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As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-algebras are not necessarily self-opposite so, as the authors point out, nothing so far prevents their example from being realized as a twisted convolution algebra. In fact, it appears that nobody knows an example of a $C^*$-algebra which is not a twisted convolution algebra!!

Among the algebras known to be (twisted or untwisted) convolution algebras one finds all Kirchberg algebras satisfying the UCT as well as all algebras in the Elliott classification program, including the somewhat elusive Jiang-Su algebra $\mathscr Z$.

On the other hand, the vast majority of useful tools available to the study of (untwisted) groupoid $C^*$-algebras apply just as well for twisted ones, so perhaps the more relevant question is which $C^*$-algebras are isomorphic to a twisted convolution algebra.

As mentioned by @Simon, the best results so far leading to a "Fourier-like" decomposition of a $C^*$-algebra as a convolution algebra (of an etale groupoid) are based on the assumption that we already know the abelian subalgebra of functions supported on the object space. This abelian subalgebra is sometimes called a Cartan algebra, given the similarities with the homonymous concept from the theory of Lie groups.

One of the stickiest points in this subject is whether or not there exists a conditional expectation onto the given Cartan subalgebra (one of the main assumptions in Renault's result). The reason one might prefer not to require this condition is that there are many groupoids whose convolution algebra does not admit such a conditional expectation due to the fact that the underlying groupoid is not Hausdorff. Examples of this situation are very common such as groupoids arising from foliations and certain dynamical systems.

Another relevant question is whether or not the Cartan subalgebra is maximal abelian, a crucial feature in the Lie algebra version of this concept, as well as an important assumption in Renault's theory. Maximal commutativity is closely related to topological freenes of the associated groupoid (a concept borrowed from those dynamical systems in which most points have trivial isotropy group). In particular, in the example given by @Simon of $C^*(B\mathbb Z)$, the natural Cartan subalgebra is $\mathbb C$, which is clearly not maximal commutative at the same time that the action of the group $\mathbb Z$ on a point has too much isotropy!

Working with David Pitts (arXiv:1901.09683) we have found a characterization of twisted groupoid $C^*$-algebras which neither assumes conditional expectations nor maximal commutativity. Perhaps one of the simplest examples where our result applies is $$ C(S^1)\subseteq C([0,1]) $$ where we identify $C(S^1)$ as the subalgebra of periodic functions. Surprisingly, there is a twisted groupoid description of this "Cartan pair".

Significantly, the lack of a conditional expectation from $C([0,1])$ to $C(S^1)$ prevents the groupoid from being Hausdorff. In fact, the topological space underlying this groupoid is essentially the best known example of a non-Hausdorff topological space in which one takes the unit interval $[0,1]$ with a duplicate copy of the point 1, except that we instead duplicate a chosen point of $S^1$. The groupoid structure is such that the two duplicate points form a copy of the 2-element group, while all other points are considered to be objects. The twist is non trivial but it turns out to be the most natural nontrivial twist one can think of.

As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-algebras are not necessarily self-opposite so, as the authors point out, nothing so far prevents their example from being realized as a twisted convolution algebra. In fact, it appears that nobody knows an example of a $C^*$-algebra which is not a twisted convolution algebra!!

Among the algebras known to be (twisted or untwisted) convolution algebras one finds all Kirchberg algebras satisfying the UCT as well as all algebras in the Elliott classification program, including the somewhat elusive Jiang-Su algebra $\mathscr Z$.

On the other hand, the vast majority of useful tools available to the study of (untwisted) groupoid $C^*$-algebras apply just as well for twisted ones, so perhaps the more relevant question is which $C^*$-algebras are isomorphic to a twisted convolution algebra.

As mentioned by @Simon, the best results so far leading to a "Fourier-like" decomposition of a $C^*$-algebra as a convolution algebra (of an etale groupoid) are based on the assumption that we already know the abelian subalgebra of functions supported on the object space. This abelian subalgebra is sometimes called a Cartan algebra, given the similarities with the homonymous concept from the theory of Lie groups.

One of the stickiest points in this subject is whether or not there exists a conditional expectation onto the given Cartan subalgebra (one of the main assumptions in Renault's result). The reason one might prefer not to require this condition is that there are many groupoids whose convolution algebra does not admit such a conditional expectation due to the fact that the underlying groupoid is not Hausdorff. Examples of this situation are very common such as groupoids arising from foliations and certain dynamical systems.

Another relevant question is whether or not the Cartan subalgebra is maximal abelian, a crucial feature in the Lie algebra version of this concept, as well as an important assumption in Renault's theory. Maximal commutativity is closely related to topological freenes of the associated groupoid (a concept borrowed from those dynamical systems in which most points have trivial isotropy group). In particular, in the example given by @Simon of $C^*(B\mathbb Z)$, the natural Cartan subalgebra is $\mathbb C$, which is clearly not maximal commutative at the same time that the action of the group $\mathbb Z$ on a point has too much isotropy!

Working with David Pitts (Characterizing groupoid C*-algebras of non-Hausdorff étale groupoids, arXiv:1901.09683) we have found a characterization of twisted groupoid $C^*$-algebras which neither assumes conditional expectations nor maximal commutativity. Perhaps one of the simplest examples where our result applies is $$ C(S^1)\subseteq C([0,1]) $$ where we identify $C(S^1)$ as the subalgebra of periodic functions. Surprisingly, there is a twisted groupoid description of this "Cartan pair".

Significantly, the lack of a conditional expectation from $C([0,1])$ to $C(S^1)$ prevents the groupoid from being Hausdorff. In fact, the topological space underlying this groupoid is essentially the best known example of a non-Hausdorff topological space in which one takes the unit interval $[0,1]$ with a duplicate copy of the point 1, except that we instead duplicate a chosen point of $S^1$. The groupoid structure is such that the two duplicate points form a copy of the 2-element group, while all other points are considered to be objects. The twist is non trivial but it turns out to be the most natural nontrivial twist one can think of.

As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-algebras are not necessarily self-opposite so, as the authors point out, nothing so far prevents their example from being realized as a twisted convolution algebra. In fact, it appears that nobody knows an example of a $C^*$-algebra which is not a twisted convolution algebra!!

On the other hand, the vast majority of useful tools available to the study of (untwisted) groupoid $C^*$-algebras apply just as well for twisted ones, so perhaps the more relevant question is which $C^*$-algebras are isomorphic to a twisted convolution algebra.

As mentioned by @Simon, the best results so far leading to a "Fourier-like" decomposition of a $C^*$-algebra as a convolution algebra (of an etale groupoid) are based on the assumption that we already know the abelian subalgebra of functions supported on the object space. This abelian subalgebra is sometimes called a Cartan algebra, given the similarities with the homonymous concept from the theory of Lie groups.

One of the stickiest points in this subject is whether or not there exists a conditional expectation onto the given Cartan subalgebra (one of the main assumptions in Renault's result). The reason one might prefer not to require this condition is that there are many groupoids whose convolution algebra does not admit such a conditional expectation due to the fact that the underlying groupoid is not Hausdorff. Examples of this situation are very common such as groupoids arising from foliations and certain dynamical systems.

Another relevant question is whether or not the Cartan subalgebra is maximal abelian, a crucial feature in the Lie algebra version of this concept, as well as an important assumption in Renault's theory. Maximal commutativity is closely related to topological freenes of the associated groupoid (a concept borrowed from those dynamical systems in which most points have trivial isotropy group). In particular, in the example given by @Simon of $C^*(B\mathbb Z)$, the natural Cartan subalgebra is $\mathbb C$, which is clearly not maximal commutative at the same time that the action of the group $\mathbb Z$ on a point has too much isotropy!

Working with David Pitts (arXiv:1901.09683) we have found a characterization of twisted groupoid $C^*$-algebras which neither assumes conditional expectations nor maximal commutativity. Perhaps one of the simplest examples where our result applies is $$ C(S^1)\subseteq C([0,1]) $$ where we identify $C(S^1)$ as the subalgebra of periodic functions. Surprisingly, there is a twisted groupoid description of this "Cartan pair".

Significantly, the lack of a conditional expectation from $C([0,1])$ to $C(S^1)$ prevents the groupoid from being Hausdorff. In fact, the topological space underlying this groupoid is essentially the best known example of a non-Hausdorff topological space in which one takes the unit interval $[0,1]$ with a duplicate copy of the point 1, except that we instead duplicate a chosen point of $S^1$. The groupoid structure is such that the two duplicate points form a copy of the 2-element group, while all other points are considered to be objects. The twist is non trivial but it turns out to be the most natural nontrivial twist one can think of.

As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-algebras are not necessarily self-opposite so, as the authors point out, nothing so far prevents their example from being realized as a twisted convolution algebra. In fact, it appears that nobody knows an example of a $C^*$-algebra which is not a twisted convolution algebra!!

Among the algebras known to be (twisted or untwisted) convolution algebras one finds all Kirchberg algebras satisfying the UCT as well as all algebras in the Elliott classification program, including the somewhat elusive Jiang-Su algebra $\mathscr Z$.

On the other hand, the vast majority of useful tools available to the study of (untwisted) groupoid $C^*$-algebras apply just as well for twisted ones, so perhaps the more relevant question is which $C^*$-algebras are isomorphic to a twisted convolution algebra.

As mentioned by @Simon, the best results so far leading to a "Fourier-like" decomposition of a $C^*$-algebra as a convolution algebra (of an etale groupoid) are based on the assumption that we already know the abelian subalgebra of functions supported on the object space. This abelian subalgebra is sometimes called a Cartan algebra, given the similarities with the homonymous concept from the theory of Lie groups.

One of the stickiest points in this subject is whether or not there exists a conditional expectation onto the given Cartan subalgebra (one of the main assumptions in Renault's result). The reason one might prefer not to require this condition is that there are many groupoids whose convolution algebra does not admit such a conditional expectation due to the fact that the underlying groupoid is not Hausdorff. Examples of this situation are very common such as groupoids arising from foliations and certain dynamical systems.

Another relevant question is whether or not the Cartan subalgebra is maximal abelian, a crucial feature in the Lie algebra version of this concept, as well as an important assumption in Renault's theory. Maximal commutativity is closely related to topological freenes of the associated groupoid (a concept borrowed from those dynamical systems in which most points have trivial isotropy group). In particular, in the example given by @Simon of $C^*(B\mathbb Z)$, the natural Cartan subalgebra is $\mathbb C$, which is clearly not maximal commutative at the same time that the action of the group $\mathbb Z$ on a point has too much isotropy!

Working with David Pitts (arXiv:1901.09683) we have found a characterization of twisted groupoid $C^*$-algebras which neither assumes conditional expectations nor maximal commutativity. Perhaps one of the simplest examples where our result applies is $$ C(S^1)\subseteq C([0,1]) $$ where we identify $C(S^1)$ as the subalgebra of periodic functions. Surprisingly, there is a twisted groupoid description of this "Cartan pair".

Significantly, the lack of a conditional expectation from $C([0,1])$ to $C(S^1)$ prevents the groupoid from being Hausdorff. In fact, the topological space underlying this groupoid is essentially the best known example of a non-Hausdorff topological space in which one takes the unit interval $[0,1]$ with a duplicate copy of the point 1, except that we instead duplicate a chosen point of $S^1$. The groupoid structure is such that the two duplicate points form a copy of the 2-element group, while all other points are considered to be objects. The twist is non trivial but it turns out to be the most natural nontrivial twist one can think of.

As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-algebras are not necessarily self-opposite so, as the authors point out, nothing so far prevents their example from being realized as a twisted convolution algebra. In fact, it appears that nobody knows an example of a $C^*$-algebra which is not a twisted convolution algebra!!

On the other hand, the vast majority of useful tools available to the study of (untwisted) groupoid $C^*$-algebras apply just as well for twisted ones, so perhaps the more relevant question is which $C^*$ algebras are isomorphic to a twisted convolution algebra.

As mentioned by @Simon, the best results so far leading to a "Fourier-like" decomposition of a $C^*$-algebra as a convolution algebra (of an etale groupoid) are based on the assumption that we already know the abelian subalgebra of functions supported on the object space. This abelian subalgebra is sometimes called a Cartan algebra, given the similarities with the homonymous concept from the theory of Lie groups.

One of the stickiest points in this subject is whether or not there exists a conditional expectation onto the given Cartan subalgebra (one of the main assumptions in Renault's result). The reason one might prefer not to require this condition is that there are many groupoids whose convolution algebra does not admit such a conditional expectation due to the fact that the underlying groupoid is not Hausdorff. Examples of this situation are very common such as groupoids arising from foliations and certain dynamical systems.

Another relevant question is whether or not the Cartan subalgebra is maximal abelian, a crucial feature in the Lie algebra version of this concept, as well as an important assumption in Renault's theory. Maximal commutativity is closely related to topological freenes of the associated groupoid (a concept borrowed from those dynamical systems in which most points have trivial isotropy group). In particular, in the example given by @Simon of $C^*(B\mathbb Z)$, the natural Cartan subalgebra is $\mathbb C$, which is clearly not maximal commutative at the same time that the action of the group $\mathbb Z$ on a point has too much isotropy!

Working with David Pitts (arXiv:1901.09683) we have found a characterization of twisted groupoid $C^*$-algebras which neither assumes conditional expectations nor maximal commutativity. Perhaps one of the simplest examples where our result applies is $$ C(S^1)\subseteq C([0,1]) $$ where we identify $C(S^1)$ as the subalgebra of periodic functions. Surprisingly, there is a twisted groupoid description of this "Cartan pair".

Significantly, the lack of a conditional expectation from $C([0,1])$ to $C(S^1)$ prevents the groupoid from being Hausdorff.

(Wait! I blundered in the descriprion of the groupoid. Will fix it and come back as soon as I can!)

As already pointed out, Buss and Sims have found an example of a $C^*$-algebra which is not isomorphic to its opposite, and hence it is not a groupoid $C^*$-algebra. However twisted groupoid $C^*$-algebras are not necessarily self-opposite so, as the authors point out, nothing so far prevents their example from being realized as a twisted convolution algebra. In fact, it appears that nobody knows an example of a $C^*$-algebra which is not a twisted convolution algebra!!

On the other hand, the vast majority of useful tools available to the study of (untwisted) groupoid $C^*$-algebras apply just as well for twisted ones, so perhaps the more relevant question is which $C^*$-algebras are isomorphic to a twisted convolution algebra.

As mentioned by @Simon, the best results so far leading to a "Fourier-like" decomposition of a $C^*$-algebra as a convolution algebra (of an etale groupoid) are based on the assumption that we already know the abelian subalgebra of functions supported on the object space. This abelian subalgebra is sometimes called a Cartan algebra, given the similarities with the homonymous concept from the theory of Lie groups.

One of the stickiest points in this subject is whether or not there exists a conditional expectation onto the given Cartan subalgebra (one of the main assumptions in Renault's result). The reason one might prefer not to require this condition is that there are many groupoids whose convolution algebra does not admit such a conditional expectation due to the fact that the underlying groupoid is not Hausdorff. Examples of this situation are very common such as groupoids arising from foliations and certain dynamical systems.

Another relevant question is whether or not the Cartan subalgebra is maximal abelian, a crucial feature in the Lie algebra version of this concept, as well as an important assumption in Renault's theory. Maximal commutativity is closely related to topological freenes of the associated groupoid (a concept borrowed from those dynamical systems in which most points have trivial isotropy group). In particular, in the example given by @Simon of $C^*(B\mathbb Z)$, the natural Cartan subalgebra is $\mathbb C$, which is clearly not maximal commutative at the same time that the action of the group $\mathbb Z$ on a point has too much isotropy!

Working with David Pitts (arXiv:1901.09683) we have found a characterization of twisted groupoid $C^*$-algebras which neither assumes conditional expectations nor maximal commutativity. Perhaps one of the simplest examples where our result applies is $$ C(S^1)\subseteq C([0,1]) $$ where we identify $C(S^1)$ as the subalgebra of periodic functions. Surprisingly, there is a twisted groupoid description of this "Cartan pair".

Significantly, the lack of a conditional expectation from $C([0,1])$ to $C(S^1)$ prevents the groupoid from being Hausdorff. In fact, the topological space underlying this groupoid is essentially the best known example of a non-Hausdorff topological space in which one takes the unit interval $[0,1]$ with a duplicate copy of the point 1, except that we instead duplicate a chosen point of $S^1$. The groupoid structure is such that the two duplicate points form a copy of the 2-element group, while all other points are considered to be objects. The twist is non trivial but it turns out to be the most natural nontrivial twist one can think of.