My Research

My research lies in  the beautiful field of C*-algebras. The following is a very short introduction to the type of objects I study.

C*-algebras

The notion of a C*-algebra arises naturally by observing the structure of \(\mathcal{B(H)}\), the algebra of bounded linear operators between Hilbert spaces, and then generalizing it. In particular, a C*-algebra \(U\) is a Banach algebra with an involution operation, \(*: U \rightarrow U\), that satisfies $$(\alpha A + \beta B)^{*} = \bar{\alpha}A^{*}+\bar{\beta}B^{*},$$ $$(AB)^{*}=B^{*}A^{*},$$ and lastly $$(A^{*})^{*}=A,$$ for \(A,B \in U\) and \(\alpha,\beta \in \mathrm{C}\).

The C*-algebra \(U\) must also satisy the C*-equation $$\vert\vert{A^{*}A}\vert\vert=\vert\vert{A}\vert\vert^{2} .$$

\(\mathcal{B(H)}\) is a C*-algebra where the multiplication is composition of operators and the involution is the adjoint of an operator. Another important C*-algebra is \(C(X)\), the space of continuous functions on a compact Hausdorff space \(X\). With pointwise multiplication as multiplication, and conjugation of functions as the involution, this space admits the properties above. Whilst this algebra is Abelian, \(\mathcal{B(H)}\) is not.

A fundamental theorem, due to Gelfand and Neumark, says that every C*-algebra has a faithful representation; ie it can be imbedded into \(\mathcal{B(H)}\) for some Hilbert space \(\mathcal{H}\). The space \(\mathcal{H}\) can be obtained constructively by a method known as the Gelfand-Neumark-Segal (GNS) construction.

In the case where the algebra is unital and Abelian, it is in fact isometrically \(*\)-isomorpic to \(C(K)\) for some compact Hausdorff space \(K\), obtained as the pure state space of the original algebra.

A fantastic book on C*-algebras is \(\text{Fundamentals of the Theory of Operator Algebras}\) by Richard Kadison and John Ringrose.

 

Groupoid C*-algebras

A groupoid is a mathematical object that generalizes the notion of a group. Indeed, it is a set \(G\) containing a distinguished subset \(G^{2} \subseteq G \times G\), a product map \(p: G^{2} \rightarrow G,  (g,h) \to gh \) and an idempotent \(G \rightarrow G, \; \; g \to g^{-1}\); satisfying the following conditions:

  1. If \((g,h) \in G^{2}\) and \((h,k) \in G^{2}\) then \((gh,k)\) and  \((g,hk)\) are both in \(G^{2}\), and their images under the product map are equal.
  2. For all \(g\in G\), both \((g,g^{-1})\) and \((g^{-1},g)\) belong to  \(G^{2}\).
  3. Whenever \((g,h) \in G^{2}\) we have that \(g^{-1}gh=h\) and whenever \((h,g) \in G^{2} \) we have that \(hgg^{-1} = h\).

Alternatively, you can define a groupoid as a small category with inverses. Whilst in a group there is only one identity, in a groupoid you may have many of them: indeed the unit space \(G^{0}\) is the set \(\{gg^{-1}  :   g \in G\}\). A groupoid with a singleton unit space is a group, and of course every group is a groupoid in the natural sense.

There is an easy-to-prove classification theorem for groupoids: namely that for every groupoid \(G\) we have a groupoid isomorphism \(G \cong \bigsqcup\limits_{i \in I}G_{i} \times (X_{i} \times X_{i})\), where the \(G_{i}\)‘s are groups and the \(X_{i}\)‘s are equivalence relations – all obtained from \(G\).

Hence there is not much to say, algebraically, about groupoids. However by giving them a topology the groupoids become much richer and more interesting to study. In particular, an etale groupoid is a specific topological groupoid that makes the source and range maps, namely the maps defined by \(s(g )= g^{-1}g\)  and \(r(g) = gg^{-1}\), local homeomorphisms.

Now in the same style that one constructs C*-algebras from groups, one may construct a C*-algebra from a locally compact Hausdorff etale groupoid. Indeed, one starts by defining a multiplication (that behaves like convolution) and an involution on the space \(C_{C}(G)\) of a all complex-valued continuous compactly supported functions on \(G\). As usual, one then considers *-representations of this space and then defines the corresponding C*-algebra as the completion of this *-algebra under the universal norm.

A lot of interesting C*-algebras can be reproduced as groupoid C*-algebras. In particular AF C*-algebras, crossed product C*-algebras and group C*-algebras all have groupoid C*-algebraic reformulations. This is also true for the Toeplitz and (more generally) the Cuntz algebra.

 

Cartan subalgebras of C*-algebras

A Cartan subalgebra \(B\) of \(A\) is a C*-subalgebra that is a regular m.a.s.a.,  which contains an approximate unit of \(A\) and admits a faithful conditional expectation of \(A\) onto \(B\).

For example, the C*-algebra \(M_{n}\) of \(n\) by \(n\) matrices with complex entries has a Cartan subalgebra, namely that consisting of all the diagonal matrices. The crossed product C*-algebra \(C(X) \times_{\phi} \mathbb{Z}\), where \(X\) is a compact Hausdorff space and \(\phi\) is a homeomorphism of \(X\), has \(C(X)\) as a Cartan subalgebra.

Cartan subalgebras retain a lot of information about the C*-algebra in which they live. For the example of the crossed product C*-algebra, it turns out that simplicity of the ambient C*-algebra can be determined by the simplicity of the Cartan subalgebra. In other cases, nuclearity of the ambient C*-algebra is a result of nuclearity for the Cartan subalgebra.

It is also so that Cartan subalgebras provide an important bridge that connects C*-algebras with topological dynamical systems and geometric group theory. It is well know how to construct C*-algebras from dynamical systems and groups. But by introducing the notions of continuous orbit equivalence for dynamical systems, and quasi-isometry for geometric group theory, it turns out that Cartan subalgebras allow us to go from C*-algebras back into these fields – and hence have a total equivalence.

There is a classification theorem for Cartan subalgebras. Indeed, given a locally compact Hausdorff effective twisted etale groupoid \((G,\Sigma)\), one can construct a Cartan subalgebra, namely \(C_{0}(G^{0})\) inside \(C^{*}_{r}(G,\Sigma)\). Conversely, given any Cartan subalgebra \(B\) of \(A\), there exists a twisted etale locally compact Hausdorff effective groupoid \((G,\Sigma)\) and an isomorphism that carries \(C^{*}_{r}(G,\Sigma)\) onto \(A\) and \(C_{0}(G^{0})\) onto \(B\).

There are a lot of interesting open questions regarding Cartan subalgebras, especially considering existence and uniqueness. Some C*-algebras, like the reduced group C*-algebra formed from a non-Abelian free group, has no Cartan subalgebras. However lots of classifiable C*-algebras have a lot of Cartan subalgebras.  An open question is: does every classifiable C*-algebra has a Cartan subalgebra? Another open question is: does every simple seperable C*-algebra satisfy the UCT? We know that the simple seperable C*-algebras that have a Cartan subalgebra do, so the question can be reduced to an existence question of Cartan subalgebras.

We may also consider uniqueness. The Roe algebra, for instance, has a unique Cartan subalgebra. Voiculescu and Stratila have constructed a canonical Cartan subalgebra for AF-algebras. One may ask to what extent are such canonical Cartan subalgebras unique. Can we find generic conditions on the Cartan subalgebras which makes every such subalgebra unique, up to isomorphism?

For further information, and references, please see the Personal Notes section.