cone of the K 0 group of A. Views Read Edit View history.
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 latter case, we can use the fact that the structure of these is completely determined by the Gelfand isomorphism. Though K H does not have an identity element, a sequential approximate identity for K H can be developed.