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Does the convolution $C^*$-algebra of geometric $n$-groupoidslocally compact Hausdorff groupoids recover back the respective $n$-groupoidgroupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.

Let $X$ be a topological, Lie, localic or even algebraiclocally compact Hausdorff groupoid $n$-groupoid with any non-discrete notion of Haar measure(or Lie groupoid). One can, for $n = 1$, then, define $C := C^* (X, A)$ through convolution of functions from $X_1$ to some ring $A$ (I'm ok with just $\mathbb{C}$ here).

Question: Can one recover $X$ back from $C$ (maybe up to Morita equivalence)?

The above definition seems to make sense too for increased categorical dimension, i.e., $n$-groupoids. See, for instance, https://ncatlab.org/nlab/show/category%20algebra#convolution_algebras .

Can one recover $X$ back from $C$ (maybe up to Morita equivalence)?

As mentioned above, I'm interested in any non There are also other possible choices of geometry by taking internal ($n$)-discrete case including localic onesgroupoids inside other categories. For instance, one can still talk about algebraic or (be it internalinternal or not to some topoi). Also an answer to the case $n =1$ would already be extremely helpful. I would be very happy too to hear about generalisations for localic $n$-groupoids even if it's just something conjecturaland maybe construct a convolution algebra as long as a notion of Haar measure makes sense.

Extra question: Any idea about the case of $n$-groupoids internal to any of the mentioned above categories (topological, differentiable, algebraic etc) even if it's just something conjectural?

Thanks in advance.

Does the convolution $C^*$-algebra of geometric $n$-groupoids recover back the respective $n$-groupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.

Let $X$ be a topological, Lie, localic or even algebraic $n$-groupoid with any non-discrete notion of Haar measure. One can, for $n = 1$, then, define $C := C^* (X, A)$ through convolution of functions from $X_1$ to some ring $A$ (I'm ok with just $\mathbb{C}$ here). The definition seems to make sense too for increased categorical dimension, i.e., $n$-groupoids. See, for instance, https://ncatlab.org/nlab/show/category%20algebra#convolution_algebras .

Can one recover $X$ back from $C$ (maybe up to Morita equivalence)?

As mentioned above, I'm interested in any non-discrete case including localic ones (be it internal or not to some topoi). Also an answer to the case $n =1$ would already be extremely helpful. I would be very happy too to hear about generalisations for $n$-groupoids even if it's just something conjectural.

Thanks in advance.

Does the convolution $C^*$-algebra of locally compact Hausdorff groupoids recover back the respective groupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.

Let $X$ be a locally compact Hausdorff groupoid (or Lie groupoid). One can, then, define $C := C^* (X, A)$ through convolution of functions from $X_1$ to some ring $A$ (I'm ok with just $\mathbb{C}$ here).

Question: Can one recover $X$ back from $C$ (maybe up to Morita equivalence)?

The above definition seems to make sense too for increased categorical dimension, i.e., $n$-groupoids. See, for instance, https://ncatlab.org/nlab/show/category%20algebra#convolution_algebras . There are also other possible choices of geometry by taking internal ($n$)-groupoids inside other categories. For instance, one can still talk about algebraic or (internal or not) localic $n$-groupoids and maybe construct a convolution algebra as long as a notion of Haar measure makes sense.

Extra question: Any idea about the case of $n$-groupoids internal to any of the mentioned above categories (topological, differentiable, algebraic etc) even if it's just something conjectural?

Thanks in advance.

Source Link
user40276
user40276
  • 2.2k
  • 1
  • 17
  • 28

Does the convolution $C^*$-algebra of geometric $n$-groupoids recover back the respective $n$-groupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known.

Let $X$ be a topological, Lie, localic or even algebraic $n$-groupoid with any non-discrete notion of Haar measure. One can, for $n = 1$, then, define $C := C^* (X, A)$ through convolution of functions from $X_1$ to some ring $A$ (I'm ok with just $\mathbb{C}$ here). The definition seems to make sense too for increased categorical dimension, i.e., $n$-groupoids. See, for instance, https://ncatlab.org/nlab/show/category%20algebra#convolution_algebras .

Can one recover $X$ back from $C$ (maybe up to Morita equivalence)?

As mentioned above, I'm interested in any non-discrete case including localic ones (be it internal or not to some topoi). Also an answer to the case $n =1$ would already be extremely helpful. I would be very happy too to hear about generalisations for $n$-groupoids even if it's just something conjectural.

Thanks in advance.