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 line of research began with Werner Heisenberg ‘s matrix mechanics and in a more mathematically developed form with Pascual Jordan around Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators.
Segal in to describe norm-closed subalgebras of B Hnamely, the space of bounded operators on some Hilbert space H. In the latter case, we can use the fact that the structure of sctar is completely determined by the Gelfand isomorphism.
C-star-algebra in nLab
Elements of this cone are called non-negative or sometimes positiveeven though this cdtar conflicts with its use for elements of R. The involution is given by the conjugate transpose. More generally, one can consider finite direct sums of matrix algebras. In the language of K-theorythis vector is the positive cone of the K csrar group of A. Let H alhebra a separable infinite-dimensional Hilbert space.
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 of B H. For separable Hilbert spaces, it is the unique ideal. Let X be a locally compact Hausdorff space.
The involution is pointwise conjugation. Such functions exist by the Tietze extension theorem which applies to locally compact Hausdorff spaces.
C^*-Algebra — from Wolfram MathWorld
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.
In fact it is sufficient to consider only factor representations, i. From Wikipedia, the free encyclopedia. This article needs additional citations for verification.
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Kribs, and Raymond Laflamme.
Volume 2, Number 5, p. Riesz extension Riesz representation Open mapping Parseval’s identity Schauder fixed-point.