Noncommutative Varieties Research Articles

Noncommutative Varieties, Symmetric Weighted Fock Spaces, and Multipliers

Noncommutative Varieties, Symmetric Weighted Fock Spaces, and Multipliers

Joint similarity in admissible noncommutative polydomains

In this paper we solve several problems concerning joint similarity to tuples of bounded linear operators in noncommutative polydomains Dg−1(H)⊂B(H)n1+⋯+nk associated with admissible k-tuples g:=(g1,…,gk) of formal power series gi:=1+∑αi∈Fni+,|αi|≥1bi,αZi,α, biα>0, in noncommuting indeterminates Zi,1,…,Zi,ni, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. We obtain analogues of the Rota's model theorem for operators with spectral radius less than one, the refinement obtained by Foiaş and by de Branges and Rovnyak for similarity to strongly stable contractions, and the Sz.-Nagy characterization of operators similar to isometries (or unitary operators). The results are new even in the particular case when k=1 for certain classes of admissible noncommutative domains which were recently introduced. Joint similarity problems to tuples of operators in noncommutative varieties in admissible polydomains are also discussed. This includes, in particular, the commutative setting.

Characteristic functions and functional models on noncommutative varieties

In this paper we develop a Sz.-Nagy-Foiaş type theory of characteristic functions and functional models on noncommutative domains and varieties in \(B(\mathcal{H})^n\), where \(B(\mathcal{H})\) is the algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\), associated with admissible free holomorphic functions for operator model theory. This includes, in particular, a large class of commutative domains in \(B(\mathcal{H})^n\). In the commutative case, the characteristic functions are multipliers of reproducing kernel Hilbert spaces on domains in \(\mathbb{C}^n\). We also show that the universal operator model associated with any admissible noncommutative variety has the quasi-wandering property.

Wold decompositions for representations of C∗-algebras associated with noncommutative varieties

Given a set Q of polynomials in noncommutative indeterminates Z1,…,Zn and a regular domain Dmp(H)⊂B(H)n, m,n∈N, associated with a positive regular polynomial p∈C⟨Z1,…,Zn⟩, we consider the variety VQ(H):={X=(X1,…,Xn)∈Dmp(H):g(X)=0 for all g∈Q}. Each variety VQ(H) admits a {\it universal model} B=(B1,…,Bn). The main goal of the paper is to study the structure of the ∗-representations of the C∗-algebra C∗(VQ) generated by B1,…,Bn and the identity.

Commuting row contractions with polynomial characteristic functions

A characteristic function is a special operator-valued analytic function defined on the open unit ball of $\mathbb{C}^n$ associated with an $n$-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of $n$-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of the noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions.

Entropy of Monomial Algebras and Derived Categories

Abstract Let $A$ be a finitely presented associative monomial algebra. We study the category $\textsf{qgr}(A)$, which is a quotient of the category of graded finitely presented $A$-modules by the finite-dimensional ones. As this category plays a role of the category of coherent sheaves on the corresponding noncommutative variety, we consider its bounded derived category $\textbf{D}^b(\textsf{qgr}(A))$. We calculate the categorical entropy of the Serre twist functor on $\textbf{D}^b(\textsf{qgr}(A))$ and show that it is equal to the (natural) logarithm of the entropy of the algebra $A$ itself. Moreover, we relate these two kinds of entropy with the topological entropy of the Ufnarovski graph of $A$ and the entropy of the path algebra of the graph. If $A$ is a path algebra of some quiver, the categorical entropy is equal to the logarithm of the spectral radius of the quiver’s adjacency matrix.

Submodules in Polydomains and Noncommutative Varieties

Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of n-fold tensor products of Drury–Arveson spaces.

The Arveson boundary of a free quadrilateral is given by a noncommutative variety

Let $SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices and set $SM(\mathbb{R})^g = \cup_n SM_n(\mathbb{R})^g$. A free quadrilateral is the collection of tuples $X \in SM(\mathbb{R})^2$ which have positive semidefinite evaluation on the linear equations defining a classical quadrilateral. Such a set is closed under a rich class of convex combinations called matrix convex combination. That is, given elements $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$ of a free quadrilateral $\mathcal{Q}$, one has \[ V_1^T X V_1+V_2^T Y V_2 \in \mathcal{Q} \] for any contractions $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ and $V_2:\mathbb{R}^n \to \mathbb{R}^{n_2}$ satisfying $V_1^T V_1+V_2^T V_2=I_n$. These matrix convex combinations are a natural analogue of convex combinations in the dimension free setting. A natural class of extreme point for free quadrilaterals is free extreme points: elements of a free quadrilateral which cannot be expressed as a nontrivial matrix convex combination of elements of the free quadrilateral. These free extreme points serve as the minimal set which recovers a free quadrilateral through matrix convex combinations. In this article we show that the set of free extreme points of a free quadrilateral is determined by the zero set of a collection of noncommutative polynomials. More precisely, given a free quadrilateral $\mathcal{Q}$, we construct noncommutative polynomials $p_1,p_2,p_3,p_4$ such that a tuple $X \in SM (\mathbb{R})^2$ is a free extreme point of a $\mathcal{Q}$ if and only if $X \in \mathcal{Q}$ and $p_i(X) =0 $ for $i=1,2,3,4$ and $X$ is irreducible. In addition we establish several basic results for projective maps of free spectrahedra and for homogeneous free spectrahedra.

Constrained characteristic functions, multivariable interpolation, and invariant subspaces

In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}(mathcal{H}) in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna–Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators B_{1},ldots,B_{n}, where the n-tuple (B_{1},ldots,B_{n}) is the universal model associated with the abstract noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}.

Structure theorems for idempotent residuated lattices

In this paper we study structural properties of residuated lattices that are idempotent as monoids. We provide descriptions of the totally ordered members of this class and obtain counting theorems for the number of finite algebras in various subclasses. We also establish the finite embeddability property for certain varieties generated by classes of residuated lattices that are conservative in the sense that monoid multiplication always yields one of its arguments. We then make use of a more symmetric version of Raftery’s characterization theorem for totally ordered commutative idempotent residuated lattices to prove that the variety generated by this class has the amalgamation property. Finally, we address an open problem in the literature by giving an example of a noncommutative variety of idempotent residuated lattices that has the amalgamation property.

Algebras of noncommutative functions on subvarieties of the noncommutative ball: The bounded and completely bounded isomorphism problem

Given a noncommutative (nc) variety V in the nc unit ball Bd, we consider the algebra H∞(V) of bounded nc holomorphic functions on V. We investigate the problem of when two algebras H∞(V) and H∞(W) are isomorphic. We prove that these algebras are weak-⁎ continuously isomorphic if and only if there is an nc biholomorphism G:W˜→V˜ between the similarity envelopes that is bi-Lipschitz with respect to the free pseudo-hyperbolic metric. Moreover, such an isomorphism always has the form f↦f∘G, where G is an nc biholomorphism. These results also shed some new light on automorphisms of the noncommutative analytic Toeplitz algebras H∞(Bd) studied by Davidson–Pitts and by Popescu. In particular, we find that Aut(H∞(Bd)) is a proper subgroup of Aut(B˜d).When d<∞ and the varieties are homogeneous, we remove the weak-⁎ continuity assumption, showing that two such algebras are boundedly isomorphic if and only if there is a bi-Lipschitz nc biholomorphism between the similarity envelopes of the nc varieties. We provide two proofs. In the noncommutative setting, our main tool is the noncommutative spectral radius, about which we prove several new results. In the free commutative case, we use a new free commutative Nullstellensatz that allows us to bootstrap techniques from the fully commutative case.

On stringy Euler characteristics of Clifford non-commutative varieties

It was shown by Kuznetsov that complete intersections of n generic quadrics in P2n−1 are related by Homological Projective Duality to certain non-commutative (Clifford) varieties which are in some sense birational to double covers of Pn−1 ramified over symmetric determinantal hypersurfaces. Mirror symmetry predicts that the Hodge numbers of the complete intersections of quadrics must coincide with the appropriately defined Hodge numbers of these double covers. We observe that these numbers must be different from the well-known Batyrev's stringy Hodge numbers, else the equality fails already at the level of Euler characteristics. We define a natural modification of stringy Hodge numbers for the particular class of Clifford varieties, and prove the corresponding equality of Euler characteristics in arbitrary dimension.

Homological Projective Duality for Linear Systems with Base Locus

Abstract We show how blowing up varieties in base loci of linear systems gives a procedure for creating new homological projective duals from old. Starting with a homological projective (HP) dual pair $X,Y$ and smooth orthogonal linear sections $X_L,Y_L$, we prove that the blowup of $X$ in $X_L$ is naturally HP dual to $Y_L$. The result also holds true when $Y$ is a noncommutative variety or just a category. We extend the result to the case where the base locus $X_L$ is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of $Y_L$. Finally we give examples where, starting with a noncommutative $Y$, the above process nevertheless gives geometric HP duals.

Andô dilations and inequalities on non-commutative domains

We obtain intertwining dilation theorems for non-commutative regular domains 𝒟f and non-commutative varieties 𝒱J in B(𝓗)n, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain 𝒟f (respectively, variety 𝒱J ) as well as a Schur-type representation for the unit ball of the Hardy algebra associated with the variety 𝒱J. We provide Andô-type dilations and inequalities for bi-domains 𝒟f ×c 𝒟g consisting of all pairs (X,Y ) of tuples X := (X1,…, Xn1) ∊ 𝒟f and Y := (Y1,…, Yn2) ∊ 𝒟g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n1 = n2 = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties 𝒱J1×c 𝒱J2, where 𝒱J1 and 𝒱J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n1 = n2 = 1 are also discussed.

Invariant subspaces and operator model theory on noncommutative varieties

Let $${{\mathcal {Q}}}\subset {{\mathbb {C}}}\left $$ be an arbitrary set of polynomials in noncommutative indeterminates such that $$q(0)=0$$ for all $$q\in {{\mathcal {Q}}}$$ . The noncommutative variety $$\begin{aligned} {{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}}):=\left\{ { X}=(X_1,\ldots , X_n)\in \mathbf{D}_f^m({{\mathcal {H}}}): \ q({ X})=0 \ \text { for all } q\in {{\mathcal {Q}}}\right\} , \end{aligned}$$ where $$\mathbf{D}_f^m({{\mathcal {H}}})$$ is a noncommutative regular domain in $$B({{\mathcal {H}}})^n$$ and $$B({{\mathcal {H}}})$$ is the algebra of bounded linear operators on a Hilbert space $${{\mathcal {H}}}$$ , admits a universal model $$B^{(m)}=(B_1^{(m)},\ldots , B_n^{(m)})$$ such that $$q(B^{(m)})=0$$ , $$q\in {{\mathcal {Q}}}$$ , acting on a model space which is a subspace of the full Fock space with n generators. In this paper, we obtain a Beurling type characterization of the joint invariant subspaces under the operators $$B_1^{(m)},\ldots , B_n^{(m)}$$ , in terms of partially isometric multi-analytic operators acting on model spaces. More generaly, a Beurling-Lax-Halmos type representation is obtained and used to parameterize the wandering subspaces of the joint invariant subspaces under $$B_1^{(m)}\otimes I_{{\mathcal {E}}},\ldots , B_n^{(m)}\otimes I_{{\mathcal {E}}}$$ , and to characterize when they are generating for the corresponding invariant subspaces. Similar results are obtained for any pure n-tuple $$(X_1,\ldots , X_n)$$ in the noncommutative variety $${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$$ . We characterize the elements in the noncommutative variety $${{\mathcal {V}}}_{f,{{\mathcal {Q}}}}^m({{\mathcal {H}}})$$ which admit characteristic functions, develop an operator model theory for the completely non-coisometric elements, and prove that the characteristic function is a complete unitary invariant for this class of elements. This extends the classical Sz.-Nagy–Foias functional model for completely non-unitary contractions, based on characteristic functions. Our results apply, in particular, when $${{\mathcal {Q}}}$$ consists of the noncommutative polynomials $$Z_iZ_j-Z_jZ_i$$ , $$i,j=1,\ldots , n$$ . In this case, the model space is a symmetric weighted Fock space, which is identified with a reproducing kernel Hilbert space of holomorphic functions on a Reinhardt domain in $${{\mathbb {C}}}^n$$ , and the universal model is the n-tuple $$(M_{\lambda _1},\ldots , M_{\lambda _n})$$ of multipliers by the coordinate functions.

Andô dilations and inequalities on noncommutative varieties

Andô proved a dilation result that implies his celebrated inequality which says that if T1 and T2 are commuting contractions on a Hilbert space, then for any polynomial p in two variables,‖p(T1,T2)‖≤‖p‖D2, where D2 is the bidisk in C2. The main goal of the present paper is to find analogues of Andô's results for the elements of the bi-ball Pn1,n2 which consists of all pairs (X,Y) of row contractions X:=(X1,…,Xn1) and Y:=(Y1,…,Yn2) which commute, i.e. each entry of X commutes with each entry of Y. The results are obtained in a more general setting, namely, when X and Y belong to noncommutative varieties V1 and V2 determined by row contractions subject to constraints such asq(X1,…,Xn1)=0andr(Y1,…,Yn2)=0,q∈P,r∈R, respectively, where P and R are sets of noncommutative polynomials. We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Andô's dilation theorem for commuting contractions, Sz.-Nagy–Foiaş commutant lifting theorem, and Schur's representation for the unit ball of H∞, in the framework of noncommutative varieties and Poisson kernels on Fock spaces. This leads to one of the main results of the paper, an Andô type inequality on noncommutative varieties, which, in the particular case when n1=n2=1 and T1 and T2 are commuting contractive matrices with spectrum in the open unit disk D:={z∈C:|z|<1}, takes the form‖p(T1,T2)‖≤min⁡{‖p(B1⊗ICd1,φ1(B1))‖,‖p(φ2(B2),B2⊗ICd2)‖}, where (B1⊗ICd1,φ1(B1)) and (φ2(B2),B2⊗ICd2) are analytic dilations of (T1,T2) while B1 and B2 are the universal models associated with T1 and T2, respectively. In this setting, the inequality is sharper than Andô's inequality and Agler–McCarthy's inequality. We obtain more general inequalities for arbitrary commuting contractive matrices and improve Andô's inequality for commuting contractions when at least one of them is of class C0.We prove that there is a universal model (S⊗Iℓ2,φ(S)), where S is the unilateral shift and φ(S) is an isometric analytic Toeplitz operator on H2(D)⊗ℓ2, such that‖[prs(T1,T2)]k‖≤‖[prs(S⊗Iℓ2,φ(S))]k‖, for any commuting contractions T1 and T2 on Hilbert spaces, any k×k matrix [prs]k of polynomials in C[z,w], and any k∈N. Analogues of this result for the bi-ball Pn1,n2 and for a class of noncommutative varieties are also considered.

The Failure of The Amalgamation Property for Semilinear Varieties of Residuated Lattices

Abstract The amalgamation property (AP) is of particular interest in the study of residuated lattices due to its relationship with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the AP. The variety SemRL of semilinear residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the AP are semilinear. However, we prove that this is not the case, and in the process we establish that the same is true for the variety SemCanRL of semilinear cancellative residuated lattices. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ∧ z)w ≈ xyw ∧ xzw also fails the AP.

Physical Interpretation of Noncommutative Algebraic Varieties

The theory of algebraic varieties gives an algebraic interpretation of differential geometry, thus of our physical world. To treat, among other physical properties, the theory of entanglement, we need to generalize the space parametrizing the objects of physics. We do this by introducing noncommutative varieties.

Introduction to Non Commutative Algebraic Geometry

Ordinary commutative algebraic geometry is based on commutative polynomial algebras over an algebraically closed eld k. Here we make a natural generalization to matrix polynomial k-algebras which are non-commutative coordinate rings of non-commutative varieties

More covers of the Boolean variety of unital ℓ-groups

Within the lattice of varieties of pseudo MV-algebras, the variety \({\mathcal{B}}\) of Boolean algebras is the least nontrivial variety. Komori identified all varieties of (commutative) MV-algebras that cover \({\mathcal{B}}\) . The authors previously identified all solvable varieties of pseudo MV-algebras that cover \({\mathcal{B}}\) . We will show the existence of continuum many nonsolvable varieties of pseudo MV-algebras that cover \({\mathcal{B}}\), show that periodically primitive ul-groups cannot generate Boolean covers, and show that all noncommutative varieties that are Boolean covers must be Top Boolean.