COVARIANT REPRESENTATIONS ASSOCIATED WITH COVARIANT COMPLETELY n -POSITIVE LINEAR MAPS BETWEEN C* - ALGEBRAS

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COVARIANT REPRESENTATIONS ASSOCIATED WITH
COVARIANT COMPLETELY n -POSITIVE LINEAR
MAPS BETWEEN C - ALGEBRAS

• MARIA JOITA, University of Bucharest
• TANIA LUMINITA COSTACHE, University
  Politehnica of Bucharest
• MARIANA ZAMFIR, Technical University of Civil
  Engineering of
• Bucharest

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C-module Hilbert over C -algebra

• Definition 1 Let A be a C
  -algebra. A pre-Hilbert A -module is a complex
• vector space E which is also a right A -module,
  compatible with the complex
• algebra structure, equipped with an A -valued
  inner product ? , ? E ? E ? A
• which is C- and A linear in its second variable
  and satisfies the following
• relations
• ?? , ?? ? ? , ??, for every ? , ? ? E
• ?? , ?? ? 0, for every ? ? E
• ?? , ?? 0 if and only if ? 0.
• We say that E is a Hilbert A -module if
  E is complete with respect to the
• topology determined by the norm given by
  ? (?? , ?? )1/2.
• Notations
• If E and F are two Hilbert A -modules,
  we make the following notations
• BA(E, F) is the Banach space of all bounded
  module homomorphisms from E to F
• LA(E, F) is the set of all maps T ? BA(E, F)
  for which there is a map T ?BA(F, E)
• such that ?T? , ?? ?? , T??, for all ? ? E
  and for all ? ? F.

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C-module Hilbert over C-algebra

• In general, BA(E, F) ? LA(E, F) and so
  the theory of Hilbert C -modules
• and the theory of Hilbert spaces are different.
  
• E is the Banach space of all bounded
  module homomorphisms from E to A
• which becomes a right A -module, where the
  action of A on E is defined by
• (aT)(?) a(T?), for a ? A, T ? E , ? ? E . We
  say that E is self-dual if E E as
• right A-modules.
• If E and F are self-dual, then BA(E, F)
  LA(E, F) Prop. 3.4., Paschke, 8.
• Any bounded module homomorphism T from E
  to F extends uniquely to a
• bounded homomorphism from E to F Prop.
  3.6., Paschke, 8.
• If A is a W -algebra, E becomes a
  self-dual Hilbert A module Th. 3.2.,
• Paschke, 8

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The construction Lin, Paschke, Tsui of the
self-dual Hilbert B-module

• Let E be a Hilbert B
  -module and let B be the enveloping W -algebra
  of B.
• On the algebraic tensor product E ?alg B
  we define the action of right
• B-module by (? ? b)c ? ? bc, for ? ? E and
  b, c ? Band the B -valued
• inner-product by
  . The quotient
  module
• E ?alg B/NE , where NE ? ? E ?alg B ?,
  ? 0, becomes a pre-Hilbert
• B-module. The Hilbert C -module
  obtained by the completion
• of E ?alg B/NE with respect to the norm induced
  by the inner product , is
• called the extension of E by the C -algebra
  B.
• The self-dual Hilbert B -module
  is
  denoted by and we
• consider E as embedded in without making
  distinction.

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The construction Lin, Paschke, Tsui of the
self-dual Hilbert B -module

• Let E and F be two Hilbert modules over C
  -algebra B.
• Then any operator T ? BB(E, F) extends
  uniquely to a bounded module
• homomorphism from to such that
  T Prop. 3.6., Paschke, 8.
• If T ? LB(E, F), then .
• A -representation of a C -algebra A on
  the Hilbert B -module E is a
• ? -morphism F from A to LB(E) (meaning LB(E, E)).
  This representation induces a
• representation of A on denoted by
  , for all a ? A .
• The representation F is non-degenerate if
  F(A)E is dense in E.

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Completely positive and n positive linear map
and C -dynamical system

• Definition 2 Let A and B be two C
  -algebras and E a Hilbert B-module.
• Denote by Mn(A) the -algebra of all n ? n
  matrices over A.
• A completely positive linear map from A to
  LB(E) is a linear map ? A ? LB(E)
• such that the linear map ?(n) Mn(A) ? Mn(LB(E))
  defined by
• is positive for any positive integer n. We say
  that ? is strict if (?(e?))? is strictly
• Cauchy in LB(E), for some approximate unit (e?)?
  of A.
• Definition 3 A completely n -positive
  linear map from A to LB(E) is a n ? n
• matrix of linear maps from A to
  LB(E) such that the map
• ? Mn(A) ? Mn(LB(E)) defined by
• is completely positive
• Definition 4 A triple (G, A, a) is a
  C -dynamical system if G is a locally
• compact group, A is a C -algebra and a is a
  continuous action of G on A.

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Completely positive linear u covariant map and
non-degenerate covariant representation of a
C -dynamical system

• 
  Definition 5 Let (G, A, a) be a C -dynamical
  system, let B be a C -algebra
• and let u be a unitary representation of G on a
  Hilbert B module E.
• A completely positive linear map ? from A to
  LB(E) is u covariant with respect to
• the C -dynamical system (G, A, a) if
• ?(ag(a)) ug ?(a)
  ug, for all a ? A and g ? G.
• Definition 6 A covariant non-degenerate
  representation of a C -dynamical
• system (G, A, a) on a Hilbert B module E is a
  triple (F, v, E), where F is a
• non-degenerate continuous -representation of A
  on E, v is a unitary representation
• of G on E and
• F(ag(a)) vg F(a)
  vg, for all a ? A and g ? G.

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Completely positive linear u covariant map and
non-degenerate covariant representation of a
C -dynamical system

• Proposition Let (G, A, a) be a C
  -dynamical system, let u be
• a unitary representation of G on a Hilbert module
  E over a C -algebra
• B, let ? be a u covariant non-degenerate
  completely positive linear
• map from A to LB(E).
• Then there is a covariant representation (F?, v?,
  E?) of (G, A, a) and V? in LB(E, E?) such that
• ?(a) V?F?(a)V? , for all a ? A
• F?(a)V?? a ? A, ? ? E spans a dense submodule
  of E?
• v?gV? V?ug , for all g ? G.

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Completely positive linear u covariant map and
non-degenerate covariant representation of a
C -dynamical system

• If F is a Hilbert B module, (F, v, F) is a
  covariant representation of (G, A, a) and W is in
  LB(E, F) such that
• ?(a) WF(a)W, for all a ? A
• F(a)W? a ? A, ? ? E spans a dense submodule
  of F
• vgW Wug , for all g ? G,
• then there is a unitary operator U in LB(E?, F)
  such that
• UF?(a) F(a)U, for all a ? A
• vgU Uv?g , for all g ? G
• W UV?. Th.4.3, 2, Joita, case n 1

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The main results

• Let A be a C -algebra, let E be a C
  -module Hilbert over a C -algebra B
• and let ? A ? LB(E) be a strict completely
  positive linear u -covariant map.
• Notation
• C(?) the C -subalgebra of
  generated by
• 0, ? the set of all strict completely
  positive linear u covariant maps ?
• from A to LB(E) such that ? ? ? (that is, ? ?
  is a strict completely positive
• linear u covariant map from A to LB(E)).
• 0, I? the set of all elements T in C(?)
  such that 0 ? T ? .

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The main results

• Lemma Let T ? C(?) positive. Then the map
  ?T
• defined by
• is a strict completely positive linear u
  covariant map
• from A to LB(E).
• Theorem 1 The map T ? ?T from 0, I? to
  0, ? is
• an affine order isomorphism.

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The main results

• Theorem 2 Let (G, A, a) be a C -dynamical
  system, let u be a unitary
• representation of G on a Hilbert C -module E
  over a C -algebra B , let
• be a completely n-positive linear u -covariant
  map relative to the dynamical system
• (G, A, a) from A to LB(E).
• Then there is (F?, v?, E?) a covariant
  representation of (G, A, a) on a Hilbert
• B -module E?, an isometry V? E ? E? and
  
• such that
• , for all a ?
  A and for all i, j 1, 2, , n,
• is a
  positive element in Mn(LB(E)) and
  
• F?(a)V?? a ? A, ? ?E is dense in E?
• ,
  for all a ? A and i, j 1, 2, , n
• v?gV? V?ug, for all g ? G.

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The main results

• If (?, w, F) is another covariant representation
  of (G, A, a) on a Hilbert
• B-module F, W E ? F is an isometry and
  such
  that
• , for all a
  ? A and for all i, j 1, 2, , n,
• is a
  positive element in Mn(LB(E)) and
  
• ?(a)W? a ? A, ? ?E is dense in F
• , for all
  a ? A and i, j 1, 2, , n
• wgW Wug, for g ? G
• then there is a unitary operator U E? ? F such
  that
• ?(a) UF?(a)U, for all a ? A
• W UV?
• Sij UT?ijU, for all i, j 1, 2, , n
• wg Uv?gU, for all g ? G.

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Crossed product associated to a C -dynamical
system

• Definition 7 Let (F,
  v, E) be a covariant representation (possibly
  degenerate)
• of the dynamical system (G, A, a) on a Hilbert B
  module E. Then
• is a -representation of Cc(G, A) on E called
  the integrated form of (F, v, E).
• Definition 8 Let (G, A,
  a) be a dynamical system. For f ? Cc(G, A) we
  define
• the norm on Cc(G, A)
• f sup F ? v(f) (F, v, E)
  is a covariant representation of (G, A, a)
• called the universal norm. The completion of
  Cc(G, A) with respect to is a
• C -algebra called the crossed product of A by G
  and is denoted by A ?a G.

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The main results

• Let be a completely n positive
  linear u covariant non-degenerate map
• with respect to a C -dynamical system (G, A, a).
• By Prop. 4.5., 2, there is a uniquely
  completely n positive linear map
• from A ?a G to LB(E) such that
• for all f ? Cc(G, A) and for all i, j 1, 2, ,
  n.
• By Th. 2.2, 5 there is a representation Ff of A
  ?a G on Ef, an isometry Vf E ? Ef
• and
  such that
• for
  all f ? A ?a G and for all i, j 1, 2, , n,
• is a
  positive element in Mn(LB(E)) and
• Ff(f)Vf? f ? A ?a G, ? ?E is dense in Ef
• for
  all f ? A ?a G and for all i, j 1, 2, , n.

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The main results

• 
  By Theorem 2 there is (F?, v?, E?) a
  covariant representation of (G, A, a) on a
• Hilbert B -module E?.
• By Prop. 2.39, 10, (F? ? v?, E?) is a
  representation of A ?a G on E? such that
• for all f ? Cc(G, A), g ? G.
• Proposition Let be
  a completely n positive linear u covariant
• non-degenerate map and let be
  a uniquely completely n positive linear
• map from A ?a G to LB(E) given by Prop. 4.5.,
  2.
• Then
  and
  are unitarily
• equivalent.

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References

• Arveson, W., Subalgebras of C -algebras, Acta
  Math., 1969
• Joita, M., Completely multi-positive linear maps
  between locally C -algebras and representations
  on Hilbert modules, Studia Math., 2006
• Joita, M., A Radon - Nikodym theorem for
  completely multi positive linear maps and its
  aplications, Proceedings of the 5th
  International Conference on Topological Algebras
  and Applications, Athens, Greece, 2005 (to
  appear)
• Joita, M., A Radon - Nikodym theorem for
  completely n-positive linear maps on pro -C
  -algebras and its applications, Publicationes
  Mathematicae Debrecen
• Joita, M., Costache, T. L., Zamfir, M.,
  Representations associated with completelly
  n-positive linear maps between C -algebras,
  Stud. Cercet. Stiint., Ser. Mat., 2006

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References

• Lance, E. C., Hilbert C -module. A toolkit for
  operator algebraists, London Mathematical Society
  Lecture Note Series 210, 1995
• Lin, H., Bounded module maps and pure completely
  positive maps, J. Operator Theory, 1991
• Paschke, W. L., Inner product modules over B
  -algebras, Trans. Amer. Math. Soc., 1973
• Tsui, S. K., Completely positive module maps and
  completely positive extreme maps, Proc. Amer.
  Math. Soc., 1996
• Williams, D., Crossed products of C -algebras,
  Mathematical Surveys and monographs, 2006.