Angle preserving transformations of Hilbert bundles

Ngai-Ching Wong »ÆÒãÇà

*(National Sun Yat-sen University)*

9:00-10:00,April 11,2023 A503

__Abstract:__

Let x,y be two vectors in a (real or complex) Hilbert C*- module H over a
C*- algebra A. The angle ¡Ï(x,y) between x and y can be defined in several ways. When
A = C_0(X) is a commutative C*- algebra, in other words, H is a continuous field of Hilbert
spaces over a locally compact space X, we define the cosine of the angle, u = cos¡Ï(x,y) ¡Ê C(X)
, by the equation |hx,yi| = |x||y|u . We show that if T: H ¡ú K is a linear module map between
two Hilbert C_0(X)-modules preserving (cosines of) non-flat angles, then T = ¦ÁJ for a bounded,
strictly positive and continuous scalar function ¦Á on X and a module into isometry J : H ¡ú K.
When A is noncommutative, we discuss those transformations preserving orthogonality and
parallelism.

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