Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 ·...
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Operator algebras associated with monomial ideals
Evgenios Kakariadis and Orr Shalit
January 2015, Be’er-Sheva
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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C*-correspondences
Operator algebras of C*-correspondences I
Let E - a C*-correspondence over a C*-algebra A, ' = 'E : A ! L(E ).A representation (⇡, t) of E on H is pair• ⇡ : A ! B(H)
• t : E ! B(H) s.t. t(⇠)⇤t(⌘) = ⇡(h⇠, ⌘i), t('(a)⇠) = ⇡(a)t(⇠).
The Toeplitz-Pimsner algebra TE is the C*-algebra generated by auniversal representation, i.e., the universal C*-algebra containing E and A.
The Tensor algebra T +E is the non-selfadjoint subalgebra of TE generated
by A and E .(There is a concrete version due to Pimsner and Muhly-Solel, we’ll seebelow).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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C*-correspondences
Operator algebras of C*-correspondences I
Let E - a C*-correspondence over a C*-algebra A, ' = 'E : A ! L(E ).A representation (⇡, t) of E on H is pair• ⇡ : A ! B(H)
• t : E ! B(H) s.t. t(⇠)⇤t(⌘) = ⇡(h⇠, ⌘i), t('(a)⇠) = ⇡(a)t(⇠).
The Toeplitz-Pimsner algebra TE is the C*-algebra generated by auniversal representation, i.e., the universal C*-algebra containing E and A.
The Tensor algebra T +E is the non-selfadjoint subalgebra of TE generated
by A and E .(There is a concrete version due to Pimsner and Muhly-Solel, we’ll seebelow).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 4: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/4.jpg)
C*-correspondences
Operator algebras of C*-correspondences I
Let E - a C*-correspondence over a C*-algebra A, ' = 'E : A ! L(E ).A representation (⇡, t) of E on H is pair• ⇡ : A ! B(H)
• t : E ! B(H) s.t. t(⇠)⇤t(⌘) = ⇡(h⇠, ⌘i), t('(a)⇠) = ⇡(a)t(⇠).
The Toeplitz-Pimsner algebra TE is the C*-algebra generated by auniversal representation, i.e., the universal C*-algebra containing E and A.
The Tensor algebra T +E is the non-selfadjoint subalgebra of TE generated
by A and E .(There is a concrete version due to Pimsner and Muhly-Solel, we’ll seebelow).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 5: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/5.jpg)
C*-correspondences
Operator algebras of C*-correspondences II
With every (⇡, t) there comes another representation t : K(E )! B(H)given by
t(✓⇠,⌘) = t(⇠)t(⌘)⇤.
Let J ✓ '�1(K(E )). A representation (⇡, t) is said to be J-covariant if
⇡(a) = t('E (a)) for all a 2 J
The J-relative Cuntz-Pimnser algebra O(J,E ) is the universalC*-algebra generated by J-covariant representations.
The Cuntz-Pimsner algebra is OE := O(JE ,E ), where
JE = ker'?E \ '�1
E (K(E )).
(Here too there is a concrete version, we’ll see below)
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 6: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/6.jpg)
C*-correspondences
Operator algebras of C*-correspondences II
With every (⇡, t) there comes another representation t : K(E )! B(H)given by
t(✓⇠,⌘) = t(⇠)t(⌘)⇤.
Let J ✓ '�1(K(E )). A representation (⇡, t) is said to be J-covariant if
⇡(a) = t('E (a)) for all a 2 J
The J-relative Cuntz-Pimnser algebra O(J,E ) is the universalC*-algebra generated by J-covariant representations.
The Cuntz-Pimsner algebra is OE := O(JE ,E ), where
JE = ker'?E \ '�1
E (K(E )).
(Here too there is a concrete version, we’ll see below)
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 7: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/7.jpg)
C*-correspondences
Operator algebras of C*-correspondences II
With every (⇡, t) there comes another representation t : K(E )! B(H)given by
t(✓⇠,⌘) = t(⇠)t(⌘)⇤.
Let J ✓ '�1(K(E )). A representation (⇡, t) is said to be J-covariant if
⇡(a) = t('E (a)) for all a 2 J
The J-relative Cuntz-Pimnser algebra O(J,E ) is the universalC*-algebra generated by J-covariant representations.
The Cuntz-Pimsner algebra is OE := O(JE ,E ), where
JE = ker'?E \ '�1
E (K(E )).
(Here too there is a concrete version, we’ll see below)
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 8: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/8.jpg)
C*-correspondences
Operator algebras of C*-correspondences II
With every (⇡, t) there comes another representation t : K(E )! B(H)given by
t(✓⇠,⌘) = t(⇠)t(⌘)⇤.
Let J ✓ '�1(K(E )). A representation (⇡, t) is said to be J-covariant if
⇡(a) = t('E (a)) for all a 2 J
The J-relative Cuntz-Pimnser algebra O(J,E ) is the universalC*-algebra generated by J-covariant representations.
The Cuntz-Pimsner algebra is OE := O(JE ,E ), where
JE = ker'?E \ '�1
E (K(E )).
(Here too there is a concrete version, we’ll see below)
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 9: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/9.jpg)
C*-correspondences
The operator algebras coming from C*-correspondences allow a unifiedtreatment of a very broad spectrum of C*-algebras (graphs, dynamicalsystems, Cuntz-Krieger,...) and have a rich theory (GIUT, conditions fornuclearity, C*-envelopes,...)
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 10: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/10.jpg)
Subproduct systems
Subproduct systemsA subproduct system is a family X = {X (n)}n2N of C*-correspondences(over a C*-algebra A) such that X (0) = A and
X (m + n) ✓ X (m)⌦ X (n)
We construct the Fock space
F(E ) = A� X (1)� X (2)� . . .
and the “representation” ('1,T )
'1(a) = '(a)� ('(a)⌦ I )� ('(a)⌦ I ⌦ I )� . . .
and , for ⌘ 2 X (n)
T (⇠)⌘ = PE⌦n+1!X (n+1)⇥⇠ ⌦ ⌘⇤.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 11: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/11.jpg)
Subproduct systems
Subproduct systemsA subproduct system is a family X = {X (n)}n2N of C*-correspondences(over a C*-algebra A) such that X (0) = A and
X (m + n) ✓ X (m)⌦ X (n)
We construct the Fock space
F(E ) = A� X (1)� X (2)� . . .
and the “representation” ('1,T )
'1(a) = '(a)� ('(a)⌦ I )� ('(a)⌦ I ⌦ I )� . . .
and , for ⌘ 2 X (n)
T (⇠)⌘ = PE⌦n+1!X (n+1)⇥⇠ ⌦ ⌘⇤.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 12: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/12.jpg)
Subproduct systems
Operator algebras from subproduct systems I
The Toeplitz-Pimsner algebra of X is
T (X ) = C ⇤�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)�
The tensor algebra of X is
T +(X ) = Alg�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)
�
The Cuntz-Pimsner algebra of X is
O(X ) = T (X )/I,
where I is a certain ideal (I = K(F(X )) in nice cases).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 13: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/13.jpg)
Subproduct systems
Operator algebras from subproduct systems I
The Toeplitz-Pimsner algebra of X is
T (X ) = C ⇤�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)�
The tensor algebra of X is
T +(X ) = Alg�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)
�
The Cuntz-Pimsner algebra of X is
O(X ) = T (X )/I,
where I is a certain ideal (I = K(F(X )) in nice cases).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 14: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/14.jpg)
Subproduct systems
Operator algebras from subproduct systems I
The Toeplitz-Pimsner algebra of X is
T (X ) = C ⇤�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)�
The tensor algebra of X is
T +(X ) = Alg�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)
�
The Cuntz-Pimsner algebra of X is
O(X ) = T (X )/I,
where I is a certain ideal (I = K(F(X )) in nice cases).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 15: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/15.jpg)
Subproduct systems
Operator algebras from subproduct systems I
The Toeplitz-Pimsner algebra of X is
T (X ) = C ⇤�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)�
The tensor algebra of X is
T +(X ) = Alg�'1(a),T (⇠) : a 2 A, ⇠ 2 X (1)
�
The Cuntz-Pimsner algebra of X is
O(X ) = T (X )/I,
where I is a certain ideal (I = K(F(X )) in nice cases).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 16: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/16.jpg)
Subproduct systems
Operator algebras from subproduct systems II
If E is a C*-correspondence, then X = {E⌦n}n2N is a subproduct system.Then T (X ) ⇠= TE , T +(X ) ⇠= T +
E and O(X ) ⇠= OE , so all of the previouslyconsidered algebras fit this framewrok.
This class of operator algebras seems to go far, far beyond the operatoralgebras that fall under the umbrella of C*-correspondences.
On the other hand, for the C*-algebras there is no universal construction,likewise there is no Gauge Invariant Uniqueness Theorem. Some toolsdisappear from our tool-box, and techinal difficulties arise.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 17: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/17.jpg)
Subproduct systems
Operator algebras from subproduct systems II
If E is a C*-correspondence, then X = {E⌦n}n2N is a subproduct system.Then T (X ) ⇠= TE , T +(X ) ⇠= T +
E and O(X ) ⇠= OE , so all of the previouslyconsidered algebras fit this framewrok.
This class of operator algebras seems to go far, far beyond the operatoralgebras that fall under the umbrella of C*-correspondences.
On the other hand, for the C*-algebras there is no universal construction,likewise there is no Gauge Invariant Uniqueness Theorem. Some toolsdisappear from our tool-box, and techinal difficulties arise.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 18: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/18.jpg)
Subproduct systems
Operator algebras from subproduct systems II
If E is a C*-correspondence, then X = {E⌦n}n2N is a subproduct system.Then T (X ) ⇠= TE , T +(X ) ⇠= T +
E and O(X ) ⇠= OE , so all of the previouslyconsidered algebras fit this framewrok.
This class of operator algebras seems to go far, far beyond the operatoralgebras that fall under the umbrella of C*-correspondences.
On the other hand, for the C*-algebras there is no universal construction,likewise there is no Gauge Invariant Uniqueness Theorem. Some toolsdisappear from our tool-box, and techinal difficulties arise.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 19: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/19.jpg)
Subproduct systems
Operator algebras from subproduct systems III
Some interesting problems
• Do the algebras T (X ),O(X ) enjoy a universal property?• Under what conditions are the algebras T (X ),O(X ) nuclear?• What is the C*-envelope of T +(X )?• Classification: if X ,Y are subproduct systems, when are T +(X ) andT +(Y ) (isometrically) isomorphic?
For a subproduct system X = {E⌦n}n2N the answer to the first three isknown, and a lot is known about the fourth too.For general subproduct systems, much less is known, even when A = C.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 20: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/20.jpg)
Subproduct systems
Operator algebras from subproduct systems III
Some interesting problems• Do the algebras T (X ),O(X ) enjoy a universal property?
• Under what conditions are the algebras T (X ),O(X ) nuclear?• What is the C*-envelope of T +(X )?• Classification: if X ,Y are subproduct systems, when are T +(X ) andT +(Y ) (isometrically) isomorphic?
For a subproduct system X = {E⌦n}n2N the answer to the first three isknown, and a lot is known about the fourth too.For general subproduct systems, much less is known, even when A = C.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 21: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/21.jpg)
Subproduct systems
Operator algebras from subproduct systems III
Some interesting problems• Do the algebras T (X ),O(X ) enjoy a universal property?• Under what conditions are the algebras T (X ),O(X ) nuclear?
• What is the C*-envelope of T +(X )?• Classification: if X ,Y are subproduct systems, when are T +(X ) andT +(Y ) (isometrically) isomorphic?
For a subproduct system X = {E⌦n}n2N the answer to the first three isknown, and a lot is known about the fourth too.For general subproduct systems, much less is known, even when A = C.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 22: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/22.jpg)
Subproduct systems
Operator algebras from subproduct systems III
Some interesting problems• Do the algebras T (X ),O(X ) enjoy a universal property?• Under what conditions are the algebras T (X ),O(X ) nuclear?• What is the C*-envelope of T +(X )?
• Classification: if X ,Y are subproduct systems, when are T +(X ) andT +(Y ) (isometrically) isomorphic?
For a subproduct system X = {E⌦n}n2N the answer to the first three isknown, and a lot is known about the fourth too.For general subproduct systems, much less is known, even when A = C.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 23: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/23.jpg)
Subproduct systems
Operator algebras from subproduct systems III
Some interesting problems• Do the algebras T (X ),O(X ) enjoy a universal property?• Under what conditions are the algebras T (X ),O(X ) nuclear?• What is the C*-envelope of T +(X )?• Classification: if X ,Y are subproduct systems, when are T +(X ) andT +(Y ) (isometrically) isomorphic?
For a subproduct system X = {E⌦n}n2N the answer to the first three isknown, and a lot is known about the fourth too.For general subproduct systems, much less is known, even when A = C.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 24: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/24.jpg)
Subproduct systems
Operator algebras from subproduct systems III
Some interesting problems• Do the algebras T (X ),O(X ) enjoy a universal property?• Under what conditions are the algebras T (X ),O(X ) nuclear?• What is the C*-envelope of T +(X )?• Classification: if X ,Y are subproduct systems, when are T +(X ) andT +(Y ) (isometrically) isomorphic?
For a subproduct system X = {E⌦n}n2N the answer to the first three isknown, and a lot is known about the fourth too.For general subproduct systems, much less is known, even when A = C.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 25: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/25.jpg)
Monomial ideals
A particular class of subproduct systems
Let Chzi = Chz1, . . . , zd i denote the algebra of polynomials over C in dnoncommuting variables, and I = I(1) � I(2) � . . . a homogeneous ideal inChzi.Put E = Cd , and idenitify Chzi ✓ F(E ) = C� E � E⌦n � . . ..
Define X (0) = C andX (n) = E⌦n I(n).
X is a subproduct system that encodes very well the polynomial relations inthe ideal I:
Theorem (Popoescu, S.-Solel)
T +(X ) is the universal unital operator algebra generated by a rowcontraction satisfying the relations in I.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 26: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/26.jpg)
Monomial ideals
A particular class of subproduct systems
Let Chzi = Chz1, . . . , zd i denote the algebra of polynomials over C in dnoncommuting variables, and I = I(1) � I(2) � . . . a homogeneous ideal inChzi.Put E = Cd , and idenitify Chzi ✓ F(E ) = C� E � E⌦n � . . ..
Define X (0) = C andX (n) = E⌦n I(n).
X is a subproduct system that encodes very well the polynomial relations inthe ideal I:
Theorem (Popoescu, S.-Solel)
T +(X ) is the universal unital operator algebra generated by a rowcontraction satisfying the relations in I.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 27: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/27.jpg)
Monomial ideals
A particular class of subproduct systems
Let Chzi = Chz1, . . . , zd i denote the algebra of polynomials over C in dnoncommuting variables, and I = I(1) � I(2) � . . . a homogeneous ideal inChzi.Put E = Cd , and idenitify Chzi ✓ F(E ) = C� E � E⌦n � . . ..
Define X (0) = C andX (n) = E⌦n I(n).
X is a subproduct system that encodes very well the polynomial relations inthe ideal I:
Theorem (Popoescu, S.-Solel)
T +(X ) is the universal unital operator algebra generated by a rowcontraction satisfying the relations in I.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
The setting
We will henceforth restrict attention to the case where I is generated bymonomials.Fix a basis {e1, . . . , ed}, and denote Ti = T (ei ).To prevent confusion we denote
C ⇤(T ) = T (X ) = C ⇤(I ,T1, . . . ,Td )
andAX = AI = T +(X ) = Alg(I ,T1, . . . ,Td ).
Thus T is a row contraction and p(T ) = 0 for all p 2 I. Every such rowcontraction determines a UCC representation of AX . Finally:
C ⇤(T )/K = O(X ).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
The setting
We will henceforth restrict attention to the case where I is generated bymonomials.Fix a basis {e1, . . . , ed}, and denote Ti = T (ei ).To prevent confusion we denote
C ⇤(T ) = T (X ) = C ⇤(I ,T1, . . . ,Td )
andAX = AI = T +(X ) = Alg(I ,T1, . . . ,Td ).
Thus T is a row contraction and p(T ) = 0 for all p 2 I. Every such rowcontraction determines a UCC representation of AX .
Finally:
C ⇤(T )/K = O(X ).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
The setting
We will henceforth restrict attention to the case where I is generated bymonomials.Fix a basis {e1, . . . , ed}, and denote Ti = T (ei ).To prevent confusion we denote
C ⇤(T ) = T (X ) = C ⇤(I ,T1, . . . ,Td )
andAX = AI = T +(X ) = Alg(I ,T1, . . . ,Td ).
Thus T is a row contraction and p(T ) = 0 for all p 2 I. Every such rowcontraction determines a UCC representation of AX . Finally:
C ⇤(T )/K = O(X ).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
Some formulas
For µ = µ1 · · ·µk 2 F+d we write
zµ = zµ1 · · · zµk , eµ = eµ1 ⌦ · · ·⌦ eµk .
F(X ) is generated by eµ for µ such that zµ /2 I.
Tieµ = eiµ if z iµ /2 I, and 0 otherwise .
For µ = µ1 · · ·µk we write Tµ = Tµ1Tµ2 · · ·Tµk .Then T ⇤
µTµ is a the projection onto span{ew : Tµew 6= 0}.T ⇤µTµ commutes with T ⇤
⌫ T⌫ for all µ, ⌫.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
Some formulas
For µ = µ1 · · ·µk 2 F+d we write
zµ = zµ1 · · · zµk , eµ = eµ1 ⌦ · · ·⌦ eµk .
F(X ) is generated by eµ for µ such that zµ /2 I.
Tieµ = eiµ if z iµ /2 I, and 0 otherwise .
For µ = µ1 · · ·µk we write Tµ = Tµ1Tµ2 · · ·Tµk .Then T ⇤
µTµ is a the projection onto span{ew : Tµew 6= 0}.T ⇤µTµ commutes with T ⇤
⌫ T⌫ for all µ, ⌫.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
Some formulas
For µ = µ1 · · ·µk 2 F+d we write
zµ = zµ1 · · · zµk , eµ = eµ1 ⌦ · · ·⌦ eµk .
F(X ) is generated by eµ for µ such that zµ /2 I.
Tieµ = eiµ if z iµ /2 I, and 0 otherwise .
For µ = µ1 · · ·µk we write Tµ = Tµ1Tµ2 · · ·Tµk .Then T ⇤
µTµ is a the projection onto span{ew : Tµew 6= 0}.
T ⇤µTµ commutes with T ⇤
⌫ T⌫ for all µ, ⌫.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
Some formulas
For µ = µ1 · · ·µk 2 F+d we write
zµ = zµ1 · · · zµk , eµ = eµ1 ⌦ · · ·⌦ eµk .
F(X ) is generated by eµ for µ such that zµ /2 I.
Tieµ = eiµ if z iµ /2 I, and 0 otherwise .
For µ = µ1 · · ·µk we write Tµ = Tµ1Tµ2 · · ·Tµk .Then T ⇤
µTµ is a the projection onto span{ew : Tµew 6= 0}.T ⇤µTµ commutes with T ⇤
⌫ T⌫ for all µ, ⌫.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
Example: Subshifts
On {1, . . . , d}Z, let � denote the left shift.A shift invariant space ⇤ ✓ {1, . . . , d}Z is called a subshift.A subshift ⇤ is determined by the set of F of forbidden words.
F = forbidden words , ⇤⇤ = allowed words
From a subshift we can construct an ideal I⇤ generated by zµ, for µ 2 F.This gives rise to a subproduct system X⇤.
Not all our examples arise like this, but this covers a lot of cases of interest.
In this setting, the algebras C ⇤(T )/K were studied by Matsumoto, andwere called subshift C*-algebras.However, Matsumoto later changed his definitions (in order that histheorems remain true).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
Example: Subshifts
On {1, . . . , d}Z, let � denote the left shift.A shift invariant space ⇤ ✓ {1, . . . , d}Z is called a subshift.A subshift ⇤ is determined by the set of F of forbidden words.
F = forbidden words , ⇤⇤ = allowed words
From a subshift we can construct an ideal I⇤ generated by zµ, for µ 2 F.This gives rise to a subproduct system X⇤.
Not all our examples arise like this, but this covers a lot of cases of interest.
In this setting, the algebras C ⇤(T )/K were studied by Matsumoto, andwere called subshift C*-algebras.However, Matsumoto later changed his definitions (in order that histheorems remain true).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Monomial ideals
Example: Subshifts
On {1, . . . , d}Z, let � denote the left shift.A shift invariant space ⇤ ✓ {1, . . . , d}Z is called a subshift.A subshift ⇤ is determined by the set of F of forbidden words.
F = forbidden words , ⇤⇤ = allowed words
From a subshift we can construct an ideal I⇤ generated by zµ, for µ 2 F.This gives rise to a subproduct system X⇤.
Not all our examples arise like this, but this covers a lot of cases of interest.
In this setting, the algebras C ⇤(T )/K were studied by Matsumoto, andwere called subshift C*-algebras.However, Matsumoto later changed his definitions (in order that histheorems remain true).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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C*-correspondences versus subproduct systems
Operator algebras of C*-correspodences have been studied for almost twodecades, by many strong researchers.
Subproduct systems are very flexible, and conain interesting classes ofexamples even when the base C*-algebra is C.
The situation we are dealing with has nature particularly tractable tocomputations (e.g., T1, . . . ,Td are mutually orthogonal partial isometries),and we are able to combine the two theories.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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C*-correspondences versus subproduct systems
Operator algebras of C*-correspodences have been studied for almost twodecades, by many strong researchers.
Subproduct systems are very flexible, and conain interesting classes ofexamples even when the base C*-algebra is C.
The situation we are dealing with has nature particularly tractable tocomputations (e.g., T1, . . . ,Td are mutually orthogonal partial isometries),and we are able to combine the two theories.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 40: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/40.jpg)
C*-correspondences versus subproduct systems
Operator algebras of C*-correspodences have been studied for almost twodecades, by many strong researchers.
Subproduct systems are very flexible, and conain interesting classes ofexamples even when the base C*-algebra is C.
The situation we are dealing with has nature particularly tractable tocomputations (e.g., T1, . . . ,Td are mutually orthogonal partial isometries),and we are able to combine the two theories.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 41: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/41.jpg)
C*-correspondences versus subproduct systems
Operator algebras of C*-correspodences have been studied for almost twodecades, by many strong researchers.
Subproduct systems are very flexible, and conain interesting classes ofexamples even when the base C*-algebra is C.
The situation we are dealing with has nature particularly tractable tocomputations (e.g., T1, . . . ,Td are mutually orthogonal partial isometries),and we are able to combine the two theories.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
![Page 42: Operator algebras associated with monomial idealsdanielm/seminar/2015-Shalit.pdf · 2015-01-04 · Operator algebras associated with monomial ideals Evgenios Kakariadis and Orr Shalit](https://reader034.fdocuments.in/reader034/viewer/2022050103/5f41fe6e1529fe29ed6d8e8b/html5/thumbnails/42.jpg)
The C*-correspondence of a monomial ideal
What is the smallest C*-correspondence containing T?
If we form a C*-correspondence E over A containing T1, . . . ,Td , then
hTi ,Ti i = T ⇤i Ti 2 A.
Thus TjT ⇤i Ti 2 E , and T ⇤
i TiTj .
But a computation shows :
T ⇤i TiTj = TjT ⇤
ij Tij ,
(where Tij = TiTj) thus
hTjT ⇤ij Tij ,TjT ⇤
ij Tiji = T ⇤ij TijT ⇤
j TjT ⇤ij Tij = T ⇤
ij Tij ,
So T ⇤ij Tij 2 A.
Likewise, T ⇤µTµ 2 A for all µ.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
What is the smallest C*-correspondence containing T?
If we form a C*-correspondence E over A containing T1, . . . ,Td , then
hTi ,Ti i = T ⇤i Ti 2 A.
Thus TjT ⇤i Ti 2 E , and T ⇤
i TiTj . But a computation shows :
T ⇤i TiTj = TjT ⇤
ij Tij ,
(where Tij = TiTj) thus
hTjT ⇤ij Tij ,TjT ⇤
ij Tiji = T ⇤ij TijT ⇤
j TjT ⇤ij Tij = T ⇤
ij Tij ,
So T ⇤ij Tij 2 A.
Likewise, T ⇤µTµ 2 A for all µ.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
What is the smallest C*-correspondence containing T?
If we form a C*-correspondence E over A containing T1, . . . ,Td , then
hTi ,Ti i = T ⇤i Ti 2 A.
Thus TjT ⇤i Ti 2 E , and T ⇤
i TiTj . But a computation shows :
T ⇤i TiTj = TjT ⇤
ij Tij ,
(where Tij = TiTj) thus
hTjT ⇤ij Tij ,TjT ⇤
ij Tiji = T ⇤ij TijT ⇤
j TjT ⇤ij Tij = T ⇤
ij Tij ,
So T ⇤ij Tij 2 A.
Likewise, T ⇤µTµ 2 A for all µ.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
The C*-correspondence of a monomial ideal
We defineA = C ⇤(I ,T ⇤
µTµ : µ 2 Fd+).
andE = span{Tia : a 2 A}.
Note that A is a commutative C*-algebra.
We now cosider the algebras TE , T +E and the (relative) Cuntz-Pimsner
algebras O(J,E ).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
The C*-correspondence of a monomial ideal
We defineA = C ⇤(I ,T ⇤
µTµ : µ 2 Fd+).
andE = span{Tia : a 2 A}.
Note that A is a commutative C*-algebra.
We now cosider the algebras TE , T +E and the (relative) Cuntz-Pimsner
algebras O(J,E ).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
Orientating the algebras I
Let AEA be the C*-correspondence of a monomial ideal I in Chz1, . . . , zd i.TheoremC ⇤(T ) is the relative Cuntz-Pimsner algebra O(J,E ) for the ideal Jgenerated by {1� T ⇤
µTµ | µ 2 F+d }.
Moreover C ⇤(T )/K(FX ) is the relative Cuntz-Pimsner algebra O(A,E ).
In particular J ✓ JE ✓ A, and there are canonical ⇤-epimorphisms
TE ! C ⇤(T )! OE ! C ⇤(T )/K(FX ).
Corollary (C*-correspondences help)The algebras C ⇤(T ) = T (X ) and C ⇤(T )/K(FX ) = O(X ) are nuclear.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
Orientating the algebras I
Let AEA be the C*-correspondence of a monomial ideal I in Chz1, . . . , zd i.TheoremC ⇤(T ) is the relative Cuntz-Pimsner algebra O(J,E ) for the ideal Jgenerated by {1� T ⇤
µTµ | µ 2 F+d }.
Moreover C ⇤(T )/K(FX ) is the relative Cuntz-Pimsner algebra O(A,E ).
In particular J ✓ JE ✓ A, and there are canonical ⇤-epimorphisms
TE ! C ⇤(T )! OE ! C ⇤(T )/K(FX ).
Corollary (C*-correspondences help)The algebras C ⇤(T ) = T (X ) and C ⇤(T )/K(FX ) = O(X ) are nuclear.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
Orientating the algebras I
Let AEA be the C*-correspondence of a monomial ideal I in Chz1, . . . , zd i.TheoremC ⇤(T ) is the relative Cuntz-Pimsner algebra O(J,E ) for the ideal Jgenerated by {1� T ⇤
µTµ | µ 2 F+d }.
Moreover C ⇤(T )/K(FX ) is the relative Cuntz-Pimsner algebra O(A,E ).
In particular J ✓ JE ✓ A, and there are canonical ⇤-epimorphisms
TE ! C ⇤(T )! OE ! C ⇤(T )/K(FX ).
Corollary (C*-correspondences help)The algebras C ⇤(T ) = T (X ) and C ⇤(T )/K(FX ) = O(X ) are nuclear.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
Orientating the algebras I
Let AEA be the C*-correspondence of a monomial ideal I in Chz1, . . . , zd i.TheoremC ⇤(T ) is the relative Cuntz-Pimsner algebra O(J,E ) for the ideal Jgenerated by {1� T ⇤
µTµ | µ 2 F+d }.
Moreover C ⇤(T )/K(FX ) is the relative Cuntz-Pimsner algebra O(A,E ).
In particular J ✓ JE ✓ A, and there are canonical ⇤-epimorphisms
TE ! C ⇤(T )! OE ! C ⇤(T )/K(FX ).
Corollary (C*-correspondences help)The algebras C ⇤(T ) = T (X ) and C ⇤(T )/K(FX ) = O(X ) are nuclear.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
Orientating the algebras II
TheoremThe following diagram holds:
C
⇤(T ) 6' TE , I 6= (0) , E 6' CdC
⇤(T ) ' TE , I = 0 , E ' Cd
↵◆ker �E 6= 0 ker �E = 0
KS
↵◆
KS
↵◆
OE ' C
⇤(T )/K(FX ) ' Od , ker �E = 0
OE ' C
⇤(T ) OE ' C
⇤(T )/K(FX )
with the understanding that all ⇤-isomorphisms are canonical.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
Orientating the algebras II
TheoremThe following diagram holds:
C
⇤(T ) 6' TE , I 6= (0) , E 6' CdC
⇤(T ) ' TE , I = 0 , E ' Cd
↵◆ker �E 6= 0 ker �E = 0
KS
↵◆
KS
↵◆
OE ' C
⇤(T )/K(FX ) ' Od , ker �E = 0
OE ' C
⇤(T ) OE ' C
⇤(T )/K(FX )
We aslo have precise combinatorial conditions for when ker �E = 0.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The C*-correspondence of a monomial ideal
PropositionLet AEA be the C*-correspondence of a monomial ideal I / Chz1, . . . , zd i.The following are equivalent:
1. P; 2 A;2. ker �E = C · P;;3. ker�E 6= (0);4. for every i = 1, . . . , d there is a µi 2 ⇤⇤
such that µi i /2 ⇤⇤;5. for every i = 1, . . . , d there is a µi 2 Fd
+ such that zµi /2 I andzµi zi 2 I;
6. JE := ker �?E \ ��1(K(E )) = h1� P;i = A(1� P;);7. 1 /2 JE .
If these conditions hold then ker �E = hT ⇤µ1
Tµ1 · · ·T ⇤µd
Tµd i for any tuple ofwords (µ1, . . . , µd ) such that µi i /2 ⇤⇤ for all i = 1, . . . , d .
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-envelopes I
A theorem of Katsoulis-Kribs (following Muhly-Solel) says that
C ⇤e (T +
E ) = OE
TheoremLet E be the C*-correspondence of a monomial ideal I / Chz1, . . . , zd i.Then the tensor algebra T +
E is hyperrigid (in OE ).
Recall: A ✓ B = C ⇤(A) is said to be hyperrigid if for every unital⇡ : B ! B(H), the UCC map ⇡
��A has the uniqe extension property.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-envelopes I
A theorem of Katsoulis-Kribs (following Muhly-Solel) says that
C ⇤e (T +
E ) = OE
TheoremLet E be the C*-correspondence of a monomial ideal I / Chz1, . . . , zd i.Then the tensor algebra T +
E is hyperrigid (in OE ).
Recall: A ✓ B = C ⇤(A) is said to be hyperrigid if for every unital⇡ : B ! B(H), the UCC map ⇡
��A has the uniqe extension property.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-envelopes I
A theorem of Katsoulis-Kribs (following Muhly-Solel) says that
C ⇤e (T +
E ) = OE
TheoremLet E be the C*-correspondence of a monomial ideal I / Chz1, . . . , zd i.Then the tensor algebra T +
E is hyperrigid (in OE ).
Recall: A ✓ B = C ⇤(A) is said to be hyperrigid if for every unital⇡ : B ! B(H), the UCC map ⇡
��A has the uniqe extension property.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-envelopes II
We now turn to the algebra AX .Here there is no general theory to help us and our results are far from finalform.
TheoremLet X be the subproduct system of a monomial ideal I / Chz1, . . . , zd i of
finite type and let q : C ⇤(T )! C ⇤(T )/K(FX ). Then q(AX ) is hyperrigidin C ⇤(T )/K(FX ), hence C ⇤
e (q(AX )) = C ⇤(T )/K(FX ).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-envelopes II
We now turn to the algebra AX .Here there is no general theory to help us and our results are far from finalform.
TheoremLet X be the subproduct system of a monomial ideal I / Chz1, . . . , zd i of
finite type and let q : C ⇤(T )! C ⇤(T )/K(FX ). Then q(AX ) is hyperrigidin C ⇤(T )/K(FX ), hence C ⇤
e (q(AX )) = C ⇤(T )/K(FX ).
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-envelopes III
TheoremLet X be the subproduct system of a monomial ideal I / Chz1, . . . , zd i of
finite type and let q : C ⇤(T )! C ⇤(T )/K(FX ). Then
q|AX is not completely isometric
KS(1)↵◆
q|AX is completely isometric
KS(2)↵◆
C
⇤e (AX ) ' C
⇤(T )
(3)↵◆
KS(4)
C
⇤e (AX ) ' C
⇤(T )/K(FX )
(4)↵◆
KS(3)
8i = 1, . . . , d , 9µi 2 ⇤⇤.µi i /2 ⇤⇤ 9i 2 {1, . . . , d}, 8µ 2 ⇤⇤.µi 2 ⇤⇤
Item (4) holds under the assumption that the µi s can always be chosen tohave the same length. In particular item (4) holds when X = X⇤.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-correspondences help, again
Proof.
(1) follows from Arveson’s “Boundary Theorem”.
(2) follows from the previous theorem.
(3) (Going up). The conidition implies ker � = (0) by proposition, hence by
C ⇤(T )/K = OE = C ⇤e (T +
E )
where first equality follows from a previous theorem.But AX ,! T +
E . Thus q��AX
is completely isometric, and (3) follows.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-correspondences help, again
Proof.(1) follows from Arveson’s “Boundary Theorem”.
(2) follows from the previous theorem.
(3) (Going up). The conidition implies ker � = (0) by proposition, hence by
C ⇤(T )/K = OE = C ⇤e (T +
E )
where first equality follows from a previous theorem.But AX ,! T +
E . Thus q��AX
is completely isometric, and (3) follows.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-correspondences help, again
Proof.(1) follows from Arveson’s “Boundary Theorem”.
(2) follows from the previous theorem.
(3) (Going up). The conidition implies ker � = (0) by proposition, hence by
C ⇤(T )/K = OE = C ⇤e (T +
E )
where first equality follows from a previous theorem.But AX ,! T +
E . Thus q��AX
is completely isometric, and (3) follows.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
C*-correspondences help, again
Proof.(1) follows from Arveson’s “Boundary Theorem”.
(2) follows from the previous theorem.
(3) (Going up). The conidition implies ker � = (0) by proposition, hence by
C ⇤(T )/K = OE = C ⇤e (T +
E )
where first equality follows from a previous theorem.But AX ,! T +
E . Thus q��AX
is completely isometric, and (3) follows.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Remark
In the previous theorem we say that (under assumption of finite type)
C ⇤e (AX ) = C ⇤(T ) = T (X ) or C ⇤
e (AX ) = C ⇤(T )/K = O(X ).
In all known examples until now, we had either
C ⇤e (T +(X )) = T (X ) or C ⇤
e (T +(X )) = O(X ).
Recently Dor-On and Markiewicz showed that these are not the onlypossibilities.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Remark
In the previous theorem we say that (under assumption of finite type)
C ⇤e (AX ) = C ⇤(T ) = T (X ) or C ⇤
e (AX ) = C ⇤(T )/K = O(X ).
In all known examples until now, we had either
C ⇤e (T +(X )) = T (X ) or C ⇤
e (T +(X )) = O(X ).
Recently Dor-On and Markiewicz showed that these are not the onlypossibilities.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Universal property
TheoremLet I be a monomial ideal of finite type k. Then the algebraC ⇤(T )/K(FX ) is the universal C*-algebra generated by a row contractions = [s1, . . . , sd ] such that
1 I =Pd
i=1 si s⇤i ;2 p(s) = 0 for all p 2 I;3 s⇤i si =
Pµ2Ek
isµs⇤µ where E k
i = {µ 2 ⇤⇤k | iµ 2 ⇤⇤}, for all
i = 1, . . . , d.
Previously appeared in joint work with Solel, though proof there seems tohave a gap.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Universal property
TheoremLet I be a monomial ideal of finite type k. Then the algebraC ⇤(T )/K(FX ) is the universal C*-algebra generated by a row contractions = [s1, . . . , sd ] such that
1 I =Pd
i=1 si s⇤i ;2 p(s) = 0 for all p 2 I;3 s⇤i si =
Pµ2Ek
isµs⇤µ where E k
i = {µ 2 ⇤⇤k | iµ 2 ⇤⇤}, for all
i = 1, . . . , d.
Previously appeared in joint work with Solel, though proof there seems tohave a gap.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Application of NSA methods
Proof.Given such a tuple S1, . . . , Sd 2 B(H), need to construct
⇡ : C ⇤(T )/K! B(H) , ⇡ : q(Ti ) 7! Si .
By previous results (Popescu, S.-Solel) construct ⇡ : q(AX )! B(H),
⇡ : q(Ti ) 7! Si , i = 1, . . . , d .
Then, using techniques of previous proposition, show that ⇡ has theunique extension property.Thus ⇡ extends to ⇤-representation, as required.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Application of NSA methods
Proof.Given such a tuple S1, . . . , Sd 2 B(H), need to construct
⇡ : C ⇤(T )/K! B(H) , ⇡ : q(Ti ) 7! Si .
By previous results (Popescu, S.-Solel) construct ⇡ : q(AX )! B(H),
⇡ : q(Ti ) 7! Si , i = 1, . . . , d .
Then, using techniques of previous proposition, show that ⇡ has theunique extension property.Thus ⇡ extends to ⇤-representation, as required.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Classification I
TheoremLet X and Y be subproduct systems associated with the homogeneousideals I / Chx1, . . . , xd i and J / Chy1, . . . , yd 0i. Without loss of generalitysuppose that xi /2 I and yj /2 J for all i , j . The following are equivalent:
1. AX and AY are completely isometrically isomorphic;2. AX and AY are algebraically isomorphic;3. X and Y are similar;4. X and Y are isomorphic;
5. d = d 0 and there is a permutation on the variables y1, . . . , yd suchthat I and J are defined by the same words.
Explanation: X ' Y iff Um : Xn ! Yn (unitaries) such that
Um+n(Pm+n(xm ⌦ xn)) = Pm+n(Um(xm)⌦ Un(xn))
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Classification I
TheoremLet X and Y be subproduct systems associated with the homogeneousideals I / Chx1, . . . , xd i and J / Chy1, . . . , yd 0i. Without loss of generalitysuppose that xi /2 I and yj /2 J for all i , j . The following are equivalent:
1. AX and AY are completely isometrically isomorphic;2. AX and AY are algebraically isomorphic;3. X and Y are similar;4. X and Y are isomorphic;5. d = d 0 and there is a permutation on the variables y1, . . . , yd such
that I and J are defined by the same words.
Explanation: X ' Y iff Um : Xn ! Yn (unitaries) such that
Um+n(Pm+n(xm ⌦ xn)) = Pm+n(Um(xm)⌦ Un(xn))
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Non-selfadjoint issues
Classification I
TheoremLet X and Y be subproduct systems associated with the homogeneousideals I / Chx1, . . . , xd i and J / Chy1, . . . , yd 0i. Without loss of generalitysuppose that xi /2 I and yj /2 J for all i , j . The following are equivalent:
1. AX and AY are completely isometrically isomorphic;2. AX and AY are algebraically isomorphic;3. X and Y are similar;4. X and Y are isomorphic;5. d = d 0 and there is a permutation on the variables y1, . . . , yd such
that I and J are defined by the same words.
Explanation: X ' Y iff Um : Xn ! Yn (unitaries) such that
Um+n(Pm+n(xm ⌦ xn)) = Pm+n(Um(xm)⌦ Un(xn))
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
The quantised dynamics I
On the commutative C*-algebra A = C ⇤(T ⇤µTµ | µ 2 ⇤⇤) we define d
⇤-endomorphisms↵i : A ! A,
↵i (a) = T ⇤i aTi .
We call the system (A,↵1, . . . ,↵d ) the quantised dynamical system ofthe allowable words.
TheoremThe quatised dyamical system’s conjugacy class is a complete invariant ofthe monomial ideal.
Indeed, ↵µ1 � · · · � ↵µk (I ) = 0 determines the monomials zµ in the ideal.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
The quantised dynamics I
On the commutative C*-algebra A = C ⇤(T ⇤µTµ | µ 2 ⇤⇤) we define d
⇤-endomorphisms↵i : A ! A,
↵i (a) = T ⇤i aTi .
We call the system (A,↵1, . . . ,↵d ) the quantised dynamical system ofthe allowable words.
TheoremThe quatised dyamical system’s conjugacy class is a complete invariant ofthe monomial ideal.
Indeed, ↵µ1 � · · · � ↵µk (I ) = 0 determines the monomials zµ in the ideal.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
The quantised dynamics II
A commutative C*-dynamical system (A,↵1, . . . ,↵d ) gives rise to apartially defined classical dynamical system.
Put pi = T ⇤i Ti . Then
↵i : A = C (⌦)! piApi = C (⌦i )
induces'i : ⌦i ! ⌦
We obtain a partially defined classical dynamical system (⌦,'1, . . . ,'d ),where each 'i is only defined on ⌦i ✓ ⌦.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
The quantised dynamics II
A commutative C*-dynamical system (A,↵1, . . . ,↵d ) gives rise to apartially defined classical dynamical system.Put pi = T ⇤
i Ti . Then
↵i : A = C (⌦)! piApi = C (⌦i )
induces'i : ⌦i ! ⌦
We obtain a partially defined classical dynamical system (⌦,'1, . . . ,'d ),where each 'i is only defined on ⌦i ✓ ⌦.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
The quantised dynamics II
A commutative C*-dynamical system (A,↵1, . . . ,↵d ) gives rise to apartially defined classical dynamical system.Put pi = T ⇤
i Ti . Then
↵i : A = C (⌦)! piApi = C (⌦i )
induces'i : ⌦i ! ⌦
We obtain a partially defined classical dynamical system (⌦,'1, . . . ,'d ),where each 'i is only defined on ⌦i ✓ ⌦.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
Classification II
TheoremLet I / Chx1, . . . , xd i and J / Chy1, . . . , yd 0i be monomial ideals. Let EAand FB be the C*-correspondences associated with I and J , respectively.Furthermore let (⌦I ,') and (⌦J , )) be the corresponding quantiseddynamics. The following are equivalent:
1. T +E and T +
F are completely isometrically isomorphic;2. T +
E and T +F are isomorphic as topological algebras;
3. (⌦I ,') and (⌦J , ) are locally (piecewise) conjugate;4. E and F are unitarily equivalent.
Locally (piecewise) conjugate: Mix between the notion ofDavidson-Katsoulis for dynamical systems and the notion ofDavidson-Roydor for topological graphs.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
Classification II
TheoremLet I / Chx1, . . . , xd i and J / Chy1, . . . , yd 0i be monomial ideals. Let EAand FB be the C*-correspondences associated with I and J , respectively.Furthermore let (⌦I ,') and (⌦J , )) be the corresponding quantiseddynamics. The following are equivalent:
1. T +E and T +
F are completely isometrically isomorphic;2. T +
E and T +F are isomorphic as topological algebras;
3. (⌦I ,') and (⌦J , ) are locally (piecewise) conjugate;4. E and F are unitarily equivalent.
Locally (piecewise) conjugate: Mix between the notion ofDavidson-Katsoulis for dynamical systems and the notion ofDavidson-Roydor for topological graphs.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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The quatised dynamics
Classification II
TheoremLet I / Chx1, . . . , xd i and J / Chy1, . . . , yd 0i be monomial ideals. Let EAand FB be the C*-correspondences associated with I and J , respectively.Furthermore let (⌦I ,') and (⌦J , )) be the corresponding quantiseddynamics. The following are equivalent:
1. T +E and T +
F are completely isometrically isomorphic;2. T +
E and T +F are isomorphic as topological algebras;
3. (⌦I ,') and (⌦J , ) are locally (piecewise) conjugate;4. E and F are unitarily equivalent.
Proof:
This follows from Davidson-Roydor, because a partial dynamical system is atopological graph. We also present an alternative proof.
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals
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Thank you!
Evgenios Kakariadis and Orr Shalit Operator algebras associated with monomial ideals