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.
Elements of this cone are called non-negative or sometimes positiveeven though this terminology conflicts with its use for elements of R. This characterization is one of the motivations for the noncommutative apgebra and noncommutative geometry programs.
C^*-Algebra — from Wolfram MathWorld
Retrieved from ” https: In the language of K-theorythis vector is the positive cone of the K 0 group of A. From Wikipedia, the free encyclopedia. Let H be a separable infinite-dimensional Algebrz space. The involution is given by the conjugate transpose. Views Read Edit View history.
This page was last edited on 27 Julyat February Learn how and when to remove this template message. Kribs, and Raymond Laflamme.
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 Please help improve this article by adding citations to reliable sources.
In the latter case, we can use the fact that the allgebra of these is completely determined by the Gelfand isomorphism.
Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point. Volume 2, Number 5, p. K H is a two-sided closed ideal of B H. For algebrq Hilbert spaces, it is the unique ideal. Articles needing additional references from February All articles needing additional references Wikipedia articles needing clarification from August The involution is pointwise conjugation.
Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H. Unsourced material may be challenged and removed.
They are required to cstad closed in the weak operator topologywhich is weaker than the norm topology.
This article needs additional citations for verification. 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 fact it is sufficient to consider only factor representations, i. Though K H does not have an identity element, a sequential approximate identity for K H can be developed.