How to understand what a 'noncommutative space' is

Question

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.

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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.

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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?

Answer 1

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'.

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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'.

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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.