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Martin Mathieu

Publications and source records attributed to Martin Mathieu.

17 recordsLinked to original sources
arXiv2022

Invertibility preserving mappings onto finite C*-algebras

We prove that every surjective unital linear mapping which preserves invertible elements from a Banach algebra onto a C*-algebra carrying a faithful tracial state is a Jordan homomorphism thus generalising Aupetit's 1998 result for finite von Neumann algebras.

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arXiv2021

Exact Structures for Operator Modules

We demonstrate how exact structures can be placed on the additive category of right operator modules over an operator algebra in order to discuss global dimension for operator algebras. The properties of the Haagerup tensor product play a decisive role in this.

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arXiv2019

Type-zero ternary corners

In this paper we discuss the relationship between a TRO $\mathcal{T}$ and a sub-TRO $\mathcal{S}$ that is the range of a TRO-conditional expectation on $\mathcal{T}$, a \textit{ternary corner}, by investigating a special class $\mathcal{D}$ of bounded linear maps on~$\mathcal{T}$. We pay particular attention to the case when the TROs contain partial isometries.

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arXiv2018

Relative double commutants in coronas of separable C*-algebras

We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu later proved a C*-algebraic double commutant theorem for subalgebras of the Calkin algebra. We prove a similar result for subalgebras of a much more general class of so-called corona C*-algebras.

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arXiv2013

More elementary operators that are spectrally bounded

We discuss some necessary and some sufficient conditions for an elementary operator $x\mapsto\sum_{i=1}^n a_ixb_i$ on a Banach algebra $A$ to be spectrally bounded. In the case of length three, we obtain a complete characterisation when $A$ acts irreducibly on a Banach space of dimension greater than three.

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arXiv2013

Locally quasi-nilpotent elementary operators

Let $A$ be a unital dense algebra of linear mappings on a complex vector space $X$. Let $\phi=\sum_{i=1}^n M_{a_i,b_i}$ be a locally quasi-nilpotent elementary operator of length $n$ on $A$. We show that, if $\{a_1,\ldots,a_n\}$ is locally linearly independent, then the local dimension of $V(\phi)=\spa\{b_ia_j: 1 \leq i,j \leq n\}$ is at most $\frac{n(n-1)}{2}$. If $\lDim V(\phi)=\frac{n(n-1)}{2} $, then there exists a representation of $\phi$ as $\phi=\sum_{i=1}^n M_{u_i,v_i}$ with $v_iu_j=0$ for $i\geq j$. Moreover, we give a complete characterization of locally quasi-nilpotent elementary operators of length 3.

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arXiv2012

C*-Segal algebras with order unit

We introduce the notion of a (noncommutative) C*-Segal algebra as a Banach algebra which is a dense ideal in a C*-algebra. Several basic properties are investigated and, with the aid of the theory of multiplier modules, the structure of C*-Segal algebras with order unit is determined.

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arXiv2011

Spectral isometries on non-simple C*-algebras

We prove that unital surjective spectral isometries on certain non-simple unital C*-algebras are Jordan isomorphisms. Along the way, we establish several general facts in the setting of semisimple Banach algebras.

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arXiv2011

The second local multiplier algebra of a separable C*-algebra

Several examples of (separable) C*-algebras with the property that their second (iterated) local multiplier algebra is strictly larger than the first have been found by various groups of authors over the past few years, thus answering a question originally posed by G. K. Pedersen in 1978. This survey discusses a systematic approach by P. Ara and the author to produce such examples on the one hand; on the other hand, we present new criteria guaranteeing that the second and the first local multiplier algebra of a separable C*-algebra agree. For this class of C*-algebras, each derivation of the local multiplier algebra is inner.

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arXiv2008

The Maximal C*-Algebra of Quotients as an Operator Bimodule

We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra $A$ as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product $A\otimes_h A$ labelled by the essential closed right ideals of $A$ into $A$. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.

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arXiv2007

Maximal C*-algebras of quotients and injective envelopes of C*-algebras

A new C*-enlargement of a C*-algebra $A$ nested between the local multiplier algebra $M_{\text{loc}}(A)$ of $A$ and its injective envelope $I(A)$ is introduced. Various aspects of this maximal C*-algebra of quotients, $Q_{\text{max}}(A)$, are studied, notably in the setting of AW*-algebras. As a by-product we obtain a new example of a type I C*-algebra $A$ such that $M_{\text{loc}}(M_{\text{loc}}(A))\ne M_{\text{loc}}(A)$.

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arXiv2005

A not so simple local multiplier algebra

We construct an AF-algebra $A$ such that its local multiplier algebra $M_{\text{loc}}(A)$ does not agree with $M_{\text{loc}}(M_{\text{loc}}(A))$, thus answering a question raised by G.K. Pedersen in 1978.

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