Banach *-algebra representation (original) (raw)

Definition:

The set of all representations of 𝒜 on a Hilbert space H is denoted r⁢e⁢p⁢(𝒜,H).

Special kinds of representations:

• •
  A representation π∈r⁢e⁢p⁢(𝒜,H) is said to be nondegenerate if one of the following equivalent conditions hold:

  1. (a)
     π⁢(x)⁢ξ=0 ∀x∈𝒜⟹ξ=0, where ξ∈H.
  2. (b)
     H is the closed linear span of the set of vectors π⁢(𝒜)⁢H:={π⁢(x)⁢ξ:x∈𝒜,ξ∈H}

• •
  A representation π∈r⁢e⁢p⁢(𝒜,H) is said to be topologically irreducible (or just ) if the only closed π⁢(𝒜)-invariant of H are the trivial ones, {0} and H.

• •
  A representation π∈r⁢e⁢p⁢(𝒜,H) is said to be algebrically irreducible if the only π⁢(𝒜)-invariant of H (not necessarily closed) are the trivial ones, {0} and H.

• •
  Given two representations π1∈r⁢e⁢p⁢(𝒜,H1) and π2∈r⁢e⁢p⁢(𝒜,H2), the of π1 and π2 is the representation π1⊕π2∈r⁢e⁢p⁢(𝒜,H1⊕H2) given by π1⊕π2⁢(x):=π1⁢(x)⊕π2⁢(x),x∈𝒜.
  More generally, given a family {πi}i∈I of representations, with πi∈r⁢e⁢p⁢(𝒜,Hi), their is the representation ⊕i∈Iπi∈r⁢e⁢p⁢(𝒜,⊕i∈IHi), in the direct sum of Hilbert spaces ⊕i∈IHi, such that (⊕i∈Iπi)⁢(x):=⊕i∈Iπi⁢(x) is the direct sum of the family of bounded operators (http://planetmath.org/DirectSumOfBoundedOperatorsOnHilbertSpaces) {πi⁢(x)}i∈I.

• •
  Two representations π1∈r⁢e⁢p⁢(𝒜,H1) and π2∈r⁢e⁢p⁢(𝒜,H2) of a Banach *-algebra 𝒜 are said to be unitarily equivalent if there is a unitary U:H1⟶H2 such that

• •
  A representation π∈r⁢e⁢p⁢(𝒜,H) is said to be if there exists a vector ξ∈H such that the set
  is dense (http://planetmath.org/Dense) in H. Such a vector is called a cyclic vector for the representation π.

Linked file: http://aux.planetmath.org/files/objects/9843/BanachAlgebraRepresentation.pdf