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Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

Let (A,\mathfrak {m}) be an excellent normal domain of dimension two containing a field k \cong A/\mathfrak {m}. An \mathfrak {m}-primary ideal I is a p_g-ideal if the Rees algebra A[It] is a Cohen-Macaulay normal domain. If k is algebraically closed then Okuma, Watanabe and Yoshida proved that A has p_g-ideals and furthermore product of two p_g-ideals is a p_g ideal. Previously we showed that if k has characteristic zero then A has p_g-ideals. In this paper we prove that if k is perfect field of positive characteristic then also A has p_g ideals. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura [Arch. Math. (Basel) 110 (2018), pp. 327–332] this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev [Arch. Math. (Basel) 89 (2007), pp. 33–40]. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

Spin^h manifolds are the quaternionic analogue to \text {spin}^c manifolds. We compute the \text {spin}^h bordism groups at the prime 2 by proving a structure theorem for the cohomology of the \text {spin}^h bordism spectrum \mathrm {MSpin^h} as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of \mathrm {MSpin^h} as a wedge sum of familiar spectra. We also compute the decomposition of H^*(\mathrm {MSpin^h};\mathbb {Z}/2\mathbb {Z}) explicitly in degrees up through 30 via a counting process. [ABSTRACT FROM AUTHOR]

We prove that for any fixed unitary matrix U, any abelian self-adjoint algebra of matrices that is invariant under conjugation by U can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by U. We use this result to analyse the structure of matrices A for which A^*A commutes with AA^*, and to characterize matrices that are unitarily equivalent to weighted permutations. [ABSTRACT FROM AUTHOR]

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let \mathcal {G} be a topological group, and \mathcal {A} a unital symmetric C^*-subalgebra of \mathrm {UC}(\mathcal {G}), the algebra of bounded uniformly continuous functions on \mathcal {G}. Generalizing the notion of a stable metric, we study \mathcal {A}-metrics \delta, i.e., the function \delta (e, \cdot) belongs to \mathcal {A}; the case \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}), the algebra of weakly almost periodic functions on \mathcal {G}, recovers stability. If the topology of G is induced by a left invariant metric d, we prove that \mathcal {A} determines the topology of \mathcal {G} if and only if d is uniformly equivalent to a left invariant \mathcal {A}-metric. As an application, we show that the additive group of C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now \mathcal {G} be a metric group, and assume \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}), the algebra of bounded left uniformly continuous functions on \mathcal {G}, is a unital C^*-algebra which is the uniform closure of coefficients of representations of \mathcal {G} on members of \mathscr {F}, where \mathscr {F} is a class of Banach spaces closed under \ell _2-direct sums. We prove that \mathcal {A} determines the topology of \mathcal {G} if and only if \mathcal {G} embeds into the isometry group of a member of \mathscr {F}, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability. [ABSTRACT FROM AUTHOR]

We study the maximal abelian von Neumann algebra corresponding to L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F^2, but it can also be realized as a function algebra contained in F^2. This provides an interesting example of a C^* algebra whose elements are analytic functions. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three. [ABSTRACT FROM AUTHOR]

Let R be a finitely generated \mathbb N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P\in Ass(R/I) one defines the v-invariant v_P(I) of I at P as the least c\in \mathbb N such that P=I:f for some f\in R_c. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/I^n) is constant for large n. So it makes sense to consider a prime ideal P\in Ass(R/I^n) for all the large n and investigate how v_P(I^n) depends on n. We prove that v_P(I^n) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [ Asymptotic behaviour of the \text {v}-number of homogeneous ideals , https://arxiv.org/abs/2306.14243, 2023]. [ABSTRACT FROM AUTHOR]

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case. [ABSTRACT FROM AUTHOR]