On embeddings of full amalgamated free product C*–algebras

Armstrong, Scott N.; Dykema, Ken; Exel, Ruy; Li, Hanfeng

doi:10.1090/s0002-9939-04-07370-8

Cited by 31 publications

(32 citation statements)

References 13 publications

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“…We next prove this where the amalgamation is over any common

C^{*}

‐algebra. This generalizes [, Proposition 2.2] due to Armstrong, Dykema, Exel and Li for free products of finitely many

C^{*}

‐algebras amalgamated over a common

C^{*}

‐algebra.…”

Section: Free Products and The Unique Extension Propertysupporting

confidence: 69%

“…Trying to leverage our results to free products, we discuss some of the general theory of free products of operator algebras, and prove a joint unital completely positive extension theorem for free products of operator algebras amalgamated over any common

C^{*}

‐algebra (see Theorem ). Complete injectivity of amalgamated free products of

C^{*}

‐algebras was shown by Armstrong, Dykema, Exel and Li , and we are able to use our dilation techniques to generalize this to free products of operator algebras amalgamated over any common

C^{*}

‐subalgebra (see Proposition ).…”

Section: Introductionmentioning

confidence: 69%

“…When

{false{A_{i} false}}_{i \in I}

are all non‐unital with a common non‐unital

C^{*}

‐algebra

scriptD

, we define their free product

\underset{scriptD}{\overset{*}{true}} {scriptA}_{i}

to be the operator algebra generated by the images of

A_{i}

inside the free product of their unitization

\underset{D^{1}}{\overset{*}{true}} {scriptA}_{i}^{1}

amalgamated over the unitization

D^{1}

. By Meyer's theorem and the proof of [, Lemma 2.3], we similarly get that

\underset{D^{1}}{\overset{*}{true}} {scriptA}_{i}^{1}

coincides with the unitization

{false(\underset{D}{\hat{*}} A_{i} false)}^{1}

. Using this, it follows that the non‐unital free product shares the analogous pushout universal property and complete injectivity.…”

Section: Free Products and The Unique Extension Propertymentioning

confidence: 97%

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Full Cuntz-Krieger dilations via non-commutative boundaries

Dor-On

Salomon

2018

J. London Math. Soc.

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We apply Arveson's non‐commutative boundary theory to dilate every TCK family of a directed graph G to a full CK family for G. We do this by describing all representations of the Toeplitz algebra T(G) that have a unique extension when restricted to the tensor algebra scriptT+false(Gfalse). This yields an alternative proof to a result of Katsoulis and Kribs that the C∗‐envelope of scriptT+false(Gfalse) is the CK algebra O(G). We then generalize our dilation results further, to the context of colored directed graphs, by investigating free products of operator algebras. These generalizations rely on results of independent interest on complete injectivity and a characterization of representations with the unique extension property for free products of operator algebras.

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“…We next prove this where the amalgamation is over any common

C^{*}

‐algebra. This generalizes [, Proposition 2.2] due to Armstrong, Dykema, Exel and Li for free products of finitely many

C^{*}

‐algebras amalgamated over a common

C^{*}

‐algebra.…”

Section: Free Products and The Unique Extension Propertysupporting

confidence: 69%

C^{*}

‐algebra (see Theorem ). Complete injectivity of amalgamated free products of

C^{*}

‐algebras was shown by Armstrong, Dykema, Exel and Li , and we are able to use our dilation techniques to generalize this to free products of operator algebras amalgamated over any common

C^{*}

‐subalgebra (see Proposition ).…”

Section: Introductionmentioning

confidence: 69%

“…When

{false{A_{i} false}}_{i \in I}

are all non‐unital with a common non‐unital

C^{*}

‐algebra

scriptD

, we define their free product

\underset{scriptD}{\overset{*}{true}} {scriptA}_{i}

to be the operator algebra generated by the images of

A_{i}

inside the free product of their unitization

\underset{D^{1}}{\overset{*}{true}} {scriptA}_{i}^{1}

amalgamated over the unitization

D^{1}

. By Meyer's theorem and the proof of [, Lemma 2.3], we similarly get that

\underset{D^{1}}{\overset{*}{true}} {scriptA}_{i}^{1}

coincides with the unitization

{false(\underset{D}{\hat{*}} A_{i} false)}^{1}

. Using this, it follows that the non‐unital free product shares the analogous pushout universal property and complete injectivity.…”

Section: Free Products and The Unique Extension Propertymentioning

confidence: 97%

See 1 more Smart Citation

Full Cuntz-Krieger dilations via non-commutative boundaries

Dor-On

Salomon

2018

J. London Math. Soc.

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show abstract

“…And since all of the edges that give rise to this partial isometry are reversible none of these edges are contained in the partition that contains e.Now P v C * (G)P v , as in the proof of the second case of 2 contains a copy of C(T) generated by Z. We now have that P v C * (G, χ)P v contains a copy of C k * C C * (Z) [5] which contains a copy of C k * C C(T) which is not exact, and hence C * (G, C) is not exact. Now if (G, χ) does not contain a reversible loop then after completing Algorithm 5.2 we are left with a graph which is essentially a directed graph except potentially for those edges in E a (i.e.…”

Section: 3mentioning

confidence: 71%

When universal separated graph $C^*$-algebras are exact

Duncan¹

2020

Methods Funct. Anal. Topol.

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We consider when the universal C *-algebras associated to separated graphs are exact. Specifically, for finite separated graphs we show that the universal C *-algebra is exact if and only if the C *-algebra is isomorphic to a graph C *-algebra which occurs precisely when the universal and reduced C *-algebras of the separated graph are isomorphic. 2010 Mathematics Subject Classification. 46L05, 46L09. Key words and phrases. Edge-colored directed graph, separated graph, C *-algebra, exact.

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“…Pedersen [11] shows that if B Ă C 1 Ă A 1 and B Ă C 2 Ă A 2 are unital inclusions of C*-algebras, then the natural map of C 1ˇC2 into A 1ˇA2 given by the universal property is injective. An alternative proof was given later by Armstrong, Dykema, Exel and Li [1]. This result can be extended to free products of finitely many C*-algebras.…”

Section: Preliminariesmentioning

confidence: 85%