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Register a new userCommutators Close to the Identity in Unital C*-Algebras
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Let $mathcal{H}$ be an infinite dimensional Hilbert space and $mathcal{B}(mathcal{H})$ be the C*-algebra of all bounded linear operators on $mathcal{H}$, equipped with the operator-norm. By improving the Brown-Pearcy construction, Terence Tao in 2018, extended the result of Popa [1981] which reads as : For each $0<varepsilonleq 1/2$, there exist $D,X in mathcal{B}(mathcal{H})$ with $|[D,X]-1_{mathcal{B}(mathcal{H})}|leq varepsilon$ such that $|D||X|=Oleft(log^5frac{1}{varepsilon}right)$, where $[D,X]:= DX-XD$. In this paper, we show that Taos result still holds for certain class of unital C*-algebras which include $mathcal{B}(mathcal{H})$ as well as the Cuntz algebra $mathcal{O}_2$.
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We consider a family of dynamical systems (A,alpha,L) in which alpha is an endomorphism of a C*-algebra A and L is a transfer operator for alpha. We extend Exels construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.
We investigate the notion of tracial $mathcal Z$-stability beyond unital C*-algebras, and we prove that this notion is equivalent to $mathcal Z$-stability in the class of separable simple nuclear C*-algebras.
We answer a question of Takesaki by showing that the following can be derived from the thesis of N-T Shen: If A and B are sigma-unital hereditary C*-subalgebras of C such that ||p - q|| < 1, where p and q are the corresponding open projections, then A and B are isomorphic. We give some further elaborations and counterexamples with regard to the sigma-unitality hypothesis. We produce a natural one-to-one correspondence between complete order isomorphisms of C*-algebras and invertible left multipliers of imprimitivity bimodules. A corollary of the above two results is that any complete order isomorphism between sigma-unital C*-algebras is the composite of an isomorphism with an inner complete order isomorphism. We give a separable counterexample to a question of Akemann and Pedersen; namely, the space of quasi-multipliers is not linearly generated by left and right multipliers. But we show that the space of quasi-multipliers is multiplicatively generated by left and right multipliers in the sigma-unital case. In particular every positive quasi-multiplier is of the form T*T for T a left multiplier. We show that a Lie theory consequence of the negative result just stated is that the map sending T to T*T need not be open, even for very nice C*-algebras. We show that surjective maps between sigma-unital C*-algebras induce surjective maps on left, right, and quasi-multipliers. (The more significant similar result for multipliers is Pedersens Non-commutative Tietze extension theorem.) We elaborate the relations of the above results with continuous fields of Hilbert spaces and in so doing answer a question of Dixmier and Douady (yes for separable fields, no in general). We discuss the relationship of our results to the theory of perturbations of C*-algebras.
Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $Aotimes Q$ can be tracially approximated by unital Elliott-Thomsen algebras with trivial $textrm{K}_1$-group, where $Q$ is the universal UHF algebra. In particular, it follows that $A$ is classifiable by the Elliott invariant if $A$ is Jiang-Su stable.
In this work we characterise the C*-algebras A generated by projections with the property that every pair of projections in A has positive angle, as certain extensions of abelian algebras by algebras of compact operators. We show that this property is equivalent to a lattice theoretic property of projections and also to the property that the set of finite-dimensional *-subalgebras of A is directed.
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