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- W2969706095 abstract "Let $mathcal{A}$ be a unital C$^*$-algebra. Given a faithful representation $mathcal{A}subsetmathcal B(mathcal{L})$ in a Hilbert space $mathcal{L}$, the set $G^+subsetmathcal{A}$ of positive invertible elements can be thought of as the set of inner products in $mathcal{L}$, related to $mathcal{A}$, 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 $mathcal{A}$, with an invariant Finsler metric. In the present paper we study the tangent bundle $TG^+$ of $G^+$, as a homogeneous Finsler space of a natural group of invertible matrices in $M_2(mathcal{A})$, identifying $TG^+$ with the Poincaré half-space $mathcal H$ of $mathcal{A}$, $$ mathcal H={hinmathcal{A}: {rm Im}(h)ge 0, {rm Im}(h) hbox{ invertible}}. $$ We show that $mathcal Hsimeq TG^+$ has properties similar to those of a space of non-positive constant curvature." @default.
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- W2969706095 date "2019-08-20" @default.
- W2969706095 modified "2023-09-25" @default.
- W2969706095 title "The Poincaré half-space of a C$^*$-algebra" @default.
- W2969706095 doi "https://doi.org/10.4171/rmi/1117" @default.
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