Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators
Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and suff...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Ay, Serdar [verfasserIn] |
|---|
| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer International Publishing 2017 |
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| Übergeordnetes Werk: |
Enthalten in: Integral equations and operator theory - Springer International Publishing, 1978, 87(2017), 2 vom: Feb., Seite 263-307 |
|---|---|
| Übergeordnetes Werk: |
volume:87 ; year:2017 ; number:2 ; month:02 ; pages:263-307 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00020-017-2346-1 |
|---|
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OLC2069402460 |
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| 520 | |a Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. | ||
| 650 | 4 | |a Ordered | |
| 650 | 4 | |a -space | |
| 650 | 4 | |a Admissible space | |
| 650 | 4 | |a VH-space | |
| 650 | 4 | |a Positive semidefinite kernel | |
| 650 | 4 | |a -semigroup | |
| 650 | 4 | |a Invariant kernel | |
| 650 | 4 | |a Linearisation | |
| 650 | 4 | |a Reproducing kernel | |
| 650 | 4 | |a -representation | |
| 650 | 4 | |a Locally | |
| 650 | 4 | |a -algebra | |
| 650 | 4 | |a Hilbert locally | |
| 650 | 4 | |a -module | |
| 650 | 4 | |a Completely positive map | |
| 700 | 1 | |a Gheondea, Aurelian |4 aut | |
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10.1007/s00020-017-2346-1 doi (DE-627)OLC2069402460 (DE-He213)s00020-017-2346-1-p DE-627 ger DE-627 rakwb eng 510 004 VZ 17,1 ssgn Ay, Serdar verfasserin aut Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2017 Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. Ordered -space Admissible space VH-space Positive semidefinite kernel -semigroup Invariant kernel Linearisation Reproducing kernel -representation Locally -algebra Hilbert locally -module Completely positive map Gheondea, Aurelian aut Enthalten in Integral equations and operator theory Springer International Publishing, 1978 87(2017), 2 vom: Feb., Seite 263-307 (DE-627)129859184 (DE-600)282475-9 (DE-576)015166651 0378-620X nnns volume:87 year:2017 number:2 month:02 pages:263-307 https://doi.org/10.1007/s00020-017-2346-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4318 AR 87 2017 2 02 263-307 |
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10.1007/s00020-017-2346-1 doi (DE-627)OLC2069402460 (DE-He213)s00020-017-2346-1-p DE-627 ger DE-627 rakwb eng 510 004 VZ 17,1 ssgn Ay, Serdar verfasserin aut Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2017 Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. Ordered -space Admissible space VH-space Positive semidefinite kernel -semigroup Invariant kernel Linearisation Reproducing kernel -representation Locally -algebra Hilbert locally -module Completely positive map Gheondea, Aurelian aut Enthalten in Integral equations and operator theory Springer International Publishing, 1978 87(2017), 2 vom: Feb., Seite 263-307 (DE-627)129859184 (DE-600)282475-9 (DE-576)015166651 0378-620X nnns volume:87 year:2017 number:2 month:02 pages:263-307 https://doi.org/10.1007/s00020-017-2346-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4318 AR 87 2017 2 02 263-307 |
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10.1007/s00020-017-2346-1 doi (DE-627)OLC2069402460 (DE-He213)s00020-017-2346-1-p DE-627 ger DE-627 rakwb eng 510 004 VZ 17,1 ssgn Ay, Serdar verfasserin aut Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2017 Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. Ordered -space Admissible space VH-space Positive semidefinite kernel -semigroup Invariant kernel Linearisation Reproducing kernel -representation Locally -algebra Hilbert locally -module Completely positive map Gheondea, Aurelian aut Enthalten in Integral equations and operator theory Springer International Publishing, 1978 87(2017), 2 vom: Feb., Seite 263-307 (DE-627)129859184 (DE-600)282475-9 (DE-576)015166651 0378-620X nnns volume:87 year:2017 number:2 month:02 pages:263-307 https://doi.org/10.1007/s00020-017-2346-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4318 AR 87 2017 2 02 263-307 |
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10.1007/s00020-017-2346-1 doi (DE-627)OLC2069402460 (DE-He213)s00020-017-2346-1-p DE-627 ger DE-627 rakwb eng 510 004 VZ 17,1 ssgn Ay, Serdar verfasserin aut Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2017 Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. Ordered -space Admissible space VH-space Positive semidefinite kernel -semigroup Invariant kernel Linearisation Reproducing kernel -representation Locally -algebra Hilbert locally -module Completely positive map Gheondea, Aurelian aut Enthalten in Integral equations and operator theory Springer International Publishing, 1978 87(2017), 2 vom: Feb., Seite 263-307 (DE-627)129859184 (DE-600)282475-9 (DE-576)015166651 0378-620X nnns volume:87 year:2017 number:2 month:02 pages:263-307 https://doi.org/10.1007/s00020-017-2346-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4318 AR 87 2017 2 02 263-307 |
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10.1007/s00020-017-2346-1 doi (DE-627)OLC2069402460 (DE-He213)s00020-017-2346-1-p DE-627 ger DE-627 rakwb eng 510 004 VZ 17,1 ssgn Ay, Serdar verfasserin aut Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2017 Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. Ordered -space Admissible space VH-space Positive semidefinite kernel -semigroup Invariant kernel Linearisation Reproducing kernel -representation Locally -algebra Hilbert locally -module Completely positive map Gheondea, Aurelian aut Enthalten in Integral equations and operator theory Springer International Publishing, 1978 87(2017), 2 vom: Feb., Seite 263-307 (DE-627)129859184 (DE-600)282475-9 (DE-576)015166651 0378-620X nnns volume:87 year:2017 number:2 month:02 pages:263-307 https://doi.org/10.1007/s00020-017-2346-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_267 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4318 AR 87 2017 2 02 263-307 |
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Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators |
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Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators |
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Ay, Serdar |
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Integral equations and operator theory |
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Integral equations and operator theory |
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2017 |
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Ay, Serdar Gheondea, Aurelian |
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10.1007/s00020-017-2346-1 |
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representations of $$\varvec{*}$$-semigroups associated to invariant kernels with values continuously adjointable operators |
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Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators |
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Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. © Springer International Publishing 2017 |
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Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. © Springer International Publishing 2017 |
| abstract_unstemmed |
Abstract We consider positive semidefinite kernels valued in the $$*$$-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of $$*$$-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist $$*$$-representations of the underlying $$*$$-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally $$C^*$$-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring–Kasparov type dilation theorem for completely positive maps on locally $$C^*$$-algebras and with values adjointable operators on Hilbert modules over locally $$C^*$$-algebras. © Springer International Publishing 2017 |
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Representations of $$\varvec{*}$$-Semigroups Associated to Invariant Kernels with Values Continuously Adjointable Operators |
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