Theory Of Operator Algebras Research Articles

Algebra environments II. Algebra homomorphisms and derivations

Algebra environments provide requisites for studying objects of interest in operator algebra theory, group representation theory, spin geometry, Clifford analysis, and several variable operator theory. The concept is analyzed by developing an algebraic geometry approach. Specific algebraic sets, called structure manifolds of algebra environments, and their Zariski tangent spaces are introduced and described by using as critical tools derivations on algebras. Structure manifolds of tensor environments in particular yield spaces of algebra homomorphisms. Consequently, such spaces could be investigated as algebraic manifolds. Related issues include characterizations of their Zariski tangent spaces and of derivations that preserve algebra homomorphisms.

New tensor products of C*-algebras and characterization of type I C*-algebras as rigidly symmetric C*-algebras

Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors ( A , B ) ↦ A ⊗ α B (A,B)\mapsto A\otimes _{\alpha } B , where A ⊗ α B A\otimes _\alpha B is a cross norm completion of A ⊙ B A\odot B for each pair of C*-algebras A A and B B . For the first class of bifunctors considered ( A , B ) ↦ A ⊗ p B (A,B)\mapsto A{\otimes _p} B ( 1 ≤ p ≤ ∞ 1\leq p\leq \infty ), A ⊗ p B A{\otimes _p} B is a Banach algebra cross-norm completion of A ⊙ B A\odot B constructed in a fashion similar to p p -pseudofunctions PF p ∗ ( G ) \text {PF}^*_p(G) of a locally compact group. Taking a cue from the recently introduced symmetrized p p -pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider ⊗ p , q {\otimes _{p,q}} for Hölder conjugate p , q ∈ [ 1 , ∞ ] p,q\in [1,\infty ] – a Banach ∗ * -algebra analogue of the tensor product ⊗ p , q {\otimes _{p,q}} . By taking enveloping C*-algebras of A ⊗ p , q B A{\otimes _{p,q}} B , we arrive at a third bifunctor ( A , B ) ↦ A ⊗ C p , q ∗ B (A,B)\mapsto A{\otimes _{\mathrm C^*_{p,q}}} B where the resulting algebra A ⊗ C p , q ∗ B A{\otimes _{\mathrm C^*_{p,q}}} B is a C*-algebra. For G 1 G_1 and G 2 G_2 belonging to a large class of discrete groups, we show that the tensor products C r ∗ ( G 1 ) ⊗ C p , q ∗ C r ∗ ( G 2 ) \mathrm C^*_{\mathrm r}(G_1){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G_2) coincide with a Brown-Guentner type C*-completion of ℓ 1 ( G 1 × G 2 ) \mathrm \ell ^1(G_1\times G_2) and conclude that if 2 ≤ p ′ > p ≤ ∞ 2\leq p’>p\leq \infty , then the canonical quotient map C r ∗ ( G ) ⊗ C p , q ∗ C r ∗ ( G ) → C r ∗ ( G ) ⊗ C p , q ∗ C r ∗ ( G ) \mathrm C^*_{\mathrm r}(G){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G)\to \mathrm C^*_{\mathrm r}(G){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G) is not injective for a large class of non-amenable discrete groups possessing both the rapid decay property and Haagerup’s approximation property. A Banach ∗ * -algebra A A is symmetric if the spectrum S p A ( a ∗ a ) \mathrm {Sp}_A(a^*a) is contained in [ 0 , ∞ ) [0,\infty ) for every a ∈ A a\in A , and rigidly symmetric if A ⊗ γ B A\otimes _{\gamma } B is symmetric for every C*-algebra B B . A theorem of Kügler asserts that every type I C*-algebra is rigidly symmetric. Leveraging our new constructions, we establish the converse of Kügler’s theorem by showing for C*-algebras A A and B B that A ⊗ γ B A\otimes _{\gamma }B is symmetric if and only if A A or B B is type I. In particular, a C*-algebra is rigidly symmetric if and only if it is type I. This strongly settles a question of Leptin and Poguntke from 1979 [J. Functional Analysis 33 (1979), pp. 119—134] and corrects an error in the literature.

Wedge domains in non-compactly causal symmetric spaces

This article is part of an ongoing project aiming at the connections between causal structures on homogeneous spaces, Algebraic Quantum Field Theory, modular theory of operator algebras and unitary representations of Lie groups. In this article we concentrate on non-compactly causal symmetric spaces G/H. This class contains de Sitter space but also other spaces with invariant partial ordering. The central ingredient is an Euler element h in the Lie algebra of $${{\mathfrak {g}}}$$ . We define three different kinds of wedge domains depending on h and the causal structure on G/H. Our main result is that the connected component containing the base point eH of these seemingly different domains all agree. Furthermore we discuss the connectedness of those wedge domains. We show that each of these spaces has a natural extension to a non-compactly causal symmetric space of the form $$G_{{\mathbb {C}}}/G^c$$ where $$G^c$$ is a certain real form of the complexification $$G_{{\mathbb {C}}}$$ of G. As $$G_{{\mathbb {C}}}/G^c$$ is non-compactly causal, it also contains three types of wedge domains. Our results says that the intersection of these domains with G/H agree with the wedge domains in G/H.

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Rigorous results about entropies in quantum field theory

This is an expanded written version of a plenary talk delivered at ICMP 2021. We describe some rigorous results in quantum field theory that have been obtained in recent years, with particular emphasis on those results on relative entropies in the setting of conformal field theory. These results are motivated in part by recent work of physicists which, however, depends on heuristic arguments—such as introducing cutoff and using path integrals and replica trick—which are hard to justify mathematically. Our main technical tools are from the theory of operator algebras, such as modular theory and the theory of subfactors. A discussion about open problems is presented at the end the paper.

Maximal Injective Real W∗-Subalgebras of Tensor Products of Real W∗-Algebras

It is known that injective (complex or real) W∗-algebras with particular factors have been studied well enough. In the arbitrary cases, i.e., in noninjective case, to investigate (up to isomorphism) W∗-algebras is hard enough, in particular, there exist continuum pairwise nonisomorphic noninjective factors of type II. Therefore, it seems interesting to study maximal injective W∗-subalgebras and subfactors. On the other hand, the study of maximal injective W∗-subalgebras and subfactors is also related to the well-known von Neumann’s bicommutant theorem. In the complex case, such subalgebras were investigated by S. Popa, L. Ge, R. Kadison, J. Fang, and J. Shen. In recent years, studies have also begun in the real case. Let us briefly recall the relevance of considering the real case. It is known that in the works of D. Topping and E. Stormer, it was shown that the study of JW-algebras (nonassociative real analogues of von Neumann algebras) of types II and III is essentially reduced to the study of real W∗-algebras of the corresponding type. It turned out that the structure of real W∗-algebras, generally speaking, differs essentially in the complex case. For example, in the finite-dimensional case, in addition to complex and real matrix algebras, quaternions also arise, i.e., matrix algebras over quaternions. In the infinite-dimensional case, it is proved that there exist, up to isomorphism, two real injective factors of typeIIIλ(0<λ<1), and a countable number of pairwise nonisomorphic real injective factors of typeIII0, whose enveloping (complex) W∗-factors are isomorphic, is constructed. It follows from the above that the study of the real analogue of problems in the theory of operator algebras is topical. Moreover, the real analogue is a generalization of the complex case, since the class of real linear operators is much wider than the class of complex linear operators. In this paper, the maximal injective real W∗-subalgebras of real W∗-algebras or real factors are investigated. For real factorsQ⊂R, it is proven that ifQ+iQis a maximal injective W∗-subalgebra inR+iR, thenQalso is a maximal injective real W∗-subalgebra inR. The converse is proved in the case “II1”-factors, that is, it is shown that ifRis a real factor of typeII1, then the maximal injectivity ofQimplies the maximal injectivity ofQ+iQ. Moreover, it is proven that a maximal injective real subfactorQof a real factorRis a maximal injective real W∗-subalgebra inRif and only ifQis irreducible inR, i.e.,Q′∩R=ℝIwhereIis the unit. The “splitting theorem” of Ge-Kadison in the real case is also proven, namely, ifR1is a finite real factor,R2is a finite real W∗-algebra, andRis a real W∗-subalgebra ofR1⊗¯R2containingR1⊗¯ℝI, then there is some real W∗-subalgebraQ2⊂R2such thatR=R1⊗¯Q2. Moreover, it is given some affirmative answers to the question of S. Popa for the real case.

Conformal Field Theories with Sporadic Group Symmetry

The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $c=\frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the Goddard-Kent-Olive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators, modular linear differential equations, and Rademacher sums, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups ${}^2{E}_6(2)$ and ${F}_4(2)$ of Lie type. Many of these examples are naturally associated to McKay's $\widehat{E_8}$ correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.

Real Operator Algebras and Real Positive Maps

We present some foundations for a theory of real operator algebras and real Jordan operator algebras, and the various morphisms between these. A common theme is the ingredient of real positivity from papers of the first author with Read, Neal, Wang, and other coauthors, which we import to the real scalar setting here.

Choquet order and hyperrigidity for function systems

We establish a dilation-theoretic characterization of the Choquet order on the space of measures on a compact convex set using ideas from the theory of operator algebras. This yields an extension of Cartier's dilation theorem to the non-separable case, as well as a non-separable version of Šaškin's theorem from approximation theory. We show that a slight variant of this order characterizes the representations of a commutative C*-algebras that have the unique extension property relative to a set of generators. This reduces the commutative case of Arveson's hyperrigidity conjecture to the question of whether measures that are maximal with respect to the classical Choquet order are also maximal with respect to this new order. An example shows that these orders are not the same in general.

Two-dimensional topological order and operator algebras

We review recent interactions between mathematical theory of two-dimensional topological order and operator algebras, particularly the Jones theory of subfactors. The role of representation theory in terms of tensor categories is emphasized. Connections to two-dimensional conformal field theory are also presented. In particular, we discuss anyon condensation, gapped domain walls and matrix product operators in terms of operator algebras.

A topological invariant for continuous fields of Cuntz algebras

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.

Mini-Workshop: Operator Algebraic Quantum Groups

This mini-workshop brought together a rich and varied cross-section of young and active researchers working on operator algebraic aspects of quantum group theory. The primary goals of this meeting were to highlight the state-of-the-art results on the subject and to trigger new research by advertising some of the main open directions in operator algebraic quantum group theory: classification problems for C*- and von Neumann algebras, relations to free/non-commutative probability, applications in quantum information theory, and the creation of new quantum groups and potential classification results for subclasses of quantum groups.

Fusion rules for ℤ2-orbifolds of affine and parafermion vertex operator algebras

This paper is about the orbifold theory of affine and parafermion vertex operator algebras. It is known that the parafermion vertex operator algebra K(sl2,k) associated to the integrable highest weight modules for the affine Kac—Moody algebra $$A_1^{(1)}$$ is the building block of the general parafermion vertex operator K( $$K(\mathfrak{g},k)$$ ,k) for any finite-dimensional simple Lie algebra $$\mathfrak{g}$$ and any positive integer k. We first classify the irreducible modules of ℤ2-orbifold of the simple affine vertex operator algebra of type $$A_1^{(1)}$$ and determine their fusion rules. Then we study the representations of the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k). The quantum dimensions, and more technically, fusion rules for the ℤ2-orbifold of the parafermion vertex operator algebra K(sl2, k) are completely determined.

On Popa’s factorial commutant embedding problem

An open question of Sorin Popa asks whether or not every R U \mathcal {R}^{\mathcal {U}} -embeddable factor admits an embedding into R U \mathcal {R}^{\mathcal {U}} with factorial relative commutant. We show that there is a locally universal McDuff II 1 _1 factor M M such that every property (T) factor admits an embedding into M U M^{\mathcal {U}} with factorial relative commutant. We also discuss how our strategy could be used to settle Popa’s question for property (T) factors if a certain open question in the model theory of operator algebras has a positive solution.

Edge-following topological states

We prove that Chern insulators have topologically protected edge states which not only propagate unidirectionally along a straight line boundary, but also swerve around arbitrary-angled corners and geometric imperfections of the material boundary. This is a physical manifestation of the index theory of certain semigroup operator algebras.

One-way LOCC indistinguishable lattice states via operator structures

Lattice states are a class of quantum states that naturally generalize the fundamental set of Bell states. We apply recent results from quantum error correction and from one-way local operations and classical communication (LOCC) theory that are built on the structure theory of operator systems and operator algebras, to develop a technique for the construction of relatively small sets of lattice states not distinguishable by one-way LOCC schemes. We also present examples, show the construction extends to generalized Pauli states, and compare the construction to other recent work.

Finite‐dimensional approximation properties for uniform Roe algebras

We study property A for metric spaces $X$ with bounded geometry introduced by Guoliang Yu. Property A is an amenability-type condition, which is less restrictive than amenability for groups. The property has a connection with finite-dimensional approximation properties in the theory of operator algebras. It has been already known that property A of a metric space $X$ with bounded geometry is equivalent to nuclearity of the uniform Roe algebra C$^*_u(X)$. We prove that exactness and local reflexivity of C$^*_u(X)$ also characterize property A of $X$.

Reduced spectral synthesis and compact operator synthesis

We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it to other exceptional sets in operator algebra theory, studied previously. We show that a closed subset E of a second countable locally compact group G satisfies reduced local spectral synthesis if and only if the subset E⁎={(s,t):ts−1∈E} of G×G satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten p-classes.

Lp-operator algebras with approximate identities, I

We initiate an investigation into how much the existing theory of (nonselfadjoint) operator algebras on a Hilbert space generalizes to algebras acting on L^p spaces. In particular we investigate the applicability of the theory of real positivity, which has recently been useful in the study of L^2-operator algebras and Banach algebras, to algebras of bounded operators on Lp spaces.

Tensor algebras of product systems and their C⁎-envelopes

Let (G,P) be an abelian, lattice ordered group and let X be a compactly aligned product system over P with coefficients in A. We show that the C*-envelope of the Nica tensor algebra NTX+ coincides with both Sehnem's covariance algebra A×XP and the co-universal C⁎-algebra NOXr for injective, gauge-compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of NOXr, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C⁎-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C⁎-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where (G,P)=(Z,N).