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Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries (assume that the resulting representation is strongly continuous).

Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) on Borel subsets of $B_X$, the unit ball of $X$, that is invariant under this action? If not in general then under what assumptions?

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries.

Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) on Borel subsets of $B_X$, the unit ball of $X$, that is invariant under this action? If not in general then under what assumptions?

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries (assume that the resulting representation is strongly continuous).

Is there a non-atomic probability measure on Borel subsets of $B_X$, the unit ball of $X$, that is invariant under this action? If not in general then under what assumptions?

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries (assume that the resulting representation is strongly continuous).

Is there a probability measure (non-atomic, not supported on a finite dimensional subspace) on Borel subsets of $B_X$, the unit ball of $X$, that is invariant under this action? If not in general then under what assumptions?

Let $\Gamma$ be a discrete group acting on an infinite dimensional Banach space $X$ by linear isometries (assume that the resulting representation is a strongly continuous).

Is there an probability measure on $B_X$, the unit ball of $X$, that is invariant under this action? If not in general then under what assumptions?

Let $\Gamma$ be a discrete group acting on an infinite-dimensional Banach space $X$ by linear isometries (assume that the resulting representation is strongly continuous).

Is there a non-atomic probability measure on Borel subsets of $B_X$, the unit ball of $X$, that is invariant under this action? If not in general then under what assumptions?