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Whilst acquainting myself with the fundamentals of $C^*$-algebras and their $K$-theory, I read about how Gelfand Duality allows various $C^*$-algebraic concepts to be directly translated into a geometrical/topological context and vice versa. I briefly relay my current understanding of Gelfand duality below for the sake of clarity.


Each commutative $C^*$-algebra $A$ has a spectrum $\Delta$ consisting of its non-zero complex homomorphisms, which is a locally compact Hausdorff space with the Gelfand topology. Conversely, for each locally compact Hausdorff space $X$, the algebra of continuous complex-valued functions on $X$ that vanish at infinity forms a commutative $C^*$-algebra $C_0(X)$ w.r.t. pointwise conjugation and the supremum norm $f\mapsto\|f\|_\infty$.

The Abstract Spectral Theorem (Theorem 2.13 here) states that each commutative $C^*$-algebra $A$ with (necessarily non-empty) spectrum $\Delta$ is canonically isomorphic to $C_0(\Delta)$. This isomorphism is induced by the Gelfand transforms. Conversely, mapping each $x\in X$ to its point evaluation $f\mapsto f(x):C_0(X)\to\mathbb{C}$ gives a canonical homeomorphism $X\to \Delta(C_0(X))$. In a similar manner, we can match up proper continuous maps between LCH spaces with $*$-homomorphisms, and find that $C_0$ is a two-way contravariant functor from the category of LCH spaces to the category of commutative $C^*$-algebras.


The following table [Basic Noncommutative Geometry, Khalkhali, pg. 7] describes some correspondences that result from the Gelfand duality.

Space Algebra
compact unital
1-point compactification unitization
Stone–Cech compactification multiplier algebra
closed subspace; inclusion closed ideal; quotient algebra
surjection injection
injection surjection
homeomorphism automorphism
Borel measure positive functional
probability measure state
disjoint union direct sum
Cartesian product minimal tensor product

So commutative $C^*$-algebras correspond with locally compact Hausdorff spaces. Based on my rather limited understanding of noncommutative geometry, it would appear that Gelfand duality may be extended to arbitrary (not necessarily commutative) $C^*$-algebras, whose geometrical analogue is a noncommutative space.

But what exactly is a noncommutative space? I am aware that it might be difficult or downright impossible to give an explicit definition, so I simply hope to gain a better intuition here. How can I make sense of noncommutative spaces?

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    $\begingroup$ I'll be curious to see what real noncommutative geometers have to say, but my understanding is that a noncommutative space is the imaginary nonexistent thing whose nonexistent properties allow us to fill in the left hand column of that wonderful table in the noncommutative case. $\endgroup$
    – Lee Mosher
    Jan 9, 2022 at 16:29
  • $\begingroup$ The game is to use one's imagination to extend that table: what ought to go in the right hand column, for each possible extension of the left hand column? $\endgroup$
    – Lee Mosher
    Jan 9, 2022 at 16:31
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    $\begingroup$ @LeeMosher I'm by no means an expert in the topic but what you say in these comments seems to be exactly correct to me. $\endgroup$
    – J. De Ro
    Jan 9, 2022 at 19:21
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    $\begingroup$ If you just want to gain geometric intuition, restrict to the class of non-commutative spaces coming from foliations. The latter are quite geometric. $\endgroup$ Jan 11, 2022 at 20:01
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    $\begingroup$ Indeed, noncommutative orbit spaces (i.e., crossed product algebras) and noncommutative leaf spaces (i.e., groupoid algebras) already give a ton of insight into how “noncommutative spaces” arise in practice. The story of deformation quantisation is another important source of insight. I strongly recommend reading up on the irrational noncommutative $2$-torus/irrational rotation algebra as both a deformation quantisation of the $2$-torus and as the noncommutative orbit space of an irrational rotation of the circle, respectively. $\endgroup$ Jan 14, 2022 at 17:04

1 Answer 1

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It's important to realise that the phrase "non-commutative space" is not well-defined, in the sense that this name will mean different things to different people.

However, I will try to tell something more about $C^*$-algebras.

For convenience, I will stick with the unital case in this answer, but everything I say has non-unital counterparts.

Gelfand duality gives us a duality $$\mathrm{Compact \ Hausdorff \ spaces} \leftrightarrow \mathrm{Commutative \ unital \ C^* algebras}.$$ This allows us to think of commutative unital $C^*$-algebras as being classical compact Hausdorff spaces. Extending this to non-commutative $C^*$-algebras, a $C^*$-algebra is sometimes called a "non-commutative topological space" and the theory of $C^*$-algebras is referred to as non-commutative topology. Some authors sometimes take this a bit further, and denote an arbitrary $C^*$-algebra $A$ by $A = C(\mathbb{X})$ where $\mathbb{X}$ can be thought of as an "imaginary topological space". So, as a purely formal object, $\mathbb{X}$ does not exist or does not make sense, yet we think of the $C^*$-algebra $A=C(\mathbb{X})$ as being functions on the imaginary space $\mathbb{X}$.

There is a similar duality between commutative von Neumann algebras and certain measure spaces, which is why the theory of von Neumann algebras is sometimes referred to as being 'non-commutative measure theory'.


Next, I will tell something more about "non-commutative compact topological groups" where the non-commutative is in the sense of non-commutative geometry. We have a forgetful functor $$\mathrm{Compact \ Hausdorff \ groups}\to \mathrm{Compact \ Hausdorff \ spaces}$$ so it makes sense to ask how we can complete the correspondence $$\mathrm{Compact \ Hausdorff \ \ groups} \leftrightarrow \quad ???$$ in a way that the right hand side corresponds to certain commutative unital $C^*$-algebras (with extra structure encoding the multiplication of the group).

At the end of the eighties, Woronowicz realised that the right hand side should correspond to what is now called a (unital) Woronowicz $C^*$-algebra, i.e. a unital $C^*$-algebra $A$ together with a unital $*$-homomorphism $\Delta: A \to A \otimes_{\operatorname{min}} A$ that is coassociative, $$(\Delta \otimes \iota)\circ \Delta = (\iota \otimes \Delta)\circ \Delta,$$ and such that $$\overline{\Delta(A)(1 \otimes A)}^{\|\cdot\|}= A \otimes_{\operatorname{min}} A = \overline{\Delta(A)(A \otimes 1)}^{\|\cdot\|}\quad (*)$$

Where does this come from? Well, given a compact Hausdorff group $X$, the multiplication $X \times X \to X$ dualises to a comultiplication $$\Delta_X: C(X) \to C(X \times X) \cong C(X) \otimes C(X)$$ defined by $\Delta_X(f)(x,y) = f(xy)$ where $f \in C(X)$ and $x,y \in X$. The coassociativity of $\Delta_X$ corresponds to the associativity of the multiplication in $X$ and the condition $(*)$ corresponds to the fact that a group has inverses (which is why $(*)$ is referred to as 'quantum cancellation rules').

If the Woronowicz $C^*$-algebra $(A, \Delta)$ is commutative, it is necessarily of the form $(A, \Delta) \cong (C(X), \Delta_X)$ for some compact Hausdorff group $X$ and because of this, an arbitrary Woronowicz $C^*$-algebra $A$ is often denoted by $A= C(\mathbb{X})$ where $\mathbb{X}$ is again some imaginary object, which we refer to as being a compact quantum group. Again, as a formal object this does not exist or make sense, but it conveys the right intuition we have from the classical related theories to keep talking about objects $\mathbb{X}$ instead of talking about their related 'function algebras'.


I hope this answer helped a bit. Talking about imaginary objects can be somewhat counterintuitive and confusing at first.

Many other examples of this phenomenon occur in the world of (non-commutative) algebraic geometry, but I'll leave it to someone who actually knows about this to write an answer about this.

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  • $\begingroup$ Excellent answer! You said there is a forgetful functor from compact Hd groups to Compact Hd spaces. What do you mean by the former? Isn't it just the compact Hd topological groups? Because later you ask ther is a forgetful functor from Compact Hd top groups to what? I dont know the difference between Compact Hd group and Compact Hd Top group. $\endgroup$
    – Muddana
    Jun 6, 2023 at 15:48
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    $\begingroup$ @AksharaPrasad Edited my answer. With the forgetful functor, I mean you forget the group structure on the compact Hausdorff group, so that you end up with a compact Hausdorff topological space. $\endgroup$
    – J. De Ro
    Jun 6, 2023 at 20:54

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