Poncar\'e half-space of a C*-algebra

TL;DR

This paper explores the geometric structure of the set of positive invertible elements in a C*-algebra, revealing its relation to a Poincaré half-space and properties akin to non-positive constant curvature spaces.

Contribution

It identifies the tangent bundle of the positive invertible elements as a Poincaré half-space within a C*-algebra, establishing its geometric properties and curvature characteristics.

Findings

The tangent bundle of positive invertible elements is isomorphic to a Poincaré half-space.

The space exhibits properties similar to non-positive constant curvature spaces.

The study extends geometric understanding of C*-algebra structures.

Abstract

Let $A$ be a C$*^$-algebra. Given a representation $A\subset B(L)$ in a Hilbert space $L$, the set $G^+\subset A$ of positive invertible elements can be thought as the set of inner products in $L$, related to $A$, which are equivalent to the original inner product. The set $G^+$ has a rich geometry, it is a homogeneous space of the invertible group $G$ of $A$, with an invariant Finsler metric. In the present paper we study the tangent bundle $TG^+$ of $G^+$, as a homogenous Finsler space of a natural group of invertible matrices in $M_2(A)$, identifying $TG^+$ with the {\it Poincar\'e halfspace} $H$ of $A$, $$ H=\{h\in A: Im(h)\ge 0, Im(h) \hbox{ invertible}\}. $$ We show that $\h\simeq TG^+$ has properties similar to those of a space of non-positive constant curvature.