topologically irreducible representations are algebrically irreducible for $C^{*}$-algebras

Theorem - Every topologically irreducible representation (http://planetmath.org/BanachAlgebraRepresentation) of a $C^{*}$-algebra (http://planetmath.org/CAlgebra) is algebraically irreducible.

$Remark-$ It follows easily from the definition that algebraically irreducible of any Banach *-algebra are always topologically irreducible. The above theorem says that for $C^{*}$-algebras the converse is also true.

Title	topologically irreducible representations are algebrically irreducible for $C^{*}$-algebras
Canonical name	TopologicallyIrreducibleRepresentationsAreAlgebricallyIrreducibleForCalgebras
Date of creation	2013-03-22 17:27:47
Last modified on	2013-03-22 17:27:47
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	5
Author	asteroid (17536)
Entry type	Theorem
Classification	msc 46L05