Department Colloquium

Location: RI 208

Speaker: Sushil Singla, University of Regina

Title:  Non-Linear Classification of Finite-dimensional C*-Algebras

Abstract:

One of the most important concepts in Euclidean geometry is that of orthogonality. This concept not only appears in the axioms of Euclidean geometry, but also in many fundamental theorems such as the Pythagoras theorem. Birkhoff-James orthogonality is a generalization of usual orthogonality in a normed space \(V\). An element \(v \in V\) is said to be Birkhoff-James orthogonal to a subspace \(W\) of \(V\) if \(\|v\| \leq \|v − w\|\) for all w ∈ W. Let \(A\) be a (real or complex) \(C^*\)-algebra. Let \(a \in A\) and \(B\) be a subspace of \(A\). We show that \(a\) is Birkhoff-James orthogonal to \(B\) if and only if there exists a state \(\Phi\) on \(A\) such that

\(\Phi(a^* a) = \|a\|^2\) and \(\Phi(a^*b) = 0\) for all \(b \in B\).

The notions of Birkhoff-James orthogonality and the best approximations go hand in hand. The above can be rewritten as follows. If \(b_0\) is a best approximation to \(a\) in \(B\) (equivalently dist(\(a, B) = \|a − b_0 \|\)), then there exists a state \(\Phi\) on \(A\) such that

\(\Phi(a^* a) + \Phi(b^* _0 b_0) = \|a − b_0\|^2\),

which is an analogue of the Pythagoras theorem in \(C^*\)-algebras.

We will use the notation \(a \bot_{BJ}\, b\) when \(a\) is Birkhoff-James orthogonal to the one-dimensional subspace generated by \(b\). We say two \(C^*\)-algebras \(A_1\) and \(A_2\) are BJ-isomorphic if there exists a bijection \(T : A_1 \rightarrow A_2\) which strongly preserves BJ orthogonality, equivalently, \(u \bot_{BJ}\, v \Leftrightarrow T(u) \bot_{BJ} T(v)\). We will conclude the talk by proving the following. Suppose two finite-dimensional \(C^*\)-algebras \(A_1\) and \(A_2\) of dimensions greater than or equal to two, are BJ-isomorphic. Then, they have the same underlying fields and are \(C^*\)-isomorphic.