Coherence and strictification for self-similarity

This paper studies questions of coherence and strictification related to self-similarity - the identity $S\cong S\otimes S$ in a (semi-)monoidal category. Based on Saavedra's theory of units, we first demonstrate that strict self-similarity cannot simultaneously occur with strict associativity -- i.e. no monoid may have a strictly associative (semi-)monoidal tensor, although many monoids have a semi-monoidal tensor associative up to isomorphism. We then give a simple coherence result for the arrows exhibiting self-similarity and use this to describe a `strictification procedure' that gives a semi-monoidal equivalence of categories relating strict and non-strict self-similarity, and hence monoid analogues of many categorical properties. Using this, we characterise a large class of diagrams (built from the canonical isomorphisms for the relevant tensors, together with the isomorphisms exhibiting the self-similarity) that are guaranteed to commute.Comment: Significant revisions from previous version: proofs simplified and based on Saavedra units & idempotent splitting, monoidal equivalences made explicit, expository sections significantly revised and shortened, notation and terminology revised and clarified, a clearer criterion for coherence give

Operators versus functions: from quantum dynamical semigroups to tomographic semigroups

Quantum mechanics can be formulated in terms of phase-space functions, according to Wigner's approach. A generalization of this approach consists in replacing the density operators of the standard formulation with suitable functions, the so-called generalized Wigner functions or (group-covariant) tomograms, obtained by means of group-theoretical methods. A typical problem arising in this context is to express the evolution of a quantum system in terms of tomograms. In the case of a (suitable) open quantum system, the dynamics can be described by means of a quantum dynamical semigroup 'in disguise', namely, by a semigroup of operators acting on tomograms rather than on density operators. We focus on a special class of quantum dynamical semigroups, the twirling semigroups, that have interesting applications, e.g., in quantum information science. The 'disguised counterparts' of the twirling semigroups, i.e., the corresponding semigroups acting on tomograms, form a class of semigroups of operators that we call tomographic semigroups. We show that the twirling semigroups and the tomographic semigroups can be encompassed in a unique theoretical framework, a class of semigroups of operators including also the probability semigroups of classical probability theory, so achieving a deeper insight into both the mathematical and the physical aspects of the problem.Comment: 12 page

Low Complexity Blind Equalization for OFDM Systems with General Constellations

This paper proposes a low-complexity algorithm for blind equalization of data in OFDM-based wireless systems with general constellations. The proposed algorithm is able to recover data even when the channel changes on a symbol-by-symbol basis, making it suitable for fast fading channels. The proposed algorithm does not require any statistical information of the channel and thus does not suffer from latency normally associated with blind methods. We also demonstrate how to reduce the complexity of the algorithm, which becomes especially low at high SNR. Specifically, we show that in the high SNR regime, the number of operations is of the order O(LN), where L is the cyclic prefix length and N is the total number of subcarriers. Simulation results confirm the favorable performance of our algorithm

Group algebras acting on LpL^p-spaces

For $p\in [1,\infty)$ we study representations of a locally compact group $G$ on $L^p$-spaces and $QSL^p$-spaces. The universal completions $F^p(G)$ and $F^p_{\mathrm{QS}}(G)$ of $L^1(G)$ with respect to these classes of representations (which were first considered by Phillips and Runde, respectively), can be regarded as analogs of the full group \ca{} of $G$ (which is the case $p=2$). We study these completions of $L^1(G)$ in relation to the algebra $F^p_\lambda(G)$ of $p$-pseudofunctions. We prove a characterization of group amenability in terms of certain canonical maps between these universal Banach algebras. In particular, $G$ is amenable if and only if $F^p_{\mathrm{QS}}(G)=F^p(G)=F^p_\lambda(G)$. One of our main results is that for $1\leq p< q\leq 2$, there is a canonical map $\gamma_{p,q}\colon F^p(G)\to F^q(G)$ which is contractive and has dense range. When $G$ is amenable, $\gamma_{p,q}$ is injective, and it is never surjective unless $G$ is finite. We use the maps $\gamma_{p,q}$ to show that when $G$ is discrete, all (or one) of the universal completions of $L^1(G)$ are amenable as a Banach algebras if and only if $G$ is amenable. Finally, we exhibit a family of examples showing that the characterizations of group amenability mentioned above cannot be extended to $L^p$-operator crossed products of topological spaces.Comment: Version 1: 27 pages. Version 2: lots of minor corrections, and we got rid of the second-countability assumption on the groups. 31 page

Cyclic-Coded Integer-Forcing Equalization

A discrete-time intersymbol interference channel with additive Gaussian noise is considered, where only the receiver has knowledge of the channel impulse response. An approach for combining decision-feedback equalization with channel coding is proposed, where decoding precedes the removal of intersymbol interference. This is accomplished by combining the recently proposed integer-forcing equalization approach with cyclic block codes. The channel impulse response is linearly equalized to an integer-valued response. This is then utilized by leveraging the property that a cyclic code is closed under (cyclic) integer-valued convolution. Explicit bounds on the performance of the proposed scheme are also derived