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AbstractSuppose R and C represent two unital *-algebras, with R containing a nontrivial projection. In the present article, we demonstrate, under certain restrictions that if a bijective map Δ:R→C satisfies Δ(K°N•W)=Δ(K)°Δ(N)•Δ(W) for all K,N,W∈R, then Δ is a *-preserving ring isomorphism. Also, this result is investigated in factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

In this paper, we study the non-commutative Choquet boundary for systems of unbounded operators on a Hilbert space. To do this, we introduce the notion of local boundary representations for local operator systems in locally C ∗ -algebras and provide some characterizations on Fréchet quantized domains. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, it is shown that every nonlinear bijective map ψ preserving mixed Jordan triple η-products with η≠−1 from a factor von Neumann algebra M with dim M>1 into another factor von Neumann algebra N with dim N>1 is additive. Also, for α∈{−1,1} , we establish that αψ is a linear *-isomorphism, and is a conjugate linear *-isomorphism only when η∈R . [ABSTRACT FROM AUTHOR]

Let 풜 {\mathcal{A}} be a unital ∗ {\ast} -algebra over the complex field ℂ {\mathbb{C}} . For any μ 1 , μ 2 , μ 3 , … , μ n ∈ 풜 {\mu_{1},\mu_{2},\mu_{3},\ldots,\mu_{n}\in\mathcal{A}} , a product μ 1 ∘ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 \mu_{1}\circ\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1} is called Jordan product and μ 1 ∙ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 ∗ {\mu_{1}\bullet\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1}^{\ast}} is called skew Jordan product. Define P 3 ⁢ ( μ 1 , μ 2 , μ 3 ) = μ 1 ∘ μ 2 ∙ μ 3 P_{3}(\mu_{1},\mu_{2},\mu_{3})=\mu_{1}\circ\mu_{2}\bullet\mu_{3} (called mixed Jordan triple product) and P n ⁢ ( μ 1 , μ 2 , … , μ n ) = μ 1 ∘ μ 2 ∘ ⋯ ∙ μ n P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n})=\mu_{1}\circ\mu_{2}\circ\cdots\bullet\mu% _{n} (called mixed Jordan n-product) for all integer n ≥ 3 {n\geq 3} . In this article, it is shown that a map (called nonlinear mixed Jordan n-derivation) φ : 풜 → 풜 {\varphi:\mathcal{A}\rightarrow\mathcal{A}} satisfies φ ⁢ ( P n ⁢ ( μ 1 , μ 2 , … , μ n ) ) = ∑ i = 1 n P n ⁢ ( μ 1 , … , ν i - 1 , φ ⁢ ( ν i ) , ν i + 1 , … , ν n ) {\varphi(P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n}))=\sum_{i=1}^{n}P_{n}(\mu_{1},% \ldots,\nu_{i-1},\varphi(\nu_{i}),\nu_{i+1},\ldots,\nu_{n})} for all ν 1 , ν 2 , … , ν n ∈ 풜 {\nu_{1},\nu_{2},\ldots,\nu_{n}\in\mathcal{A}} if and only if φ is an additive ∗ {\ast} -derivation. As applications, our main result is applied to several special classes of unital ∗ {\ast} -algebras such as prime ∗ {\ast} -algebras, factor von Neumann algebras and von Neumann algebras with no central summands of type I 1 {I_{1}} . [ABSTRACT FROM AUTHOR]

In 1988, Haagerup and Størmer conjectured that every pointwise inner automorphism of a type ${\rm III_1}$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that every type ${\rm III_1}$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer's conjecture holds in full generality if Connes' bicentralizer problem has an affirmative answer. Our proof is based on Popa's intertwining theory and Marrakchi's recent work on relative bicentralizers. [ABSTRACT FROM AUTHOR]

Let A be a unital C*-algebra and B a unital C*-algebra with a faithful trace τ. Let n be a positive integer. We give the definition of weakly approximate diagonalization (with respect to τ) of a unital homomorphism ϕ : A → M n (B) . We give an equivalent characterization of McDuff II1 factors. We show that, if A is a unital nuclear C*-algebra and B is a type II1 factor with faithful trace τ, then every unital *-homomorphism ϕ : A → M n (B) is weakly approximately diagonalizable. If B is a unital simple infinite dimensional separable nuclear C*-algebra, then any finitely many elements in M n (B) can be simultaneously weakly approximately diagonalized while the elements in the diagonals can be required to be the same. [ABSTRACT FROM AUTHOR]

We prove that a large class of nonamenable almost periodic type III 1 factors M, including all McDuff factors that tensorially absorb R ∞ and all free Araki–Woods factors, satisfy Haagerup–Størmer's conjecture (1988): any pointwise inner automorphism of M is the composition of an inner and a modular automorphism. [ABSTRACT FROM AUTHOR]

Let A be a unital prime *-algebra (over the complex field C ) with a nontrivial projection. For any S 1 , S 2 , ... , S n ∈ A , define q 1 (S 1) = S 1 , q 2 (S 1 , S 2) = [ S 1 , S 2 ] ⋄ = S 1 S 2 − S 2 S 1 ∗ and q n (S 1 , S 2 , ... , S n) = [ q n − 1 (S 1 , S 2 , ... , S n − 1) , S n ] ⋄ for all integers n ≥ 3. In this article, it is shown that if a map ξ : A → A (not necessarily linear) satisfies ξ (q n (S 1 , S 2 , ... , S n)) = ∑ i = 1 n q n (S 1 , ... , S i − 1 , ξ (S i) , S i + 1 , ... , S n) (n ≥ 3) for all S 1 , S 2 , ... , S n ∈ A with S 1 ∗ S 2 ∗ ⋯ S n ∗ = 0 , then ξ is an additive *-derivation. Finally, this result is applied to standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

Let M and N be factor von Neumann algebras, and let Ψ : M → N be a bijective map preserving [ [ G , S ] , T ] ⋄. In this paper, we will study the characterization of this map when dim M > 1. [ABSTRACT FROM AUTHOR]

The quantum injectivity problem classifies frames which are injective with respect to self-adjoint Hilbert-Schmidt operators. In this paper, we aim to analyze quantum injective frames in terms of the excess of frame elements in R n . Especially, we investigate the connection between the injectivity, full spark and phase retrieval frames. In addition, we detect injective (alternate) dual frames and show that the family of quantum injective dual frames is open and dense in the set of all dual frames. Finally, the stability of quantum injective frames will be addressed. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of regular orthogonally additive operators defined on a lattice-normed space (X , E) and taking values in a vector lattice F. We show that the vector space O A r (X , F) of all regular orthogonally additive operators from a d-decomposable lattice-normed space (X , E) to a Dedekind complete vector lattice F is a Dedekind complete vector lattice and the lattice operations can be calculated by the Riesz–Kantorovich formulas. We find necessary and sufficient conditions for an orthogonally additive operator T : X → F to be dominated and obtain a criterion of the positivity of a nonlinear superposition operator T N : E (X) → E defined on Köthe–Bochner space E(X) and taking values in Köthe-*Banach space E. Finally, we state some open problems. [ABSTRACT FROM AUTHOR]

Let be a factor von Neumann algebra with dim. In this paper, it is shown that if a map satisfies for all , then ϕ is an additive ∗-derivation, where. [ABSTRACT FROM AUTHOR]

Every positive multilinear map between C ∗ -algebras is separately weak ∗ -continuous. We show that the joint weak ∗ -continuity is equivalent to the joint weak ∗ -continuity of the multiplications of the C ∗ -algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron–Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general C ∗ -algebras enjoys a decomposition Φ = φ 2 ∘ φ 1 , in which φ 1 is a tracial positive linear map with the commutative range and φ 2 is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map Φ between C ∗ -algebra is a von Neumann algebra, then Φ has a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics for arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented. [ABSTRACT FROM AUTHOR]

We establish some Lie–Trotter formulae for unital complex Jordan–Banach algebras, showing that for any elements a, b, c in a unital complex Jordan–Banach algebra A the identities lim n → ∞ e a n ∘ e b n n = e a + b , lim n → ∞ U e a n e b n n = e 2 a + b , and lim n → ∞ U e a n , e c n e b n n = e a + b + c hold. These formulae are actually deduced from a more general result involving holomorphic functions with values in A . These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals f : A → C satisfying f (U x (y)) = U f (x) f (y) , for all x , y ∈ A . We prove that for any such a functional f, there exists a unique continuous (Jordan-) multiplicative linear functional ψ : A → C such that f (x) = ψ (x) , for every x in the connected component of the set of all invertible elements of A containing the unit element. If we additionally assume that A is a JB ∗ -algebra and f is continuous, then f is a linear multiplicative functional on A . The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Schulz, and Touré. [ABSTRACT FROM AUTHOR]

In this paper, we consider certain relations among Catalan numbers, and study free distributions of mutually free, multi semicircular elements. We not only characterize the joint free distributions of given multi semicircular elements, but also provide estimations for such free-distributional data by using our relations among Catalan numbers. As application, asymptotic behaviors of the joint free distribution of mutually free, multi semicircular elements are considered. [ABSTRACT FROM AUTHOR]

Let U be an associative unital Banach algebra endowed with the Lie product [ x , y ] = x y - y x ( x , y ∈ U ) and create a Lie algebra. In this article, we are going to study the additive maps on U that act at idempotent-products such as centralizers on the Lie structure of U . More precisely, we consider the subsequent condition on an additive map φ on a unital Banach algebra U with a non-trivial idempotent p: x , y ∈ U , x y = p ⟹ φ ([ x , y ]) = [ φ (x) , y ] = [ x , φ (y) ] , and we show under certain conditions that φ (x) = c x + μ (x) for all x ∈ U , where c ∈ Z (U) , μ : U → Z (U) ( Z (U) is the center of U ) is an additive map in which μ ([ x , y ]) = 0 for any x , y ∈ U with x y = p . The obtained results will be used for some Banach algebras, especially, for von Neumann algebras. [ABSTRACT FROM AUTHOR]

Let A be a ∗ -algebra over the complex field C. For any T 1 , T 2 , … , T n ∈ A , define q 1 (T 1) = T 1 , q 2 (T 1 , T 2) = T 1 ⋄ T 2 = T 1 T 2 ∗ + T 2 T 1 ∗ and q n (T 1 , T 2 , … , T n) = q n - 1 (T 1 , T 2 , … , T n - 1) ⋄ T n for all integers n ≥ 2. In this article, it is shown that under certain assumptions a map D : A → A is a nonlinear bi-skew Jordan n-derivation (that is, D satisfies D (q n (T 1 , T 2 , … , T n)) = ∑ i = 1 n q n (T 1 , … , T i - 1 , D (T i) , T i + 1 , … , T n) for all T 1 , T 2 , … , T n ∈ A ) if and only if D is an additive ∗ -derivation. Further, we introduce the notion of a nonlinear generalized bi-skew Jordan n-derivation of a ∗ -algebra and prove that every nonlinear generalized bi-skew Jordan n-derivation G : A → A is of the form G (T) = Z T + D (T) for all T ∈ A , where Z ∈ Z (A) and D : A → A is an additive ∗ -derivation. As applications, we apply our main results to some special classes of ∗ -algebras such as prime ∗ -algebras, factor von Neumann algebras, von Neumann algebras with no central summonds of type I 1 . [ABSTRACT FROM AUTHOR]

Let X , Y be real, infinite-dimensional Banach spaces. Let ℒ ⁢ (X , Y) be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at A ∈ ℒ ⁢ (X , Y) is to estimate the limit (which always exists) lim t → 0 + ⁡ ∥ A + t ⁢ B ∥ - ∥ A ∥ t for ⁢ B ∈ ℒ ⁢ (X , Y) , using the values of B on the state space S A = { τ ∈ ℒ ⁢ (X , Y) ∗ : τ ⁢ (A) = ∥ A ∥ , ∥ τ ∥ = 1 }. In this paper, we give several examples of Banach spaces, including the ℓ p spaces (for 1 < p < ∞ ) where a more tangible estimate is possible, under additional hypotheses on A. We also use the notion of norm-weak upper-semi-continuity (usc, for short) of the preduality map to achieve this. Our results also show that the operator subdifferential limit is related to the corresponding subdifferential limit of the vectors in the range space, when A ∗ ∗ attains its norm. [ABSTRACT FROM AUTHOR]

Let 풜 and ℬ be two von Neumann algebras. In this paper, we show that a bijective unital map Φ : 풜 → ℬ which preserves the bi-skew Jordan product is an additive ∗ -isomorphism. Moreover, we show that Φ is a linear ∗ -isomorphism or a conjugate linear ∗ -isomorphism. [ABSTRACT FROM AUTHOR]

Let M be a semifinite von Neumann algebra. We show that an operator T from the predual L 1 (M , τ) of M into a Banach space X is strongly Dunford–Pettis if and only if T ∘ i : L 1 (M , τ) ∩ M → i L 1 (M , τ) → T X is compact. We also show that for a finite measure space (A , Σ , ν) and a reflexive Banach space X, a linear bounded operator T : L 1 (ν , X) → c 0 is a Dunford–Pettis operator if and only if T is a dominated operator. We also prove that all bounded operators T from L p (0 , 1) into the Schatten–von Neumann r-class C r are necessarily narrow whenever 1 ≤ p < 2 , 1 ≤ r ≤ p , which answers a question raised in Huang et al. (Mediterr. J Math 19, 2022). Finally, we show that for a reflexive Banach space X all Dunford–Pettis operators from T : L 1 (ν , X) → c 0 are narrow. [ABSTRACT FROM AUTHOR]

We prove novel asymptotic freeness results in tracial ultraproduct von Neumann algebras. In particular, we show that whenever $M = M_1 \ast M_2$ is a tracial free product von Neumann algebra and $u_1 \in \mathscr U(M_1)$ , $u_2 \in \mathscr U(M_2)$ are Haar unitaries, the relative commutants $\{u_1\}' \cap M^{\mathcal U}$ and $\{u_2\}' \cap M^{\mathcal U}$ are freely independent in the ultraproduct $M^{\mathcal U}$ . Our proof relies on Mei–Ricard’s results [MR16] regarding $\operatorname {L}^p$ -boundedness (for all $1 < p < +\infty $ ) of certain Fourier multipliers in tracial amalgamated free products von Neumann algebras. We derive two applications. Firstly, we obtain a general absorption result in tracial amalgamated free products that recovers several previous maximal amenability/Gamma absorption results. Secondly, we prove a new lifting theorem which we combine with our asymptotic freeness results and Chifan–Ioana–Kunnawalkam Elayavalli’s recent construction [CIKE22] to provide the first example of a $\mathrm {II_1}$ factor that does not have property Gamma and is not elementary equivalent to any free product of diffuse tracial von Neumann algebras.

We show that if Γ is a finitely generated finitely presented sofic group with zero first L2-Betti number, then the von Neumann algebra L(Γ) is strongly 1-bounded in the sense of Jung. In particular, L(Γ)≆L(Λ) if Λ is any group with free entropy dimension >1, for example a free group. The key technical result is a short proof of an estimate of Jung

We show that if $\Gamma$ is an infinite finitely generated finitely presentedsofic group with zero first $L^{2}$ Betti number then the von Neumann algebra$L(\Gamma)$ is strongly $1$-bounded in the sense of Jung. In particular,$L(\Gamma)\not\cong L(\Lambda)$ if $\Lambda$ is any group with free entropydimension $>1$, for example a free group. The key technical result is a shortproof of an estimate of Jung using non-microstates entropy techniques.

In this paper, we extend to the setting of positive orthogonally bi-additive operators several results from Aliprantis and Burkinshaw (Math Z 185: 245-257, 1983), de Pagter (Math Anal Appl 472: 238-245, 2019), Pliev and Popov (Siberian Math J 57: 552-557, 2016), Pliev and Ramdane (Mediter J Math 15(2): 55, 2018). First, we show that a positive order bounded orthogonally bi-additive map T : I → W defined on a lateral ideal of a Cartesian product of vector lattices E and F and taking values in a Dedekind complete vector lattice W can be extended to the whole space E × F . Then we prove that a positive orthogonally bi-additive operator T : E × F → W is laterally-to-order continuous if and only if the kernel of each S with 0 ≤ S ≤ T is laterally closed. Finally, we calculate the laterally-to-order continuous part of a positive orthogonally bi-additive operator T : E × F → W . [ABSTRACT FROM AUTHOR]

In this paper, we study certain free random variables having their free distributions, dictated by the semicircular law, in a certain sense. In particular, we are interested in operators T, which are not self-adjoint, but whose free distributions are followed by the semicircular law in the sense that: all n-th joint free moments of T , T ∗ are identical to the n 2 -th Catalan numbers c n 2 , for all n ∈ N , where c n 2 = 0 , whenever n 2 ∉ N . The construction of such free random variables, itself, is one of the main results of this paper. We consider free-distributional data determined by such operators. [ABSTRACT FROM AUTHOR]

We introduce a notion of teleportation scheme between subalgebras of semi-finite von Neumann algebras in the commuting operator model of locality. Using techniques from subfactor theory, we present unbiased teleportation schemes for relative commutants N ′ ∩ M of a large class of finite-index inclusions N ⊆ M of tracial von Neumann algebras, where the unbiased condition means that no information about the teleported observables is contained in the classical communication sent between the parties. For a large class of subalgebras N of matrix algebras M n (C) , including those relevant to hybrid classical/quantum codes, we show that any tight teleportation scheme for N necessarily arises from an orthonormal unitary Pimsner–Popa basis of M n (C) over N ′ , generalising work of Werner (J Phys A 34(35):7081–7094, 2001). Combining our techniques with those of Brannan–Ganesan–Harris (J Math Phys 63(11): 112204, 2022) we compute quantum chromatic numbers for a variety of quantum graphs arising from finite-dimensional inclusions N ⊆ M . [ABSTRACT FROM AUTHOR]

The quantum detection problem was considered in some recent references for both discrete and continuous frames. The purpose of this paper is to presenting several versions of characterizations for S p -injective continuous frames in terms of discrete representations of continuous frames. This work answers the quantum detection problem in a more general setting and gives the characterizations for a larger class of injective continuous frames. With the help of these characterizations, we obtain some concrete examples of injective continuous frames. Moreover, the stability of S p -injective continuous frames will be addressed. [ABSTRACT FROM AUTHOR]

We prove that a surjective map (on the positive cones of unital C*-algebras) preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra. We represent weighted geometric mean preserving bijective maps on the positive cones of prime C*-algebras in terms of Jordan *-isomorphisms of the algebras. We also characterize order isomorphisms and orthoisomorphisms of the projection lattice of the von Neumann algebra of all bounded linear operators on a Hilbert space, answering an open question arisen by Dye. Finally, we give a description for Fuglede—Kadison determinant preserving maps on the positive cone of a finite von Neumann algebra and improve Gaal and Nayak's work on this topic. [ABSTRACT FROM AUTHOR]

Let A and B be two unital prime complex ∗ -algebras such that A has a nontrivial projection. In this paper, we study the structure of a bijective map Φ : A → B preserving the sum of triple products α 1 a b c + α 2 a ∗ c b + α 3 b a ∗ c + α 4 c a b + α 5 b c a + α 6 c b a ∗ , for all elements a , b , c ∈ A , where the scalars { α k } k = 1 6 are complex numbers satisfying some conditions. Applications of obtained results are presented. [ABSTRACT FROM AUTHOR]