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Many important physical processes can be described by differential equations. The solutions of such equations are often formulated in terms of operators on smooth manifolds. A natural question is to determine whether differential structures defined on fractals can be realized as a metric limit of differential structures on their approximating finite graphs. One of the fundamental tools of noncommutative geometry is Alain Connes’ spectral triple. Because spectral triples generalize differential structure, they open up promising avenues for extending analytic methods from mathematical physics to fractal spaces. The Gromov-Hausdorff distance is an important tool of Riemannian geometry and building on the earlier work of Marc Rieffel, Frederic Latremoliere introduced a generalization of the Gromov-Hausdorff distance that was recently extended to spectral triples. The class of piecewise $C^1$-fractal curves was first characterized by Michel Lapidus and Jonathan Sarhad as a generalized setting for the spectral triple construction developed by Christensen, Ivan, and Lapidus in the context of the Sierpinski gasket. We provide an analytic framework for the metric approximation of the Lapidus-Sarhad spectral triple on a piecewise $C^1$-fractal curve by spectral triples defined on an approximating sequence of finite graphs which exhibit properties motivated by the setting of the Sierpinski gasket.

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be maximal subdiagonal algebra of $\mathcal{M}$. We prove Stein-Weiss type interpolation theorem of Haagerup noncommutative $H^{p}$-spaces associated with $\A$., Comment: 16. arXiv admin note: text overlap with arXiv:2209.08542

We show that if the cochain complex computing Ext groups (in the category of modules over Hopf algebroids) admits a cocyclic structure, then the noncommutative Cartan calculus structure on Tor over Ext dualises in a cyclic sense to a calculus on Coext over Cotor. More precisely, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functors lead to complexes that compute the more exotic ones, giving a cyclic opposite module over an operad with multiplication that induce operations such as a Lie derivative, a cap product (or contraction), and a (cyclic) differential, along with higher homotopy operators defining a noncommutative Cartan calculus up to homotopy. In particular, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley-Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness conditions or the use of topological tensor products., Comment: 31 pages; to appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class., 45 pages

We give a noncommutative geometric description of the internal Hom dg-category in the homotopy category of dg-categories between two noncommutative projective schemes in the style of Artin-Zhang. As an immediate application, we give a noncommutative projective derived Morita statement along lines of Rickard and Orlov.

Let R be an algebra. Given a noncommutative polynomial f, let f(R) stand for the additive subgroup of R generated by the image of f. For a unital or an affine algebra R, $$S_k(R)$$ is completely determined for any standard polynomial $$S_k$$ when R is generated by $$S_k(R)$$ as an ideal. Motivated by Bresar’s paper [Adv. Math. 374 (2020), 107346, 21 pp] and Robert’s paper [J. Oper. Theory 75 (2016), 387–408], under certain conditions we also prove that f(R) is equal to either [R, R] or the whole ring R. We obtain these results by studying the structure of Lie ideals L of a ring R whenever R is generated by [R, L] as an ideal.

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of noncommutative matrix factorization for an arbitrary nonzero non-unit element of a ring. First we show that the category of noncommutative graded matrix factorizations is invariant under the operation called twist (this result is a generalization of the result by Cassidy-Conner-Kirkman-Moore). Then we give two category equivalences involving noncommutative matrix factorizations and totally reflexive modules (this result is analogous to the famous result by Eisenbud for commutative hypersurfaces). As an application, we describe indecomposable noncommutative graded matrix factorizations over skew exterior algebras., 25 pages

In this note, we answer two questions about Gurariĭ spaces asked in the literature in the affirmative. We also prove the analogue of one of the results for the noncommutative Gurariĭ space.

We consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $U\to u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated to functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$., 52 pages, minor changes to text

We study the noncommutative Poincare duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich's deformation quantization as well as Koszul duality preserve the corresponding Poincare duality. As a corollary, the Batalin-Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.

Classically, Jensen’s Inequality asserts that if $X$ is a compact convex set, and $f:K\to {\mathbb {R}}$ is a convex function, then for any probability measure $\mu $ on $K$, that $f(\text {bar}(\mu ))\le \int f\; \text {d}\mu $, where $\text {bar}(\mu )$ is the barycenter of $\mu $. Recently, Davidson and Kennedy proved a noncommutative (“nc”) version of Jensen’s inequality that applies to nc convex functions, which take matrix values, with probability measures replaced by ucp maps. In the classical case, if $f$ is only a separately convex function, then $f$ still satisfies the Jensen inequality for any probability measure that is a product measure. We prove a noncommutative Jensen inequality for functions that are separately nc convex in each variable. The inequality holds for a large class of ucp maps that satisfy a noncommutative analogue of Fubini’s theorem. This class of ucp maps includes any free product of ucp maps built from Boca’s theorem, or any ucp map that is conditionally free in the free-probabilistic sense of Młotkowski. As an application to free probability, we obtain some operator inequalities for conditionally free ucp maps applied to free semicircular families.

We study multiplication modules. The rings are not assumed to be commutative. Several criteria with some applications are given for a direct sum of modules to be a multiplication module.

We obtain a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m({\bf D})$, as well as Eschmeier and Langendorfer extension to the unit ball of ${\bf C}^n$. All our results are proved in the more general setting of noncommutative poly-hyperballs ${\bf D_n^m}(H)$, ${\bf n,m}\in {\bf N}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on poly-hyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel $$ \kappa_{\bf m}(z,w):=\prod_{i=1}^k \frac{1}{(1-\bar z_i w_i)^{m_i}},\qquad z,w\in {\bf D}^k, $$ where $m_i\geq 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

Let$\mathcal {M}$be a semifinite von Nemann algebra equipped with an increasing filtration$(\mathcal {M}_n)_{n\geq 1}$of (semifinite) von Neumann subalgebras of$\mathcal {M}$. For$0, let$\mathsf {h}_p^c(\mathcal {M})$denote the noncommutative column conditioned martingale Hardy space and$\mathsf {bmo}^c(\mathcal {M})$denote the column “little” martingale BMO space associated with the filtration$(\mathcal {M}_n)_{n\geq 1}$.We prove the following real interpolation identity: if$0and$0, then for$1/r=(1-\theta )/p$,$$ \begin{align*} \big(\mathsf{h}_p^c(\mathcal{M}), \mathsf{bmo}^c(\mathcal{M})\big)_{\theta, r}=\mathsf{h}_{r}^c(\mathcal{M}), \end{align*} $$with equivalent quasi norms.For the case of complex interpolation, we obtain that if$0and$0, then for$1/r =(1-\theta )/p +\theta /q$,$$ \begin{align*} \big[\mathsf{h}_p^c(\mathcal{M}), \mathsf{h}_q^c(\mathcal{M})\big]_{\theta}=\mathsf{h}_{r}^c(\mathcal{M}) \end{align*} $$with equivalent quasi norms.These extend previously known results from$p\geq 1$to the full range$0. Other related spaces such as spaces of adapted sequences and Junge’s noncommutative conditioned$L_p$-spaces are also shown to form interpolation scale for the full range$0when either the real method or the complex method is used. Our method of proof is based on a new algebraic atomic decomposition for Orlicz space version of Junge’s noncommutative conditioned$L_p$-spaces.We apply these results to derive various inequalities for martingales in noncommutative symmetric quasi-Banach spaces.

Fuzzy tori are finite dimensional C*-algebras endowed with an appropriate notion of noncommutative geometry inherited from an ergodic action of a finite closed subgroup of the torus, which are meant as finite dimensional approximations of tori and more generally, quantum tori. A mean to specify the geometry of a noncommutative space is by constructing over it a spectral triple. We prove in this paper that we can construct spectral triples on fuzzy tori which, as the dimension grow to infinity and under other natural conditions, converge to a natural spectral triple on quantum tori, in the sense of the spectral propinquity. This provides a formal assertion that indeed, fuzzy tori approximate quantum tori, not only as quantum metric spaces, but as noncommutative differentiable manifolds -- including convergence of the state spaces as metric spaces and of the quantum dynamics generated by the Dirac operators of the spectral triples, in an appropriate sense., v. 4: 75 pages, various typos corrections, presentation improved

The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focussed on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras., 20 pages

We fully describe the doubly stochastic orbit of a self-adjoint element in the noncommutative $$L_1$$ -space affiliated with a semifinite von Neumann algebra, which answers a problem posed by Alberti and Uhlmann (Stochasticity and partial order: doubly stochastic maps and unitary mixing. VEB Deutscher Verlag der Wissenschaften, Berlin, 1982) in the 1980s, extending several results in the literature. It follows further from our methods that, for any $$\sigma $$ -finite von Neumann algebra $${{\mathcal {M}}}$$ equipped with a semifinite infinite faithful normal trace $$\tau $$ , there exists a self-adjoint operator $$y\in L_1({{\mathcal {M}}},\tau )$$ such that the doubly stochastic orbit of y does not coincide with the orbit of y in the sense of Hardy–Littlewood–Polya, which confirms a conjecture by Hiai (J Math Anal Appl 127:18–48, 1987). However, we show that Hiai’s conjecture fails for non- $$\sigma $$ -finite von Neumann algebras. The main result of the present paper also answers the (noncommutative) infinite counterparts of problems due to Luxemburg (Proc Symp Anal Queen’s univ 83–144, 1967) and Ryff (Pac J Math 13:1379–1386, 1963) in the 1960s.

In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$ , as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$ . The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.

Let R be a noncommutative prime ring equipped with an antiautomorphism f such that (i) [f(xk), xm] = 0, all x ∈ R, where k, m are fixed positive integers. In this article, we establish that from (i...

By using a Wick symbol transform, a characterization theorem is proven for the white noise operators in terms of a new spaces of (p, q)-entire functions with certain growth rates determined by Young functions and a suitable (p, q)-exponential map. Next, we investigate the main aim of this paper which is to give unitary equivalent representations of the (p, q)-square oscillator white noise algebra in the basis of noncommutative white noise theory.

In this work we consider all metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \C and \R up to dimension 5 and all metric Lie algebras over \C in dimension 6. We introduce cyclic and reduced cyclic cohomology to identify their metric deformations., Comment: 27 pages, 1 figure

Let D be a noncommutative division ring with centre F and V a right vector space over D. Among results, we show that the automorphism group of V does not satisfy any generalized group identity of l...

In the present article, we construct the super fuzzy Dirac and chirality operators on super fuzzy sphere SF(2|2) . Using the super fuzzy Ginsparg-Wilson algebra, we study the super gauged fuzzy Dirac and chirality operators in instanton sector. We show that they have correct commutative limit in the limit case when noncommutative parameter l tends to infinity.

We introduce the class of split Malcev Poisson algebras as the natural extension of split (noncommutative) Poisson algebras. We show that if $ P $ is a split Malcev Poisson algebra then $ P=\oplus_{j\in J}I_{j} $ with $ I_{j} $ a nonzero ideal of $ P $ such that $ \{I_{j_{1}},I_{j_{2}}\}=I_{j_{1}}I_{j_{2}}=0 $ for $ j_{1}\neq j_{2} $ . Under some conditions, the above decomposition of $ P $ involves a family of the simple ideals of $ P $ .

In this paper we investigate a semi-linear degenerate elliptic equation Δ L u + h ( ξ ) u p + g ( ξ ) u q = 0 related to the generalized Baouendi-Grushin vector fields, where p and q satisfy certain conditions. The idea is to extend the Obata method applying elliptic equations to the degenerate elliptic equation. In consideration of noncommutative relations of the generalized Baouendi-Grushin vector fields, an identity corresponding to the degenerate elliptic equation is established. By using a special truncated function, the identity deformation and accurate estimates, we derive Liouville theorems which show that all nonnegative solutions to the equation are trivial.

We employ some techniques involving projections in von Neumann algebras to establish some maximal inequalities such as the strong and weak symmetrization, Levy, Levy–Skorohod, and Ottaviani inequal...

Random noncommutative geometry can be seen as a Euclidean path-integral approach to the quantization of the theory defined by the Spectral Action in noncommutative geometry (NCG). With the aim of investigating phase transitions in random NCG of arbitrary dimension, we study the non-perturbative Functional Renormalization Group for multimatrix models whose action consists of noncommutative polynomials in Hermitian and anti-Hermitian matrices. Such structure is dictated by the Spectral Action for the Dirac operator in Barrett's spectral triple formulation of fuzzy spaces.The present mathematically rigorous treatment puts forward "coordinate-free" language that might be useful also elsewhere, all the more so because our approach holds for general multimatrix models. The toolkit is a noncommutative calculus on the free algebra that allows to describe the generator of the renormalization group flow -- a noncommutative Laplacian introduced here -- in terms of Voiculescu's cyclic gradient and Rota-Sagan-Stein noncommutative derivative. We explore the algebraic structure of the Functional Renormalization Group Equation and, as an application of this formalism, we find the $\beta$-functions, identify the fixed points in the large-$N$ limit and obtain the critical exponents of $2$-dimensional geometries in two different signatures., Comment: 50 pages + glossary and four appendices. Four figures and some tables. v5: Conform with the Annales Henri Poincar\'e version (except, that the "supplementary material" there is part of the appendices here)

This article illustrates the dynamical concept of $homomesy$ in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and shows the relationship between these three settings. In particular, we show how the rowmotion and promotion operations of Striker and Williams can be lifted to (continuous) piecewise-linear operations on the order polytope of Stanley, and then lifted to birational operations on the positive orthant in $\mathbb{R}^{|P|}$ and indeed to a dense subset of $\mathbb{C}^{|P|}$. When the poset $P$ is a product of a chain of length $a$ and a chain of length $b$, these lifted operations have order $a+b$, and exhibit the homomesy phenomenon: the time-averages of various quantities are the same in all orbits. One important tool is a concrete realization of the conjugacy between rowmotion and promotion found by Striker and Williams; this $recombination$ $map$ allows us to use homomesy for promotion to deduce homomesy for rowmotion. NOTE: An earlier draft showed that Stanley's transfer map between the order polytope and the chain polytope arises as the tropicalization of an analogous map in the bilinear realm; in 2020 we removed this material for the sake of brevity, especially after Joseph and Roby generalized our proof to the noncommutative realm (see arXiv:1909.09658v3). Readers who nonetheless wish to see our proof can find the September 2018 draft of this preprint through the arXiv.

We give a series of infinite dimensional noncommutative and noncocommutative pointed Hopf algebras, which are Artin-Schelter Gorenstein Hopf algebras with injective dimensions 3. Radford’s Hopf alg...

We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley-Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.

Bilinear Logic of Lambek amounts to Noncommutative MALL of Abrusci. Lambek proves the cut–elimination theorem for a one-sided (in fact, left-sided) sequent system for this logic. Here we prove an analogous result for the nonassociative version of this logic. Like Lambek, we consider a left-sided system, but the result also holds for its right-sided version, by a natural symmetry. The treatment of nonassociative sequent systems involves some subtleties, not appearing in associative logics. We also prove the PTime complexity of the multiplicative fragment of NBL.

In this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let $S$ be a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree $1$ and $f =x_1^2 + \cdots +x_n^2 \in S$. We prove that the stable category $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ of graded maximal Cohen--Macaulay module over $S/(f)$ can be completely computed using the four graphical operations. As a consequence, $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is equivalent to the derived category $\mathsf{D^b}(\operatorname{\mathsf{mod}} k^{2^r})$, and this $r$ is obtained as the nullity of a certain matrix over ${\mathbb F}_2$. Using the properties of Stanley--Reisner ideals, we also show that the number of irreducible components of the point scheme of $S$ that are isomorphic to ${\mathbb P}^1$ is less than or equal to $\binom{r+1}{2}$., Comment: 10 pages, v2: minor changes

Let R be a noncommutative prime ring of characteristic different from 2 and 3, C the extended centroid of R, F and G two generalized derivations of R, d a nonzero derivation of R and $$f(x_1,\ldots ,x_n)$$ a multilinear polynomial over C. Suppose that I is a nonzero ideal of R and $$f(I)=\{f(x_1,\ldots ,x_n)| x_1,\ldots ,x_n\in I\}$$ . If $$f(x_1,\ldots ,x_n)$$ is not central valued on R and $$\begin{aligned} d(F^2(u)u-G(u^2))=0 \end{aligned}$$ for all $$u\in f(I)$$ , then all possible forms of the maps F, G and d are described in the present paper.

We give an explicit formula showing how the double Poisson algebra introduced in [14] appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on A ⊕ A ⁎ . Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra A and emphasises the special role of the fourth component of pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces ( Rep n A ) G l n for any associative algebra A.

Let E be a separable symmetric space on 0 , ∞ and E M the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than E M and obtain the following interpolation result: let E ^ M be the space of all bounded E M -quasimartingales and 1 < p < p E < q E < q < ∞ . Then, there exists a symmetric space F on 0 , ∞ with nontrivial Boyd indices such that E ^ M = L ^ p M , L ^ q M F , K .

The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

We wish to investigate continuous fields of the Cuntz algebras. The Cuntz algebras $${\mathcal {O}}_{n+1}, n\ge 1$$ play an important role in the theory of operator algebras, and they are characterized by their K-groups $$K_0({\mathcal {O}}_{n+1})={\mathbb {Z}}_n$$ , the cyclic groups of order $$n\ge 1$$ . Since the mod n K-group for a compact Hausdorff space can be realized by the K-group of the trivial continuous field of $${\mathcal {O}}_{n+1}$$ over the space, one can regard $${\mathcal {O}}_{n+1}$$ as a noncommutative analogue of the Moore space of $${\mathbb {Z}}_n$$ , and classifying continuous fields of the Cuntz algebras is an interesting problem. M. Dadarlat classifies these fields which are constructed from the vector bundles of rank $$n+1$$ , and he also showed that not every continuous field comes from the vector bundle. For a continuous field of $${\mathcal {O}}_{n+1}$$ over a finite CW complex, we introduce a topological invariant, which is an element in Dadarlat–Pennig’s generalized cohomology group, and prove that the invariant is trivial if and only if the field comes from a vector bundle via Pimsner’s construction.