Spectrum of a C*-algebra | Wikipedia audio article
Channel: wikipedia tts
Date: 06/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Spectrum_of_a_C*-algebra


00:01:35 1 Primitive spectrum
00:04:03 2 Examples
00:04:12 2.1 Commutative C*-algebras
00:05:41 2.2 The C*-algebra of bounded operators
00:06:00 2.3 Finite-dimensional C*-algebras
00:06:42 3 Other characterizations of the spectrum
00:07:43 3.1 The space Irrsubin/i/sub(iA/i)
00:09:50 4 Mackey–Borel structure
00:12:18 5 Algebraic primitive spectra
00:14:02 6 References



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SUMMARY
=======
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of  irreducible *-representations of A. A *-representation π of A on a Hilbert space H is irreducible if, and only if, there is no closed subspace K different from H and {0} which is invariant under all operators π(x) with x ∈ A. We implicitly assume that irreducible representation means non-null irreducible representation, thus excluding trivial (i.e. identically 0) representations on one-dimensional spaces. As explained below, the spectrum Â is also naturally a topological space; this is similar to the notion of the spectrum of a ring.
One of the most important applications of this concept is to provide a notion of dual object for any locally compact group.  This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I.  The resulting duality theory for locally compact groups is however much weaker than the Tannaka–Krein duality theory for compact topological groups or Pontryagin duality for locally compact abelian groups, both of which are complete invariants. That the dual is not a complete invariant is easily seen as the  dual of any finite-dimensional full matrix algebra Mn(C) consists of a single point.