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1. Lebesgue Decomposition of Non-Commutative Measures

   https://doi.org/10.1093/imrn/rnaa231  

   Jury, Michael T; Martin, Robert T (September 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $$C^{\ast }-$$algebra, the $$C^{\ast }-$$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $$C^{\ast }-$$algebras; any *−representation of the Cuntz–Toeplitz $$C^{\ast }-$$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory. 

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2. A NON-COMMUTATIVE F. & M. RIESZ THEOREM

   Jury, Michael T.; Martin, Robert TW; Timko, Edward E (October 2023, Journal of Operator Theory)

   We extend results on complex analytic measures on the complex unit circle to a non-commutative multivariate setting. Identifying continuous linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz C ∗ - algebra, the free disk operator system, with non-commutative (NC) analogues of complex measures, we refine a previously developed Lebesgue decompo- sition for positive NC measures to establish an NC version of the Frigyes and Marcel Riesz Theorem for “analytic” measures, i.e. complex measures with vanishing positive moments. The proof relies on novel results on the order properties of positive NC measures that we develop and extend from classical measure theory. 

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3. Toeplitz quotient C*-algebras and ratio limits for random walks

   Dor-On, Adam (January 2021, Documenta mathematica)

   We study quotients of the Toeplitz C*-algebra of a random walk, similar to those studied by the author and Markiewicz for finite stochastic matrices. We introduce a new Cuntz-type quotient C*-algebra for random walks that have convergent ratios of transition probabilities. These C*-algebras give rise to new notions of ratio limit space and boundary for such random walks, which are computed by appealing to a companion paper by Woess. Our combined results are leveraged to identify a unique symmetry-equivariant quotient C*-algebra for any symmetric random walk on a hyperbolic group, shedding light on a question of Viselter on C*-algebras of subproduct systems. 

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4. Unitary groups and augmented Cuntz semigroups of separable simple Z-Stable C∗-algebras

   https://doi.org/10.1142/S0129167X22500185  

   Lin, Huaxin (February 2022, International journal of mathematics)

   Let A be a separable simple exact Z-stable C∗-algebra. We show that the unitary group of \tilde{A} has the cancellation property. If A has continuous scale then the Cuntz semigroup of A has strict comparison property and a weak cancellation property. Let C be a 1-dimensional noncommutative CW complex with K1(C) = {0}. Suppose that λ : Cu∼(C) → Cu∼(A) is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms φn : C → A such that limn→∞ Cu∼(φn) = λ. This result leads to the proof that every separable amenable simple C∗-algebra in the UCT class has rationally generalized tracial rank at most one. 

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5. Operator realizations of non-commutative analytic functions

   https://doi.org/10.1017/fms.2025.10038  

   Augat, Méric L; Martin, Robert T_W; Shamovich, Eli (January 2025, Forum of Mathematics, Sigma)

   A realization is a triple, (A,b,c), consisting of a d−tuple, A=(A1,⋯,Ad), d∈N, of bounded linear operators on a separable, complex Hilbert space, H, and vectors b,c∈H. Any such realization defines an analytic non-commutative (NC) function in an open neighbourhood of the origin, 0:=(0,⋯,0), of the NC universe of d−tuples of square matrices of any fixed size. For example, a univariate realization, i.e., where A is a single bounded linear operator, defines a holomorphic function of a single complex variable, z, in an open neighbourhood of the origin via the realization formula b∗(I−zA)−1c . It is well known that an NC function has a finite-dimensional realization if and only if it is a non-commutative rational function that is defined at 0 . Such finite realizations contain valuable information about the NC rational functions they generate. By extending to infinite-dimensional realizations, we construct, study and characterize more general classes of analytic NC functions. In particular, we show that an NC function is (uniformly) entire if and only if it has a jointly compact and quasinilpotent realization. Restricting our results to one variable shows that a formal Taylor series extends globally to an entire or meromorphic function in the complex plane, C, if and only if it has a realization whose component operator is compact and quasinilpotent, or compact, respectively. This motivates our definition of the field of global (uniformly) meromorphic NC functions as the field of fractions generated by NC rational expressions in the ring of NC functions with jointly compact realizations. This definition recovers the field of meromorphic functions in C when restricted to one variable. 

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