Strong Approximation of $L^p$ Convolution Operators

Definitions and mathematical statement

For a Banach space $X$, let $\mathcal B(X)$ denote the bounded linear operators on $X$. The strong operator topology on $\mathcal B(X)$ is the topology of pointwise norm convergence: a net $(T_i)$ converges strongly to $T$ if $\|T_i\xi-T\xi\|_X\to0$ for every $\xi\in X$.

The supplied problem asks whether every bounded left convolution operator $\lambda_p(f)$ on $L^p(G)$ satisfies \[ \lambda_p(f)\in\overline{\lambda_p({\mathbb{C}}\Gamma)}^{\mathrm{SOT}}. \] Here ${\mathbb{C}}\Gamma$ denotes the complex group algebra, whose elements are finitely supported functions on the group $\Gamma$. Provided the operators in this display are well defined on the same Banach space $L^p(G)$, the requested membership means that, for every integer $m\geq1$, every $\xi_1,\ldots,\xi_m\in L^p(G)$, and every $\varepsilon>0$, there exists $a\in{\mathbb{C}}\Gamma$ such that \[ \max_{1\leq j\leq m} \|\lambda_p(f)\xi_j-\lambda_p(a)\xi_j\|_{L^p(G)}<\varepsilon. \] The approximating element $a$ may depend on all the test vectors and on $\varepsilon$.

Unresolved qualifications. The supplied statement uses both $G$ and $\Gamma$, without identifying them or specifying how $\Gamma$ acts on $L^p(G)$. It supplies neither a range for $p$, a measure defining $L^p(G)$, nor the admissible class of kernels $f$ and the meaning of their bounded convolution operators. The report title suggests a discrete-group context, but does not resolve these omissions. Consequently, the displayed approximation condition precisely explains the stated closure question only after these data are supplied; it does not select an intended group, exponent range, or convolution class.

Short English statement

The question asks whether each specified bounded convolution operator can be approximated simultaneously on any finite collection of $L^p$ vectors by convolution operators with finitely supported coefficients. The relation between $G$ and $\Gamma$, the allowed exponents, and the convolution domain remain unspecified in the supplied record.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30001791$, title Strong Approximation of $L^p$ Convolution Operators, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2011; no individual proposer is named.
• Supplied report citation: Finite-dimensional Approximations of Discrete Groups (2011), Oberwolfach Reports, pp. 1429--1467, DOI 10.4171/owr/2011/26. Source URL and literature-source link labelled "original problem": https://doi.org/10.4171/owr/2011/26.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open and reports that the strong operator closure question remains open outside classes such as weakly amenable groups or groups with the approximation property. This is supplied dated curation, not an independently verified assessment; its time-sensitive status, cited claims, and applicable hypotheses require independent verification.
• The registry marks the statement corrected_verified and records a manual semantic, typographic, and LaTeX review claiming a self-contained, meaning-preserving cleanup. This supplied review label does not resolve the formulation omissions identified here or establish mathematical correctness.