A cornerstone of stability theory is Voiculescu's 1983 counterexample: he constructed a sequence of pairs of unitary matrices whose commutators converge to zero in the operator norm, but whose distances from the set of commuting unitary pairs remain bounded away from zero. Namely, the group $\mathbb{Z}^2$ is not stable in the operator norm. We prove, somewhat surprisingly, that stability is restored after an asymptotically negligible enlargement of the dimension. That is, the group $\mathbb{Z}^2$ is flexibly stable in the operator norm. This provides the first example, in any context, of a flexibly stable group that is not stable. Building on a construction of Eckhardt, who produced finitely generated amenable groups that are very-flexibly stable but not flexibly stable in the normalized Hilbert-Schmidt norm, we show that the same groups exhibit the analogous separation in the operator norm: they are very-flexibly stable but not flexibly stable.
We present an example that shows that the double Hilbert transform $\H$ can have the characteristic function of an open set of finite measure in $\mathbb{R} \times \mathbb{R}$ as an approximate eigenvector for the eigenvalue $1$. The same holds for the dyadic model of $\H$, the double dyadic shift. We further construct...
E. Abakumov, K. Domelevo, S. Petermichl et al.· 0 citations
We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified in every case exc...
Dominique Guillot, Himanshu Gupta, P. K. Vishwakarma et al.· 0 citations
We construct a function $\Phi $ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight $\alpha =12$. Moreover, we show that $\Phi$ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by pro...
This survey paper provides a comprehensive study of unitarily invariant norms on the algebra of $n \times n$ matrices. This investigation leads naturally to the theory of symmetric gauge functions, a class of norms on $\mathbb{R}^n$ characterized by invariance and monotonicity properties. We develop the necessary frame...
S. Garcia, J. Mashreghi, M. Ptak et al.· 0 citations
For pairs $(G,H)$, where $G$ is a locally compact group and $H$ is a closed subgroup of $G$, we introduce analogous versions of the full group $\mathrm{C}^*$-algebra and the Fourier--Stieltjes algebra for uniformly bounded representations. We use these objects to characterise relative property (T) in this more general...
Smoothness of a function on the real line is reflected in the decay of its Fourier transform, which suggests that smoothness of a function in $L^2(G)$ for a group $G$ should mean concentration of the Fourier coefficients at low frequency. Such a reading presupposes an ordering of the irreducible representations of $G$,...
Zachary P. Bradshaw· 0 citations
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