Fernando Lledó (UC·M-ICMAT)
Finite dimensional approximations in two classes of operator algebras
Abstract: In this talk I will present finite dimensional matrix approximations in two classes of operator algebras.
Seminar (Operator Algebras)
Eduard Vilalta (Universitat Autònoma de Barcelona)
Nowhere scattered multiplier algebras
Abstract: A natural assumption that ensures sufficient noncommutativity of a C*-algebra is nowhere scatteredness, which in one of its many formulations asks the algebra to contain no nonzero elementary ideal-quotients. This notion enjoys many good permanence properties, but fails to pass to certain unitizations. For example, no minimal unitization of a non-unital C*-algebra (nowhere scattered or not) can ever be nowhere scattered. However, it is unclear when a nowhere scattered C*-algebra has a nowhere scattered multiplier algebra.
In this talk, I will give sufficient conditions under which this happens. It will follow from the main result of the talk that a $\sigma$-unital C*-algebra of finite nuclear dimension, or of real rank zero, or of stable rank one and k-comparison, is nowhere scattered if and only if its multiplier algebra is. I will also give some examples of nowhere scattered C*-algebras whose multiplier algebra is not nowhere scattered.
Seminar (Operator Algebras)
Joachim Zacharias (University of Glasgow)
On a finite section method to approximate exact C*-algebras
Abstract: Exact C*-algebras are an important class of C*-algebras which is closed under subalgebras and contains all nuclear C*-algebras. A basic result due to Kirchberg asserts that any such separable C*-algebra is a sub-quotient of a UHF-algebra.
We give a short survey on exact C*-algebras, indicating a simplified ‘finite-section’ approach to Kirchberg’s basic result and outline possible applications, including a Stone-Weierstrass type Theorem for exact C*-algebras.
Seminar (Operator Algebras)
Francesc Perera (Universitat Autònoma de Barcelona)
The dynamical Cuntz semigroup and crossed products
Abstract: In this talk I shall discuss the definition of dynamical subequivalence for open subsets of a compact topological space and its natural counterpart involving Cuntz subequivalence. This will lead to the definition of the dynamical Cuntz semigroup. I will mention how this semigroup is related to the construction of crossed products in various categories. This is part of joint work with J. Bosa, J. Wu, and J. Zacharias, and also R. Antoine and H. Thiel.
Seminar (Ring Theory)
Guillem Quingles (Universitat Autònoma de Barcelona)
Finiteness properties of local cohomology modules
Abstract: Local cohomology modules were introduced by Grothendieck in 1961 and they quickly became an important tool in commutative algebra. They have been studied by a number of authors, but the structure of these modules is still quite unknown. When a local cohomology module $H^i(\Gamma_I(M*))$ is nonzero, it is rarely finitely generated, even if $M$ is. So it is not clear whether they satisfy finiteness properties that finitely generated modules do. Huneke proposed a list of problems on local cohomology which guided the study of local cohomology modules. One of the questions on the list asks the following: Is the number of associated primes of $H^i(\Gamma_I(R*))$ finite? Are all the Bass numbers of $H^i(\Gamma_I(R*))$ finite? Lyubeznik conjectured that the answer is affirmative when $R$ is a Noetherian regular commutative ring with unit. Substantial progress has been made on this conjecture. If the regular ring has prime positive characteristic $p$, then the conjecture was completely settled by Huneke and Sharp. Lyubeznik proved the conjecture for regular rings containing a field of characteristic zero. For complete unramified regular local rings of mixed characteristic, the conjecture was also settled by Lyubeznik. The finiteness of associated primes of local cohomology was also proved by Bhatt, Blickle, Lyubeznik, Singh and Zhang for smooth $Z$-algebras. The conjecture is still open when $R$ is a ramified regular local ring of mixed characteristic.
In this talk I will explain the concepts and tools needed to understand the problem of the finiteness of the set of associated primes and Bass numbers of local cohomology modules $H^i(\Gamma_I(M*))$, and the techniques that lead to the proof of the cases where $R$ has positive characteristic and where $R$ is a $K$-algebra, with a field of characteristic 0.
Seminar (Operator Algebras)
Martin Mathieu (Queen’s University Belfast)
A contribution to Kaplansky’s problem
Abstract: A Jordan homomorphism between two unital, complex algebras A and B is a linear mapping T such that $T(x^2)=(Tx)^2$ for all $x\in A$. Equivalently, T preserves the Jordan product $xy+yx$. Every surjective unital Jordan homomorphism preserves invertible elements. In 1970, Kaplansky asked whether the following converse is true: Suppose $T\colon A\to B$ is a unital surjective invertibility-preserving linear mapping between unital (Jacobson) semisimple Banach algebras A and B. Does it follow that T is a Jordan homomorphism?
In the past 50 years a lot of progress has been made towards a positive solution to Kaplansky’s problem, however, as it stands, it is still open. We will report on some recent joint work with Francois Schulz (University of Johannesburg, SA) which gives a positive answer if B is a C*-algebra with faithful tracial state. Until recently, the existence of traces had been a major obstacle to a solution. Moreover, in our approach, no assumption on the existence of projections (such as real rank zero) is necessary.
I will further discuss a sharpening of Kaplansky’s problem in which the assumption on T is reduced to the preservation of the spectral radius only (a spectral isometry).