The main result of this note, Theorem 1.3, is the following: a Borel measure on the space of infinite Hermitian matrices, that is invariant and ergodic under the action of the infinite unitary group and that admits well-defined projections onto the quotient space of “corners" of finite size, must be finite. A similar result, Theorem 1.1, is also established for unitarily invariant measures on the space of all infinite complex matrices. These results imply that the infinite Hua-Pickrell measures...

We study some embeddings of suitably topologized spaces of vector-valued smooth functions on topological groups, where smoothness is defined via differentiability along continuous one-parameter subgroups. As an application, we investigate the canonical correspondences between the universal enveloping algebra, the invariant local operators, and the convolution algebra of distributions supported at the unit element of any finite-dimensional Lie group, when one passes from finite-dimensional Lie groups...

A unitary representation $\pi$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\mathtt{d}\pi \left(x\right)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$. We classify all irreducible semibounded representations of the groups ${\widehat{ℒ}}_{\phi }\left(K\right)$ which are double extensions of the twisted loop group ${ℒ}_{\phi }\left(K\right)$, where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi$ is...