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Journal articles on the topic '$K$-theory and operator algebras {Selfadjoint operator algebras}'

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Published: 4 June 2021

Last updated: 11 February 2022

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1

Saito, Kichi-Suke. "Generalized interpolation in finite maximal subdiagonal algebras." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 1 (January 1995): 11–20. http://dx.doi.org/10.1017/s0305004100072893.

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Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.

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2

Anjidani, Ehsan, and Mohammad Reza Changalvaiy. "Reverse Jensen-Mercer Type Operator Inequalities." Electronic Journal of Linear Algebra 31 (February 5, 2016): 87–99. http://dx.doi.org/10.13001/1081-3810.3058.

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Let $A$ be a selfadjoint operator on a Hilbert space $\mathcal{H}$ with spectrum in an interval $[a,b]$ and $\phi:B(\mathcal{H})\rightarrow B(\mathcal{K})$ be a unital positive linear map, where $\mathcal{K}$ is also a Hilbert space. Let $m,M\in J$ with $m

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3

Kaad, J., R. Nest, and A. Rennie. "KK-Theory and Spectral Flow in von Neumann Algebras." Journal of K-Theory 10, no. 2 (April 4, 2012): 241–77. http://dx.doi.org/10.1017/is012003003jkt185.

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AbstractWe present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1(A, K(N)). For a unitary u ∈ A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.

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4

Lance, Christopher. "Book Review: $K$-theory for operator algebras." Bulletin of the American Mathematical Society 18, no. 1 (January 1, 1988): 67–74. http://dx.doi.org/10.1090/s0273-0979-1988-15604-2.

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5

Dong, Chongying, Kefeng Liu, Xiaonan Ma, and Jian Zhou. "$K$-theory associated to vertex operator algebras." Mathematical Research Letters 11, no. 5 (2004): 629–47. http://dx.doi.org/10.4310/mrl.2004.v11.n5.a7.

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6

Rosenberg, Jonathan. "The Algebraic K-Theory of Operator Algebras." K-Theory 12, no. 1 (July 1997): 75–99. http://dx.doi.org/10.1023/a:1007736420938.

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7

Inassaridze, Hvedri, and Tamaz Kandelaki. "K-Theory of Stable Generalized Operator Algebras." K-Theory 27, no. 2 (October 2002): 103–10. http://dx.doi.org/10.1023/a:1021197520756.

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8

Hebestreit, Fabian, and Steffen Sagave. "Homotopical and operator algebraic twisted K-theory." Mathematische Annalen 378, no. 3-4 (August 15, 2020): 1021–59. http://dx.doi.org/10.1007/s00208-020-02066-6.

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Abstract Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K-theory, both in the real and complex case. We also improve several comparison results about twisted K-theory of $$C^*$$ C ∗ -algebras to include multiplicative structures. Our results can also be interpreted in the $$\infty $$ ∞ -categorical setup for parametrized spectra.

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9

Oyono-Oyono, Hervé, and Guoliang Yu. "Quantitative K-theory and the Künneth formula for operator algebras." Journal of Functional Analysis 277, no. 7 (October 2019): 2003–91. http://dx.doi.org/10.1016/j.jfa.2019.01.009.

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10

Wright, J. D. M. "K-THEORY FOR OPERATOR ALGEBRAS: (Mathematical Sciences Research Institute Publications 5)." Bulletin of the London Mathematical Society 22, no. 3 (May 1990): 307–8. http://dx.doi.org/10.1112/blms/22.3.307.

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11

Phillips, N. Christopher. "The Toeplitz operator proof of noncommutative Bott periodicity." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 53, no. 2 (October 1992): 229–51. http://dx.doi.org/10.1017/s1446788700035813.

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AbstractWe adapt the Toeplitz operator proof of Bott periodicity to give a short direct proof of Bott periodicity for the representable K-theory of σ-C*-algebras. We further show how the use of this proof and the right definitions simplifies the derivation of the basic properties of representable K-theory.

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12

PARK, EFTON, and CLAUDE SCHOCHET. "ON THE K-THEORY OF QUARTER-PLANE TOEPLITZ ALGEBRAS." International Journal of Mathematics 02, no. 02 (April 1991): 195–204. http://dx.doi.org/10.1142/s0129167x91000132.

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Given a C*-algebra A which is filtered by a collection of closed ideals Ai, there is a spectral sequence which relates the K-theory of A to the K-theory of the various quotient algebras Ai/Ai−1. The d1 differentials in this spectral sequence are familiar index invariants, but the higher differentials are not well-understood. Considering the case of Toeplitz C*-algebras associated with certain cones in Z2, it is shown that a d2 differential in the spectral sequence is non-trivial. This differential turns out to be an obstruction to a classical lifting problem in operator theory. Analysis of this obstruction leads to necessary and sufficient conditions for the lifting problem. It is hoped that this example will illuminate the role of higher differentials in the K-theory spectral sequence.

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13

DEELEY, ROBIN J., MAGNUS GOFFENG, BRAM MESLAND, and MICHAEL F. WHITTAKER. "Wieler solenoids, Cuntz–Pimsner algebras and -theory." Ergodic Theory and Dynamical Systems 38, no. 8 (May 2, 2017): 2942–88. http://dx.doi.org/10.1017/etds.2017.10.

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We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

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14

Eilers, Søren, Gunnar Restorff, and Efren Ruiz. "Non-splitting in Kirchberg's Ideal-related KK-Theory." Canadian Mathematical Bulletin 54, no. 1 (March 1, 2011): 68–81. http://dx.doi.org/10.4153/cmb-2010-083-7.

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AbstractA. Bonkat obtained a universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory in the fundamental case of a C*-algebra with one specified ideal. The universal coefficient sequence was shown to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras using a method introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify ∗-isomorphisms for purely infinite C*-algebras with one non-trivial ideal.

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15

Kaptanoğlu, H. Turgay. "Aspects of multivariable operator theory on weighted symmetric Fock spaces." Communications in Contemporary Mathematics 16, no. 05 (August 29, 2014): 1350034. http://dx.doi.org/10.1142/s021919971350034x.

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We obtain all Dirichlet spaces ℱq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂN as weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱq. We pick out the analytic Hilbert modules from among the ℱq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱq fit in exact sequences that are in the same Ext class. We identify the groups K0 and K1 of the Toeplitz algebras on the ℱq arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱb that depend on a weight sequence b.

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16

Feeman, Timothy G. "The Bourgain algebra of a nest algebra." Proceedings of the Edinburgh Mathematical Society 40, no. 1 (February 1997): 151–66. http://dx.doi.org/10.1017/s0013091500023518.

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In analogy with a construction from function theory, we herein define right, left, and two-sided Bourgain algebras associated with an operator algebra A. These algebras are defined initially in Banach space terms, using the weak-* topology on A, and our main result is to give a completely algebraic characterization of them in the case where A is a nest algebra. Specifically, if A = alg N is a nest algebra, we show that each of the Bourgain algebras defined has the form A + K ∩ B, where B is the nest algebra corresponding to a certain subnest of N. We also characterize algebraically the second-order (and higher) Bourgain algebras of a nest algebra, showing for instance that the second-order two-sided Bourgain algebra coincides with the two-sided Bourgain algebra itself in this case.

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17

Skandalis, George. "Exact Sequences for the Kasparov Groups of Graded Algebras." Canadian Journal of Mathematics 37, no. 2 (April 1, 1985): 193–216. http://dx.doi.org/10.4153/cjm-1985-013-x.

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In [11] G. G. Kasparov defined the “operator K-functor” KK(A, B) associated with the graded C*-algebras A and B. If the algebras A and B are trivially graded and A is nuclear he proves six term exact sequence theorems. He asks whether this extends to the graded case.Here we prove such “six-term exact sequence” results in the graded case. Our proof does not use nuclearity of the algebra A. This condition is replaced by a completely positive lifting condition (Theorem 1.1).Using our result we may extend the results by M. Pimsner and D. Voiculescu on the K groups of crossed products by free groups to KK groups [15]. We give however a different way of computing these groups using the equivariant KK-theory developed by G. G. Kasparov in [12]. This method also allows us to compute the KK groups of crossed products by PSL2(Z).

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18

Kohnen, Winfried, and Geoffrey Mason. "On Generalized Modular forms and their Applications." Nagoya Mathematical Journal 192 (2008): 119–36. http://dx.doi.org/10.1017/s0027763000026003.

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AbstractWe study the Fourier coefficients of generalized modular forms f(τ) of integral weight k on subgroups Γ of finite index in the modular group. We establish two Theorems asserting that f(τ) is constant if k = 0, f(τ) has empty divisor, and the Fourier coefficients have certain rationality properties. (The result is false if the rationality assumptions are dropped.) These results are applied to the case that f(τ) has a cuspidal divisor, k is arbitrary, and Γ = Γ0(N), where we show that f(τ) is modular, indeed an eta-quotient, under natural rationality assumptions on the Fourier coefficients. We also explain how these results apply to the theory of orbifold vertex operator algebras.

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19

Ciubotaru, Dan, Eric M. Opdam, and Peter E. Trapa. "Algebraic and analytic Dirac induction for graded affine Hecke algebras." Journal of the Institute of Mathematics of Jussieu 13, no. 3 (March 13, 2013): 447–86. http://dx.doi.org/10.1017/s147474801300008x.

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AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

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20

Alldridge, Alexander, Christopher Max, and Martin R. Zirnbauer. "Bulk-Boundary Correspondence for Disordered Free-Fermion Topological Phases." Communications in Mathematical Physics 377, no. 3 (October 12, 2019): 1761–821. http://dx.doi.org/10.1007/s00220-019-03581-7.

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Abstract Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C$$^*$$ ∗ -algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator $$K$$ K -theory (or $$KR$$ KR -theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these $$KR$$ KR -theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants.

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21

Sinclair, A. M. "B. Blackadar K-theory for operator algebras (Mathematical Sciences Research Institute Publications Vol. 5, Springer-Verlag, New York–Berlin, 1986), pp. 337, 3 540 96391 X, DM78." Proceedings of the Edinburgh Mathematical Society 32, no. 1 (February 1989): 167–68. http://dx.doi.org/10.1017/s0013091500007057.

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22

Moretti, Valter, and Marco Oppio. "Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry." Reviews in Mathematical Physics 29, no. 06 (June 13, 2017): 1750021. http://dx.doi.org/10.1142/s0129055x17500210.

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As earlier conjectured by several authors and much later established by Solèr (relying on partial results by Piron, Maeda–Maeda and other authors), from the lattice theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. Stückelberg provided some physical, but not mathematically rigorous, reasons for ruling out the real Hilbert space formulation, assuming that any formulation should encompass a statement of Heisenberg principle. Focusing on this issue from another — in our opinion, deeper — viewpoint, we argue that there is a general fundamental reason why elementary quantum systems are not described in real Hilbert spaces. It is their basic symmetry group. In the first part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the real one, all selfadjoint operators represent observables in accordance with Solèr’s thesis, and the standard quantum version of Noether theorem may be formulated. In the second part of this work, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. This complex structure reveals a nice interplay of Poincaré symmetry and the classification of the commutant of irreducible real von Neumann algebras.

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23

Sinclair, Allan M. "D. E. Evans and M. Takesaki Operator algebras and applications, Volume 1: Structure theory; K-theory, geometry and topology; Volume 2: Mathematical physics and subfactors (London Mathematical Society Lecture Note Series 135, 136, Cambridge University Press, Cambridge1988) Vol. 1, viii + 244 pp, paper: 0 521 36843 X, £17.50; Vol. 2, viii + 240 pp, paper: 0 521 36844 8, £17.50." Proceedings of the Edinburgh Mathematical Society 32, no. 3 (October 1989): 496–97. http://dx.doi.org/10.1017/s0013091500004752.

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24

Katsoulis, Elias G., and Christopher Ramsey. "The Non-selfadjoint Approach to the Hao–Ng Isomorphism." International Mathematics Research Notices, November 26, 2019. http://dx.doi.org/10.1093/imrn/rnz271.

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Abstract In an earlier work, the authors proposed a non-selfadjoint approach to the Hao–Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of $\mathrm{C}^*$-correspondences, each one of these conjectures is equivalent to the Hao–Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C$^*$-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of $\mathrm{C}^*$-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamana’s injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao–Ng isomorphism for the reduced crossed product and all hyperrigid $\mathrm{C}^*$-correspondences. A culmination of these results is the resolution of the Hao–Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg, and Spielberg.

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25

Todorov, I. "Gause Symmetry and Howe Duality in 4D Conformal Field Theory Models." Acta Polytechnica 50, no. 3 (January 3, 2010). http://dx.doi.org/10.14311/1193.

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It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.The paper reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren and the author.

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26

Eberhardt, Lorenz, and Tomáš Procházka. "The Grassmannian VOA." Journal of High Energy Physics 2020, no. 9 (September 2020). http://dx.doi.org/10.1007/jhep09(2020)150.

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Abstract We study the 3-parametric family of vertex operator algebras based on the Grassmannian coset CFT $$ \mathfrak{u} $$ u (M + N )k /($$ \mathfrak{u} $$ u (M )k×$$ \mathfrak{u} $$ u (N )k ). This VOA serves as a basic building block for a large class of cosets and generalizes the $$ {\mathcal{W}}_{\infty } $$ W ∞ algebra. We analyze representations and their characters in detail and find surprisingly simple character formulas for the representations in the generic parameter regime that admit an elegant combinatorial formulation. We also discuss truncations of the algebra and give a conjectural formula for the complete set of truncation curves. We develop a theory of gluing for these algebras in order to build more complicated coset and non-coset algebras. We demonstrate the power of this technology with some examples and show in particular that the $$ \mathcal{N} $$ N = 2 supersymmetric Grassmannian can be obtained by gluing three bosonic Grassmannian algebras in a loop. We finally speculate about the tantalizing possibility that this algebra is a specialization of an even larger 4-parametric family of algebras exhibiting pentality symmetry. Specializations of this conjectural family should include both the unitary Grassmannian family as well as the Lagrangian Grassmannian family of VOAs which interpolates between the unitary and the orthosymplectic cosets.

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27

Wang, Bing. "Representations and fusion rules for the orbifold vertex operator algebras L s l 2 ̂ ( k , 0 ) Z p." Communications in Algebra, October 1, 2020, 1–18. http://dx.doi.org/10.1080/00927872.2020.1817469.

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