This article extends the classical Real Nullstellensatz of Dubois and Risler to left ideals in free $\ast$-algebras $\mathbb{R} \langle x, x^\ast \rangle$ with $x = (x_1, \dots , x_n)$. First, we introduce notions of the (noncommutative) zero set of a left ideal and of a real left ideal. We prove that every element from $\mathbb{R} \langle x, x^\ast \rangle$ whose zero set contains the intersection of zero sets of elements from a finite subset $S$ of $\mathbb{R} \langle x, x^\ast \rangle$ belongs to the smallest real left ideal containing $S$. Next, we give an implementable algorithm, which for every finite $S \subset \mathbb{R} \langle x, x^\ast \rangle$, computes the smallest real left ideal containing $S$, and prove that the algorithm succeeds in a finite number of steps. Our definitions and some of our results also work for other $\ast$-algebras. As an example, we treat real left ideals in $M_n(\mathbb{R}[x_1])$.
COBISS.SI-ID: 16636249
The Bogomolov multiplier is a group theoretical invariant isomorphic to the unramified Brauer group of a given quotient space. We derive a homological version of the Bogomolov multiplier, prove a Hopf-type formula, find a five term exact sequence corresponding to this invariant, and describe the role of the Bogomolov multiplier in the theory of central extensions. A new description of the Bogomolov multiplier of a nilpotent group of class two is obtained. We define the Bogomolov multiplier within K-theory and show that proving its triviality is equivalent to solving a long-standing problem posed by Bass. An algorithm for computing the Bogomolov multiplier is developed.
COBISS.SI-ID: 16521305
Given linear matrix inequalities (LMIs) $L_1$ and $L_2$ it is natural to ask: $(Q_1)$ when does one dominate the other, that is, does $L_1(X) \succeq 0$ imply $L_2(X) \succeq 0$? $(Q_2)$ when are they mutually dominant, that is, when do they have the same solution set? The matrix cube problem of Ben-Tal and Nemirovski (SIAM J Optim 12:811-833, 2002) is an example of LMI domination. Hence such problems can be NP-hard. This paper describes a natural relaxation of an LMI, based on substituting matrices for the variables $x_j$. With this relaxation, the domination questions $(Q_1)$ and $(Q_2)$ have elegant answers, indeed reduce to constructible semidefinite programs. As an example, to test the strength of this relaxation we specialize it to the matrix cube problem and obtain essentially the relaxation given in Ben-Tal and Nemirovski (SIAM J Optim 12:811-833, 2002). Thus our relaxation could be viewed as generalizing it. Assume there is an $X$ such that $L_1(X)$ and $L_2(X)$ are both positive definite, and suppose the positivity domain of $L_1$ is bounded. For our "matrix variable" relaxation a positive answer to $(Q_1)$ is equivalent to the existence of matrices $V_j$ such that $$L_2(x) = V_1^\ast L_1 (x)V_1 + \cdots + V_\mu^\ast L_1(x) V_\mu. \qquad \qquad (A_1)$$ As for $(Q_2)$ we show that $L_1$ and $L_2$ are mutually dominant if and only if, up to certain redundancies described in the paper, $L_1$ and $L_2$ are unitarily equivalent. Algebraic certificates for positivity, such as $(A_1)$ for linear polynomials, are typically called Positivstellensätze. The paper goes on to derive a Putinar-type Positivstellensatz for polynomials with a cleaner and more powerful conclusion under the stronger hypothesis of positivity on an underlying bounded domain of the form $\{X|L(X) \succeq 0\}$. An observation at the core of the paper is that the relaxed LMI domination problem is equivalent to a classical problem. Namely, the problem of determining if a linear map $\tau$ from a subspace of matrices to a matrix algebra is "completely positive". Complete positivity is one of the main techniques of modern operator theory and the theory of operator algebras. On one hand it provides tools for studying LMIs and on the other hand, since completely positive maps are not so far from representations and generally are more tractable than their merely positive counterparts, the theory of completely positive maps provides perspective on the difficulties in solving LMI domination problems.
COBISS.SI-ID: 16592985
We prove a general archimedean positivstellensatz for hermitian operator-valued polynomials and show that it implies the multivariate Fejer-Riesz theorem of Dritschel-Rovnyak and positivstellensätze of Ambrozie-Vasilescu and Scherer-Hol. We also obtain several generalizations of these and related results. The proof of the main result depends on an extension of the abstract archimedean positivstellensatz for $\ast$-algebras that is interesting in its own right.
COBISS.SI-ID: 15997529
Let $X$ be a complex Banach space of dimension at least 2, and let $\mathcal{S}$ be a multiplicative semigroup of operators on $X$ such that the rank of $ST-TS$ is at most 1 for all $\{S,T\} \subset \mathcal{S}$. We prove that $\mathcal{S}$ has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that $\mathcal{S}$ is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997) 443-456] and [G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladič, H. Radjavi, P. Rosenthal, Invariant subspaces for semigroups of algebraic operators, J. Funct. Anal. 160 (1998) 452-465].
COBISS.SI-ID: 15167321